# Calculus/Points, paths, surfaces, and volumes

This chapter will provide an intuitive interpretation of vector calculus using simple concepts such as multi-points, multi-paths, multi-surfaces, and multi-volumes. Scalar fields will not be simply treated as a function ${\displaystyle f:\mathbb {R} ^{3}\to \mathbb {R} }$ that returns a number given an input point, and vector fields will not be simply treated as a function ${\displaystyle \mathbf {F} :\mathbb {R} ^{3}\to \mathbb {R} ^{3}}$ that returns a vector given an input point.

## Basic structures

The basic structures are multi-points, multi-paths, multi-surfaces, and multi-volumes.

### Multi-points

A point ${\displaystyle \mathbf {q} _{0}}$ is an arbitrary position. A "multi-point" is a set of point/weight pairs: ${\displaystyle \mathbf {Q} =\{(\mathbf {q} _{1},w_{1}),(\mathbf {q} _{2},w_{2}),...,(\mathbf {q} _{k},w_{k})\}}$ where ${\displaystyle w_{i}}$ is the "weight" that is assigned to point ${\displaystyle \mathbf {q} _{i}}$. Given two point/weight pairs ${\displaystyle (\mathbf {q} ,w_{1})}$ and ${\displaystyle (\mathbf {q} ,w_{2})}$ that cover the same point ${\displaystyle \mathbf {q} }$, the weights add up to get ${\displaystyle (\mathbf {q} ,w_{1}+w_{2})}$ which replaces ${\displaystyle (\mathbf {q} ,w_{1})}$ and ${\displaystyle (\mathbf {q} ,w_{2})}$. Any pair ${\displaystyle (\mathbf {q} ,0)}$ is removed. ${\displaystyle \mathbf {Q} }$ can consist of infinitely many points, and each point may have an infinitesimal weight.

An arbitrary point ${\displaystyle \mathbf {q} _{0}}$ can be described by the scalar field ${\displaystyle \delta _{0}(\mathbf {q} ;\mathbf {q} _{0})=\left\{{\begin{array}{cc}+\infty ^{3}&(\mathbf {q} =\mathbf {q} _{0})\\0&(\mathbf {q} \neq \mathbf {q} _{0})\end{array}}\right.}$. This is the "Dirac delta function" centered on point ${\displaystyle \mathbf {q} _{0}}$. The ${\displaystyle +\infty ^{3}}$ is the inverse of an infinitely small volume that wraps point ${\displaystyle \mathbf {q} _{0}}$. To further explain this, let ${\displaystyle \omega _{0}(\mathbf {q} _{0},v)}$ be a tiny volume with volume ${\displaystyle v}$ that wraps point ${\displaystyle \mathbf {q} _{0}}$. ${\displaystyle \delta _{0}(\mathbf {q} ;\mathbf {q} _{0})}$ can be approximated by ${\displaystyle \Delta _{0}(\mathbf {q} ;\mathbf {q} _{0},v)=\left\{{\begin{array}{cc}1/v&(\mathbf {q} \in \omega _{0}(\mathbf {q} _{0},v))\\0&(\mathbf {q} \notin \omega _{0}(\mathbf {q} _{0},v))\end{array}}\right.}$. A mass of 1 is being crammed into ${\displaystyle \omega _{0}(\mathbf {q} _{0},v)}$ yielding an infinitely high density. Since ${\displaystyle \delta _{0}(\mathbf {q} ;\mathbf {q} _{0})}$ is essentially a density function, it brings with it the units ${\displaystyle [{\text{length}}^{-3}]}$.

Multi-point ${\displaystyle \mathbf {Q} =\{(\mathbf {q} _{1},w_{1}),(\mathbf {q} _{2},w_{2}),...,(\mathbf {q} _{k},w_{k})\}}$ can be described by the scalar field ${\displaystyle \delta _{0}(\mathbf {q} ;\mathbf {Q} )=\sum _{i=1}^{k}w_{i}\delta _{0}(\mathbf {q} ;\mathbf {q} _{i})}$. If ${\displaystyle \mathbf {Q} }$ consists of infinitely many points with each point having infinitesimal weight, then ${\displaystyle \delta _{0}(\mathbf {q} ;\mathbf {Q} )}$ is a density function.

In the image below, the multi-point in the left panel is converted to the scalar field in the center panel by averaging the point weight over each cell. The volume of each cell should be infinitesimal. The multi-point in the right panel corresponds to the same scalar field, and is in a more canonical form where oppositely weighted points have cancelled out.

The multi-point (a collection of weighted points) on the left can be denoted by the scalar field in the middle. On the right is a more canonical multi-point with the same scalar field, where nearby points of opposite sign have cancelled out.

The image below shows how a continuous scalar field ${\displaystyle \rho :\mathbb {R} ^{3}\to \mathbb {R} }$ can be generated as a collection of points. Consider position ${\displaystyle \mathbf {q} _{0}}$ and the infinitesimal volume ${\displaystyle \omega _{0}(\mathbf {q} _{0},v)}$ with volume ${\displaystyle v}$. The total point weight contained by ${\displaystyle \omega _{0}(\mathbf {q} _{0},v)}$ is ${\displaystyle \iiint _{\mathbf {q} \in \omega _{0}(\mathbf {q} _{0},v)}\rho (\mathbf {q} )dV\approx v\cdot \rho (\mathbf {q} _{0})}$. This weight of ${\displaystyle v\cdot \rho (\mathbf {q} _{0})}$ is then split up over an arbitrarily large number of points that are scattered over the volume ${\displaystyle \omega _{0}(\mathbf {q} _{0},v)}$.

A single point of weight 1 can be "smeared out" over the volume that it sits in. The point is divided into an increasing number of points with fractional weights. After an infinite number of steps, there is an infinite number of points that fill the volume and each point has an infinitesimal weight.

In summary, a multi-point is denoted by a scalar field that quantifies the density at each point, and any scalar field that quantifies density at each point is best interpreted as a multi-point.

### Multi-paths

A simple path (also called a simple curve) ${\displaystyle C}$ is an oriented continuous curve that extends from a starting point ${\displaystyle C(0)}$ to an ending point ${\displaystyle C(1)}$. Intermediate points are indexed by ${\displaystyle t\in [0,1]}$ and are denoted by ${\displaystyle C(t)}$. A simple path should be continuous (no breaks), and may intersect or retrace itself. A "multi-path" is a set of simple-path/weight pairs: ${\displaystyle \mathbf {C} =\{(C_{1},w_{1}),(C_{2},w_{2}),...,(C_{k},w_{k})\}}$ where ${\displaystyle w_{i}}$ is the weight that is assigned to path ${\displaystyle C_{i}}$. Given two path/weight pairs ${\displaystyle (C,w_{1})}$ and ${\displaystyle (C,w_{2})}$ that cover the same path ${\displaystyle C}$, the weights add up to get ${\displaystyle (C,w_{1}+w_{2})}$ which replaces ${\displaystyle (C,w_{1})}$ and ${\displaystyle (C,w_{2})}$. Any pair ${\displaystyle (C,0)}$ is removed. In addition given two path/weight pairs ${\displaystyle (C_{1},w)}$ and ${\displaystyle (C_{2},w)}$ with the same weight ${\displaystyle w}$ and ${\displaystyle C_{1}(1)=C_{2}(0)}$, then ${\displaystyle C_{1}}$ and ${\displaystyle C_{2}}$ can be linked end-to-end to get the pair ${\displaystyle (C_{1}+C_{2},w)}$ which replaces ${\displaystyle (C_{1},w)}$ and ${\displaystyle (C_{2},w)}$. Assigning a path a negative weight effectively reverses its orientation: if ${\displaystyle -C}$ denotes path ${\displaystyle C}$ with the opposite orientation, then ${\displaystyle (C,-w)}$ is equivalent to ${\displaystyle (-C,w)}$. ${\displaystyle \mathbf {C} }$ can consist of infinitely many paths, and each path may have an infinitesimal weight.

This image depicts the Dirac delta function of a simple path. Unlike the Dirac delta function for a point which is a scalar field, the Dirac delta function for a path is a vector field.

An arbitrary curve ${\displaystyle C}$ can be described by the vector field ${\displaystyle \delta _{1}(\mathbf {q} ;C)=\left\{{\begin{array}{cc}(+\infty ^{2}){\hat {\mathbf {n} }}(\mathbf {q} ;C)&(\mathbf {q} \in C)\\\mathbf {0} &(\mathbf {q} \notin C)\end{array}}\right.}$. This is the "Dirac delta function" for the curve ${\displaystyle C}$. ${\displaystyle {\hat {\mathbf {n} }}(\mathbf {q} ;C)}$ is the unit length tangent vector to path ${\displaystyle C}$ at point ${\displaystyle \mathbf {q} \in C}$. ${\displaystyle {\hat {\mathbf {n} }}(\mathbf {q} ;C)=\mathbf {0} }$ if ${\displaystyle \mathbf {q} \notin C}$. If there are multiple tangent vectors due to ${\displaystyle C}$ intersecting itself, then ${\displaystyle {\hat {\mathbf {n} }}(\mathbf {q} ;C)}$ is the sum of these tangent vectors. The ${\displaystyle +\infty ^{2}}$ is the inverse of the cross-sectional area of an infinitely thin tube that encloses ${\displaystyle C}$. To further explain this, let ${\displaystyle \omega _{1}(C,a)}$ be a thin tube with cross-sectional area ${\displaystyle a}$ that encloses ${\displaystyle C}$. ${\displaystyle \delta _{1}(\mathbf {q} ;C)}$ can be approximated by ${\displaystyle \Delta _{1}(\mathbf {q} ;C,a)=\left\{{\begin{array}{cc}(1/a){\hat {\mathbf {n} }}_{*}(\mathbf {q} ;C,a)&(\mathbf {q} \in \omega _{1}(C,a))\\\mathbf {0} &(\mathbf {q} \notin \omega _{1}(C,a))\end{array}}\right.}$. ${\displaystyle {\hat {\mathbf {n} }}_{*}(\mathbf {q} ;C,a)}$ is the generalization of ${\displaystyle {\hat {\mathbf {n} }}(\mathbf {q} ;C)}$ to the tube ${\displaystyle \omega _{1}(C,a)}$. A path weight of 1 is being crammed into the cross-sectional area of ${\displaystyle \omega _{1}(C,a)}$ yielding an infinitely high path density. Since ${\displaystyle \delta _{1}(\mathbf {q} ;C)}$ is essentially a density over area, it brings with it the units ${\displaystyle [{\text{length}}^{-2}]}$.

The image to the right gives a depiction of the Dirac delta function for a simple curve. The vector field ${\displaystyle \delta _{1}(\mathbf {q} ;C)}$ is ${\displaystyle \mathbf {0} }$ everywhere outside of an infinitely thin tube that encloses the path. Inside the tube, the vectors are parallel to the path, and have a magnitude equal to the inverse of the cross-sectional area. The Dirac delta function is the limit as the tube becomes infinitely thin.

Multi-path ${\displaystyle \mathbf {C} =\{(C_{1},w_{1}),(C_{2},w_{2}),...,(C_{k},w_{k})\}}$ can be described by the vector field ${\displaystyle \delta _{1}(\mathbf {q} ;\mathbf {C} )=\sum _{i=1}^{k}w_{i}\delta _{1}(\mathbf {q} ;C_{i})}$. If ${\displaystyle \mathbf {C} }$ consists of infinitely many paths with each path having infinitesimal weight, then ${\displaystyle \delta _{1}(\mathbf {q} ;\mathbf {C} )}$ is a flow density function.

In the image below, the multi-path in the left panel is converted to the vector field in the center panel by computing the total displacement in each cell and averaging over the volume. The volume of each cell should be infinitesimal. The multi-path in the right panel corresponds to the same vector field, and is in a more canonical form where the individual paths do not cross each other.

The multi-path (a collection of weighted paths) on the left can be denoted by the vector field in the middle (in generating the vector field, each path was approximated to enter each cell through the middle of an edge). On the right is a more canonical multi-path with the same vector field, where nearby path segments with opposite orientations have cancelled out, and the individual paths do not cross each other.

In summary, a multi-path is denoted by a vector field that quantifies the path/flow density at each point, and any vector field that quantifies a flow density at each point (such as current density) is best interpreted as a multi-path. (Flow density is a vector that points in the net direction of a flow, and has a length equal to the flow rate per unit area through a surface that is perpendicular to the net flow.)

### Multi-surfaces

A simple surface ${\displaystyle \sigma }$ is an oriented continuous surface. A simple surface may intersect or fold back on itself. A "multi-surface" is a set of simple-surface/weight pairs: ${\displaystyle \mathbf {S} =\{(\sigma _{1},w_{1}),(\sigma _{2},w_{2}),...,(\sigma _{k},w_{k})\}}$ where ${\displaystyle w_{i}}$ is the weight that is assigned to surface ${\displaystyle \sigma _{i}}$. Given two surface/weight pairs ${\displaystyle (\sigma ,w_{1})}$ and ${\displaystyle (\sigma ,w_{2})}$ that cover the same surface ${\displaystyle \sigma }$, the weights add up to get ${\displaystyle (\sigma ,w_{1}+w_{2})}$ which replaces ${\displaystyle (\sigma ,w_{1})}$ and ${\displaystyle (\sigma ,w_{2})}$. Any pair ${\displaystyle (\sigma ,0)}$ is removed. In addition given two surface/weight pairs ${\displaystyle (\sigma _{1},w)}$ and ${\displaystyle (\sigma _{2},w)}$ with the same weight ${\displaystyle w}$, then ${\displaystyle \sigma _{1}}$ and ${\displaystyle \sigma _{2}}$ can be combined to get the pair ${\displaystyle (\sigma _{1}+\sigma _{2},w)}$ which replaces ${\displaystyle (\sigma _{1},w)}$ and ${\displaystyle (\sigma _{2},w)}$. Assigning a surface a negative weight effectively reverses its orientation: if ${\displaystyle -\sigma }$ denotes surface ${\displaystyle \sigma }$ with the opposite orientation, then ${\displaystyle (\sigma ,-w)}$ is equivalent to ${\displaystyle (-\sigma ,w)}$. ${\displaystyle \mathbf {S} }$ can consist of infinitely many surfaces, and each surface may have an infinitesimal weight.

An arbitrary surface ${\displaystyle \sigma }$ can be described by the vector field ${\displaystyle \delta _{2}(\mathbf {q} ;\sigma )=\left\{{\begin{array}{cc}(+\infty ){\hat {\mathbf {n} }}(\mathbf {q} ;\sigma )&(\mathbf {q} \in \sigma )\\\mathbf {0} &(\mathbf {q} \notin \sigma )\end{array}}\right.}$. This is the "Dirac delta function" for the surface ${\displaystyle \sigma }$. ${\displaystyle {\hat {\mathbf {n} }}(\mathbf {q} ;\sigma )}$ is the unit length normal vector to surface ${\displaystyle \sigma }$ at point ${\displaystyle \mathbf {q} \in \sigma }$. ${\displaystyle {\hat {\mathbf {n} }}(\mathbf {q} ;\sigma )=\mathbf {0} }$ if ${\displaystyle \mathbf {q} \notin \sigma }$. If there are multiple normal vectors due to ${\displaystyle \sigma }$ intersecting itself, then ${\displaystyle {\hat {\mathbf {n} }}(\mathbf {q} ;\sigma )}$ is the sum of these normal vectors. The ${\displaystyle +\infty }$ is the inverse of the thickness of an infinitely thin membrane that encloses ${\displaystyle \sigma }$. To further explain this, let ${\displaystyle \omega _{2}(\sigma ,t)}$ be a thin membrane with thickness ${\displaystyle t}$ that encloses $\displaystyle \sigma$ . ${\displaystyle \delta _{2}(\mathbf {q} ;\sigma )}$ can be approximated by ${\displaystyle \Delta _{2}(\mathbf {q} ;\sigma ,t)=\left\{{\begin{array}{cc}(1/t){\hat {\mathbf {n} }}_{*}(\mathbf {q} ;\sigma ,t)&(\mathbf {q} \in \omega _{2}(\sigma ,t))\\\mathbf {0} &(\mathbf {q} \notin \omega _{2}(\sigma ,t))\end{array}}\right.}$. ${\displaystyle {\hat {\mathbf {n} }}_{*}(\mathbf {q} ;\sigma ,t)}$ is the generalization of ${\displaystyle {\hat {\mathbf {n} }}(\mathbf {q} ;\sigma )}$ to the membrane ${\displaystyle \omega _{2}(\sigma ,t)}$. A surface weight of 1 is being sandwiched into the thickness of ${\displaystyle \omega _{2}(\sigma ,t)}$ yielding an infinitely high surface density. Since ${\displaystyle \delta _{2}(\mathbf {q} ;\sigma )}$ is essentially a density over length, it brings with it the units ${\displaystyle [{\text{length}}^{-1}]}$.

Multi-surface ${\displaystyle \mathbf {S} =\{(\sigma _{1},w_{1}),(\sigma _{2},w_{2}),...,(\sigma _{k},w_{k})\}}$ can be described by the vector field ${\displaystyle \delta _{2}(\mathbf {q} ;\mathbf {S} )=\sum _{i=1}^{k}w_{i}\delta _{2}(\mathbf {q} ;\sigma _{i})}$. If ${\displaystyle \mathbf {S} }$ consists of infinitely many surfaces with each surface having infinitesimal weight, then ${\displaystyle \delta _{2}(\mathbf {q} ;\mathbf {S} )}$ is a rate-of-gain function.

In the image below, the multi-surface in the left panel is converted to the vector field in the center panel by computing the total surface in each cell and averaging over the volume. The volume of each cell should be infinitesimal. The multi-surface in the right panel corresponds to the same vector field, and is in a more canonical form where the individual surfaces do not intersect each other.

The multi-surface (a collection of weighted surfaces) on the left can be denoted by the vector field in the middle (in generating the vector field, each surface was approximated to intersect the edge of each square in the middle). On the right is a more canonical multi-surface with the same vector field, where nearby surface segments with opposite orientations have cancelled out, and the individual surfaces do not intersect.

In summary, a multi-surface is denoted by a vector field that quantifies the rate of gain at each point. To describe the rate-of-gain, imagine that passing through a surface in the preferred direction gives "energy". The rate of gain is a vector that points in the direction that yields the greatest rate of energy increase per unit length, and has a length equal to the maximum rate of energy increase per unit length. Any vector field that quantifies a rate of gain at each point (such as a force field) is best interpreted as a multi-surface.

### Multi-volumes

A volume ${\displaystyle \Omega }$ is an arbitrary region of space. A "multi-volume" is a set of volume/weight pairs: ${\displaystyle \mathbf {U} =\{(\Omega _{1},w_{1}),(\Omega _{2},w_{2}),...,(\Omega _{k},w_{k})\}}$ where ${\displaystyle w_{i}}$ is the "weight" that is assigned to volume ${\displaystyle \Omega _{i}}$. Given two volume/weight pairs ${\displaystyle (\Omega ,w_{1})}$ and ${\displaystyle (\Omega ,w_{2})}$ that cover the same volume ${\displaystyle \Omega }$, the weights add up to get ${\displaystyle (\Omega ,w_{1}+w_{2})}$ which replaces ${\displaystyle (\Omega ,w_{1})}$ and ${\displaystyle (\Omega ,w_{2})}$. Any pair ${\displaystyle (\Omega ,0)}$ is removed. In addition given two volume/weight pairs ${\displaystyle (\Omega _{1},w)}$ and ${\displaystyle (\Omega _{2},w)}$ with the same weight ${\displaystyle w}$ and ${\displaystyle \Omega _{1}\cap \Omega _{2}=\emptyset }$, then ${\displaystyle \Omega _{1}}$ and ${\displaystyle \Omega _{2}}$ can be combined to get the pair ${\displaystyle (\Omega _{1}\cup \Omega _{2},w)}$ which replaces ${\displaystyle (\Omega _{1},w)}$ and ${\displaystyle (\Omega _{2},w)}$. ${\displaystyle \mathbf {U} }$ can consist of infinitely many volumes, and each volume may have an infinitesimal weight.

An arbitrary volume ${\displaystyle \Omega }$ can be described by the scalar field ${\displaystyle \delta _{3}(\mathbf {q} ;\Omega )=\left\{{\begin{array}{cc}1&(\mathbf {q} \in \Omega )\\0&(\mathbf {q} \notin \Omega )\end{array}}\right.}$. This is the "Dirac delta function" analog for volumes, and is essentially an indicator function that indicates whether or not a position is contained by ${\displaystyle \Omega }$ or not, 1 being yes and 0 being no. Since ${\displaystyle \delta _{3}(\mathbf {q} ;\Omega )}$ is simply an indicator function, it brings with it no units (it is dimensionless).

Multi-volume ${\displaystyle \mathbf {U} =\{(\Omega _{1},w_{1}),(\Omega _{2},w_{2}),...,(\Omega _{k},w_{k})\}}$ can be described by the scalar field ${\displaystyle \delta _{3}(\mathbf {q} ;\mathbf {U} )=\sum _{i=1}^{k}w_{i}\delta _{3}(\mathbf {q} ;\Omega _{i})}$. If ${\displaystyle \mathbf {U} }$ consists of infinitely many volumes with each volume having infinitesimal weight, then ${\displaystyle \delta _{3}(\mathbf {q} ;\mathbf {U} )}$ is a potential function.

In the image below, the multi-volume in the left panel is converted to the scalar field in the center panel by averaging the volume weight in each cell. The volume of each cell should be infinitesimal. The multi-volume in the right panel corresponds to the same scalar field, and is in a more canonical form where oppositely weighted volumes have cancelled out, and the remaining volume has diffused to fill each cell.

The multi-volume (a collection of weighted volumes) on the left can be denoted by the scalar field in the middle (in generating the scalar field, the beveled corners of each volume where ignored). On the right is a more canonical multi-volume with the same scalar field, where volumes of opposite sign have cancelled out, and the remaining volume is smeared out to fill each cell.

In summary, a multi-volume is denoted by a scalar field that quantifies a potential at each point, and any scalar field that quantifies a potential at each point is best interpreted as a multi-volume.

### At infinity

An important requirement is that all multi-points, multi-paths, multi-surfaces, and multi-volumes not extend to infinity. All structures can extend to an arbitrarily large range, as long as this range is not unbounded. Allowing the structures to extend to infinity will cause problems in the later discussions.

 Paths that extend to infinity are generally not allowed for most theorems related to vector calculus. Surfaces that extend to infinity are generally not allowed for most theorems related to vector calculus. Volumes that extend to infinity are generally not allowed for most theorems related to vector calculus.

## Totals

These sections will describe the total weight of multi-points, the total displacement of multi-paths, the total surface of multi-surfaces, and the total volumes of multi-volumes.

### Total point weight

Given a multi-point ${\displaystyle \mathbf {Q} =\{(\mathbf {q} _{1},w_{1}),(\mathbf {q} _{2},w_{2}),...,(\mathbf {q} _{k},w_{k})\}}$, the total point weight is clearly ${\displaystyle \sum _{i=1}^{k}w_{i}}$. Given a scalar field ${\displaystyle \rho }$ that denotes a multi-point, the total weight of ${\displaystyle \rho }$ is ${\displaystyle \iiint _{\mathbf {q} \in \mathbb {R} ^{3}}\rho (\mathbf {q} )dV}$. Given a simple point ${\displaystyle \mathbf {q} _{0}}$, the total weight is 1 so ${\displaystyle \iiint _{\mathbf {q} \in \mathbb {R} ^{3}}\delta _{0}(\mathbf {q} ;\mathbf {q} _{0})dV=1}$.

### Total displacement

The displacement between two points is independent of the path that connects them.

Given a simple path ${\displaystyle C}$ that starts at point ${\displaystyle C(0)}$ and ends at point ${\displaystyle C(1)}$, the total displacement generated by ${\displaystyle C}$ is ${\displaystyle \int _{\mathbf {q} \in C}d\mathbf {q} =C(1)-C(0)}$. This displacement is solely dependent on the endpoints as indicated by the top image to the right.

The displacement generated by a closed loop is ${\displaystyle \mathbf {0} }$.

Given a multi-path ${\displaystyle \mathbf {C} =\{(C_{1},w_{1}),(C_{2},w_{2}),...,(C_{k},w_{k})\}}$, the total displacement generated by ${\displaystyle \mathbf {C} }$ is ${\displaystyle \sum _{i=1}^{k}w_{i}\int _{\mathbf {q} \in C_{i}}d\mathbf {q} =\sum _{i=1}^{k}w_{i}(C_{i}(1)-C_{i}(0))}$.

Given a vector field ${\displaystyle \mathbf {J} }$ that denotes a multi-path, the total displacement generated by ${\displaystyle \mathbf {J} }$ is ${\displaystyle \iiint _{\mathbf {q} \in \mathbb {R} ^{3}}\mathbf {J} (\mathbf {q} )dV}$. Since the displacement generated by a simple path ${\displaystyle C}$ is ${\displaystyle \int _{\mathbf {q} \in C}d\mathbf {q} =C(1)-C(0)}$, it is the case that ${\displaystyle \iiint _{\mathbf {q} \in \mathbb {R} ^{3}}\delta _{1}(\mathbf {q} ;C)dV=\int _{\mathbf {q} \in C}d\mathbf {q} =C(1)-C(0)}$.

A path integral can be converted to a volume integral be replacing the displacement differential dq with the shown expression that is proportional to the volume differential dV. As is shown, the path is diffused to fill a thin tube. The integrand of the volume integral at all points outside of this thin tube is 0.

One important observation from ${\displaystyle \int _{\mathbf {q} \in C}d\mathbf {q} =\iiint _{\mathbf {q} \in \mathbb {R} ^{3}}\delta _{1}(\mathbf {q} ;C)dV}$ is that given a path integral over path ${\displaystyle C}$, the differential ${\displaystyle d\mathbf {q} }$ is equal to ${\displaystyle \delta _{1}(\mathbf {q} ;C)dV}$ in a volume integral: ${\displaystyle \int _{\mathbf {q} \in C}f(\mathbf {q} ,d\mathbf {q} )=\iiint _{\mathbf {q} \in \mathbb {R} ^{3}}f(\mathbf {q} ,\delta _{1}(\mathbf {q} ;C)dV)}$ provided that function ${\displaystyle f}$ is linear in the second parameter. In the lower image to the right, the displacement differential ${\displaystyle d\mathbf {q} ={\hat {\mathbf {n} }}\cdot \Delta l}$ is equated to the volume differential ${\displaystyle \left({\frac {\hat {\mathbf {n} }}{\Delta A}}\right)dV=\delta _{1}(\mathbf {q} ;C)dV}$ by diffusing the path over an infintely thin cross-sectional area ${\displaystyle \Delta A}$. The integrand at points outside of the infinitely thin tube is 0: for all points ${\displaystyle \mathbf {q} \notin C}$, ${\displaystyle f(\mathbf {q} ,\delta _{1}(\mathbf {q} ;C)dV)=f(\mathbf {q} ,\mathbf {0} )=0}$.

### Total surface vector

A flat surface with area "A", a counter-clockwise boundary denoted by the arrows, and an orientation out of the plane is depicted by this image. Normal vector "n" has a length of 1, is perpendicular to the surface, and is oriented out of the plane as shown. The surface itself can be described by the vector "A n". The length is the area, and the direction is the orientation.

Given an arbitrary oriented surface ${\displaystyle \sigma }$, its "counter-clockwise boundary", denoted by ${\displaystyle \partial \sigma }$, is the boundary of ${\displaystyle \sigma }$ whose orientation is determined in the following manner: Looking at ${\displaystyle \sigma }$ so that the preferred direction through ${\displaystyle \sigma }$ is oriented towards the viewer, the boundary ${\displaystyle \partial \sigma }$ wraps ${\displaystyle \sigma }$ in a counter-clockwise direction.

Given a flat surface as shown in the image to the right, this surface can be quantified by the "surface vector" which is a vector that is perpendicular (normal) to the surface in the preferred orientation, and has a length equal to the area of the surface. In the image to the right, a flat surface has an area of ${\displaystyle A}$ and is oriented to be perpendicular to unit-length normal vector ${\displaystyle {\hat {\mathbf {n} }}}$. The "surface vector" of this surface is ${\displaystyle A\cdot {\hat {\mathbf {n} }}}$.

Given a non-flat surface ${\displaystyle \sigma }$, the total surface vector of ${\displaystyle \sigma }$ is computed by summing the surface vectors of each infinitesimal portion of ${\displaystyle \sigma }$. The total surface vector is ${\displaystyle \mathbf {S} =\iint _{\mathbf {q} \in \sigma }d\mathbf {S} }$.

In a manner similar to how the total displacement of a path is solely a function of the endpoints, the total surface vector of a surface is solely a function of its counter-clockwise boundary. This is not intuitive, and will be explained in greater detail below using two approaches:

Two different surfaces are shown. Both surfaces have identical counter-clockwise boundaries, and because of this, the "total surface vector" for each surface are the same. Similar to how the total displacement along a path is purely a function of its endpoints, the total surface vector of a surface is purely a function of its boundary.

#### Generalizing from surfaces in 2D space

Below are shown two images related to surface vectors in 2D space. The image to the left shows surface vectors in 2D space. In 2 dimensions, surfaces are called 1D surfaces and are similar to paths. The boundary of a 1D surface consists of 2 points. The surface vector of a 1D surface segment is a 90 degree rotation of the segment and is oriented in the direction of the surface's orientation. The total surface vector of a 1D surface is the sum of all of the surface vectors of the individual components. For each component of the surface, the surface vector is a 90 degree rotation of the displacement that traverses the component, so the total surface vector is a 90 degree rotation of the displacement between the points that form the boundary of the surface. This proves that in two dimensions, the total surface vector depends only on the boundary of the 1D surface.

The image to the right extrudes the 1D surfaces in two dimensional space into 2D "ribbons" in 3 dimensional space. At the top a closed "ribbon" is shown. This "ribbon" is a surface that is always parallel to the vertical dimension, and whose boundary forms two identical loops that are vertically displaced from each other. The boundary loops are also perpendicular to the vertical dimension. The ribbon itself is partitioned into tiny rectangles whose height is equal to that of the ribbon. To the bottom left, a view of the same ribbon from the top down is shown. It can be seen that the length of the each surface vector is proportional to the length of the corresponding rectangular segment, since the heights are all uniform. To the bottom right, by rotating the surface vectors 90 degrees around the vertical dimension, the surface vectors now sum to ${\displaystyle \mathbf {0} }$, so the sum of the unrotated surface vectors is also ${\displaystyle \mathbf {0} }$.

 This image depicts how in 2 dimensions, the total surface vector of a 1D surface is a 90 degree rotation of the displacement between the two endpoints (the boundary of a 1D surface), and is therefore purely a function of the endpoints. In the left panel, a 1D surface is a sequence of black line segments, and the surface vectors of each segment are denoted by the dashed red arrows. Each surface vector is a 90 degree rotation of the displacement along the surface. The long grey line is the net displacement between the endpoints of the surface, and the dashed pink arrow is a 90 degree rotation of this net displacement. In the right panel, the pink arrow is shown as the sum of the dashed red arrow vectors, hence the "total surface" is purely a function of the 1D surface's endpoints. This image demonstrates that the total surface vector of a surface that is a closed ribbon is 0. The top image shows a surface that is a closed ribbon where the width of the ribbon is constant, the width is always parallel to the vertical dimension, and the edge is always perpendicular to the vertical dimension. The surface is sub-divided into tiny rectangular portions, the surface vectors of which are shown. The lower-left image shows the same surface from a top down perspective. In the lower-right image, the surface vectors are all rotated 90 degrees counter-clockwise around the vertical dimension and clearly sum to 0.

The fact that the total surface vector of a closed ribbon is ${\displaystyle \mathbf {0} }$ means that if relief is added to a surface without changing its boundaries, the total surface vector is conserved. The two left images below give examples of distorting the interior of a surface by hammering in relief. The vertical surfaces introduced by the relief are ribbons which contribute ${\displaystyle \mathbf {0} }$ to the total surface vector, while the horizontal surfaces are simply displaced vertically be the relief. The rightmost image below shows how the total surface vector is preserved if the "texture" of the surface at infinitesimal scales is converted from "steps" (a union of horizontal and vertical surfaces) to "smooth slopes" and vice versa. The surface formed from the red and green planes is a step, while the surface formed from the blue plane is a slope. These two surfaces can be seen to have equal total surface vectors from the right-angled triangle at the right side of the image.

 Adding elevation (depression in this image) or relief to a surface does not change the total surface vector. The red colored horizontal surfaces are clearly conserved, albeit at different elevations. The green colored vertical surfaces sum to 0 at each tier/elevation. Adding elevation (depression in this image) or relief to a surface does not change the total surface vector. The horizontal surfaces are clearly conserved, albeit at different elevations. The green colored vertical surfaces sum to their initial value above the lower red surface, and sum to 0 beneath the lower red surface. In this image there are two surfaces. The first surface is the union of the red and green planes, and the counter-clockwise boundary is shown by the thick black line. The second surface is the blue plane and the counter-clockwise boundary is shown by the dashed blue line. The surface vectors of the red, green, and blue planes are shown. The total surface vector of the first surface is the sum of the surface vectors of the red and green planes, and is equal to the surface vector of the blue plane. This all implies that the total surface vector of a sloped flat surface is unchanged by replacing the surface with its horizontal and vertical components (projections).

#### Generalizing from displacement vectors

The total displacement along a simple oriented curve can be used to compute the net displacement in a specific direction. Given a simple oriented curve ${\displaystyle C}$ and an oriented straight line with the direction indicated by normal vector ${\displaystyle {\hat {\mathbf {n} }}}$, the total displacement ${\displaystyle \Delta \mathbf {q} }$ along ${\displaystyle C}$ can be used to compute the net displacement in the direction indicated by the line. This displacement is ${\displaystyle {\hat {\mathbf {n} }}\cdot \Delta \mathbf {q} }$, and depends only on the endpoints of the curve.

In a direct analogy, given a simple oriented surface ${\displaystyle \sigma }$ with counter-clockwise boundary ${\displaystyle \partial \sigma }$, and an oriented flat plane whose surface normal is ${\displaystyle {\hat {\mathbf {n} }}}$, a quantity of interest is the total signed area of ${\displaystyle \sigma }$ perpendicularly projected onto the plane. The signed area that is projected by a flat infinitesimal portion of ${\displaystyle \sigma }$ with surface vector ${\displaystyle d\mathbf {S} }$ is ${\displaystyle {\hat {\mathbf {n} }}\cdot d\mathbf {S} }$, and the total signed area is ${\displaystyle \iint _{\mathbf {q} \in \sigma }{\hat {\mathbf {n} }}\cdot d\mathbf {S} ={\hat {\mathbf {n} }}\cdot \iint _{\mathbf {q} \in \sigma }d\mathbf {S} ={\hat {\mathbf {n} }}\cdot \mathbf {S} }$ where ${\displaystyle \mathbf {S} }$ is the total surface vector of ${\displaystyle \sigma }$.

The total signed projected area ${\displaystyle {\hat {\mathbf {n} }}\cdot \mathbf {S} }$ onto the plane is purely a function of the boundary ${\displaystyle \partial \sigma }$, and does not depend on how ${\displaystyle \sigma }$ fills its boundary ${\displaystyle \partial \sigma }$. This is much more obvious and clearer than the claim that the total surface vector ${\displaystyle \mathbf {S} }$ is only a function of ${\displaystyle \partial \sigma }$: the area enclosed by a boundary in 2D space is purely a function of that boundary. Since the projected area is signed, "upside down" surfaces project negative area, and folds and overhangs cancel each other out.

Since ${\displaystyle {\hat {\mathbf {n} }}\cdot \mathbf {S} }$ is purely a function of ${\displaystyle \partial \sigma }$ for all choices of plane orientation ${\displaystyle {\hat {\mathbf {n} }}}$, then the total surface vector ${\displaystyle \mathbf {S} }$ is purely a function of ${\displaystyle \partial \sigma }$.

 Given an arbitrary oriented path, the total displacement covered by the perpendicularly projected path onto an oriented straight line does not depend on the placement of the interior points of the path. The displacement only depends on the endpoints. Since this is true no matter the choice of straight line, the total 3D displacement vector generated by an oriented curve is purely a function of its endpoints, and does not change if the interior points are moved. The total signed area of the projection of an oriented surface onto an oriented flat plane depends only on the boundary and not on any of the interior points. The "shadow" does not change if the interior points are moved around. If the surface is deformed so that there is an "overhang" where some projected points fall outside of the projected boundary, such as in the example of the right, these points cancel out with the points on the opposite side (top or bottom) of the overhang. An upside down surface projects negative area, and in the example on the right, all negative projected area is cancelled out with the positive area projected by the upright surface on top of the overhang.
 Computing the signed projected area of a flat surface onto a flat plane is equivalent to computing the signed projected length of the surface vector onto the line that is perpendicular to the plane.

#### Summary

The total surface vector generated by a closed surface is ${\displaystyle \mathbf {0} }$.

Given a multi-surface ${\displaystyle \mathbf {S} =\{(\sigma _{1},w_{1}),(\sigma _{2},w_{2}),...,(\sigma _{k},w_{k})\}}$ the total surface vector generated by ${\displaystyle \mathbf {S} }$ is ${\displaystyle \sum _{i=1}^{k}w_{i}\iint _{\mathbf {q} \in \sigma _{i}}d\mathbf {S} }$.

Given a vector field ${\displaystyle \mathbf {F} }$ that denotes a multi-surface, the total surface vector generated by ${\displaystyle \mathbf {F} }$ is ${\displaystyle \iiint _{\mathbf {q} \in \mathbb {R} ^{3}}\mathbf {F} (\mathbf {q} )dV}$. Since the surface vector generated by simple surface ${\displaystyle \sigma }$ is ${\displaystyle \iint _{\mathbf {q} \in \sigma }d\mathbf {S} }$, it is the case that ${\displaystyle \iiint _{\mathbf {q} \in \mathbb {R} ^{3}}\delta _{2}(\mathbf {q} ;\sigma )dV=\iint _{\mathbf {q} \in \sigma }d\mathbf {S} }$. One important observation is that given a surface integral over ${\displaystyle \sigma }$, the differential ${\displaystyle d\mathbf {S} }$ is equal to ${\displaystyle \delta _{2}(\mathbf {q} ;\sigma )dV}$ in a volume integral: ${\displaystyle \iint _{\mathbf {q} \in \sigma }f(\mathbf {q} ,d\mathbf {S} )=\iiint _{\mathbf {q} \in \mathbb {R} ^{3}}f(\mathbf {q} ,\delta _{2}(\mathbf {q} ;\sigma )dV)}$ provided that function ${\displaystyle f}$ is a linear in the second parameter.

### Total volume

Consider a multi-volume ${\displaystyle \mathbf {U} =\{(\Omega _{1},w_{1}),(\Omega _{2},w_{2}),...,(\Omega _{k},w_{k})\}}$, where the volumes of ${\displaystyle \Omega _{1},\Omega _{2},...,\Omega _{k}}$ are respectively ${\displaystyle V_{1},V_{2},...,V_{k}}$, then the total volume of ${\displaystyle \mathbf {U} }$ is ${\displaystyle \sum _{i=1}^{k}w_{i}V_{i}}$. Each volume ${\displaystyle V_{i}}$ can be computed by ${\displaystyle V_{i}=\iiint _{\mathbf {q} \in \Omega _{i}}dV=\iiint _{\mathbf {q} \in \mathbb {R} ^{3}}\delta _{3}(\mathbf {q} ;\Omega _{i})dV}$. The total volume of ${\displaystyle \mathbf {U} }$ is ${\displaystyle V=\sum _{i=1}^{k}w_{i}V_{i}=\sum _{i=1}^{k}w_{i}\iiint _{\mathbf {q} \in \Omega _{i}}dV=\sum _{i=1}^{k}w_{i}\iiint _{\mathbf {q} \in \mathbb {R} ^{3}}\delta _{3}(\mathbf {q} ;\Omega _{i})dV}$ ${\displaystyle =\iiint _{\mathbf {q} \in \mathbb {R} ^{3}}\left(\sum _{i=1}^{k}w_{i}\delta _{3}(\mathbf {q} ;\Omega _{i})\right)dV=\iiint _{\mathbf {q} \in \mathbb {R} ^{3}}\delta _{3}(\mathbf {q} ;\mathbf {U} )dV}$.

If a multi-volume ${\displaystyle \mathbf {U} }$ can be denoted by scalar field ${\displaystyle U}$, then the volume of ${\displaystyle \mathbf {U} }$ is ${\displaystyle \iiint _{\mathbf {q} \in \mathbb {R} ^{3}}U(\mathbf {q} )dV}$.

Given an arbitrary volume ${\displaystyle \Omega }$, a volume integral over ${\displaystyle \Omega }$ can be converted to a volume integral over ${\displaystyle \mathbb {R} ^{3}}$ by replacing the differential ${\displaystyle dV}$ with ${\displaystyle \delta _{3}(\mathbf {q} ;\Omega )dV}$:

${\displaystyle \iiint _{\mathbf {q} \in \Omega }f(\mathbf {q} ,dV)=\iiint _{\mathbf {q} \in \mathbb {R} ^{3}}f(\mathbf {q} ,\delta _{3}(\mathbf {q} ;\Omega )dV)}$ provided that ${\displaystyle f}$ is linear in the second parameter.

## Intersections

The union of two multi-points denoted by scalar fields ${\displaystyle \rho _{1}}$ and ${\displaystyle \rho _{2}}$ is simply ${\displaystyle \rho _{1}+\rho _{2}}$, and the same is true for the union of two multi-paths, the union of two multi-surfaces, and the union of two multi-volumes. The union of two structures with different types, such as a multi-point with a multi-path, is forbidden however.

Unions
structure multi-point ${\displaystyle \rho _{2}}$ multi-path ${\displaystyle \mathbf {J} _{2}}$ multi-surface ${\displaystyle \mathbf {F} _{2}}$ multi-volume ${\displaystyle U_{2}}$
multi-point ${\displaystyle \rho _{1}}$ multi-point ${\displaystyle \rho _{1}+\rho _{2}}$ n/a n/a n/a
multi-path ${\displaystyle \mathbf {J} _{1}}$ n/a multi-path ${\displaystyle \mathbf {J} _{1}+\mathbf {J} _{2}}$ n/a n/a
multi-surface ${\displaystyle \mathbf {F} _{1}}$ n/a n/a multi-surface ${\displaystyle \mathbf {F} _{1}+\mathbf {F} _{2}}$ n/a
multi-volume ${\displaystyle U_{1}}$ n/a n/a n/a multi-volume ${\displaystyle U_{1}+U_{2}}$

The intersection on the other hand, is less trivial and can occur between structures of different types.

### Point-Volume intersections

When a point ${\displaystyle \mathbf {q} }$ with weight ${\displaystyle w_{1}}$ intersects a volume ${\displaystyle \Omega }$ with weight ${\displaystyle w_{2}}$, then the intersection is point ${\displaystyle \mathbf {q} }$ with weight ${\displaystyle w_{1}w_{2}}$. Given a multi-point and a multi-volume, the intersection is the sum of the pair-wise intersections of each simple point with each simple volume. The image below gives an example of the intersection of a multi-point with a multi-volume.

The left panel depicts both a multi-point and a multi-volume. The right panel depicts the intersection between the multi-point and the multi-volume, which is itself a multi-point. Note that points that intersect a volume with weight -1 have their weights flipped to their negative.

Given a multi-point with scalar field ${\displaystyle \rho }$, and a multi-volume with scalar field ${\displaystyle U}$, then the intersection is a multi-point with scalar field ${\displaystyle \rho U}$.

The total intersection between a multi-point ${\displaystyle \rho }$ and a multi-volume ${\displaystyle U}$ is ${\displaystyle \iiint _{\mathbf {q} \in \mathbb {R} ^{3}}\rho (\mathbf {q} )U(\mathbf {q} )dV}$.

If ${\displaystyle \rho }$ denotes a simple point ${\displaystyle \mathbf {q} _{0}}$, then the total intersection is ${\displaystyle \iiint _{\mathbf {q} \in \mathbb {R} ^{3}}\delta _{0}(\mathbf {q} ;\mathbf {q} _{0})U(\mathbf {q} )dV=U(\mathbf {q} _{0})}$.

If ${\displaystyle U}$ denotes a simple volume ${\displaystyle \Omega }$, then the total intersection is ${\displaystyle \iiint _{\mathbf {q} \in \mathbb {R} ^{3}}\rho (\mathbf {q} )\delta _{3}(\mathbf {q} ;\Omega )dV=\iiint _{\mathbf {q} \in \Omega }\rho (\mathbf {q} )dV}$.

### Path-Surface intersections

When a path ${\displaystyle C}$ with weight ${\displaystyle w_{1}}$ intersects a surface ${\displaystyle \sigma }$ with weight ${\displaystyle w_{2}}$ at point ${\displaystyle \mathbf {q} }$, then the intersection is point ${\displaystyle \mathbf {q} }$ with weight ${\displaystyle \pm w_{1}w_{2}}$. The weight is ${\displaystyle +w_{1}w_{2}}$ if ${\displaystyle C}$ passes through ${\displaystyle \sigma }$ in the direction in which ${\displaystyle \sigma }$ is oriented. The weight is ${\displaystyle -w_{1}w_{2}}$ if ${\displaystyle C}$ passes through ${\displaystyle \sigma }$ opposite the direction in which ${\displaystyle \sigma }$ is oriented. Given a multi-path and a multi-surface, the intersection is the sum of the pair-wise intersections of each simple path with each simple surface. The images below give examples of the intersections of a multi-path with a multi-surface.

 A 2D image showing the intersection of a multi-path (dark blue dashed curves) with a multi-surface (dark red solid curves). Positive intersection points (red circles) occur when a path intersects a surface in the preferred direction. Negative intersection points (teal circles) occur when a path intersects a surface in the opposite direction. The intersection is effectively a multi-point. A 3D image showing the intersection of a simple path (red curve) with a simple surface (green surface with the counter-clockwise boundary highlighted). The positive intersection points are denoted by red "+" signs, and the negative intersection points are denoted by blue "-" signs. The intersection between a multi-path shown as a blue tube with a multi-surface shown as layers of red sheets. Vector F is the flow density through the blue tube. Vector G is the surface density in the red sheets. The green parallelogram is a 2D projection of the volume of the intersection. The intersection points become more dilute as the angle theta increases, so the intersection point density is the dot product of F and G.

In the image above to the far right, the multi-path is denoted by a vector field which has the value ${\displaystyle \mathbf {F} }$ inside the blue tube, and is ${\displaystyle \mathbf {0} }$ everywhere else. The multi-surface is denoted by a vector field which has the value ${\displaystyle \mathbf {G} }$ among the red sheets, and is ${\displaystyle \mathbf {0} }$ everywhere else. The total path weight in the blue tube is ${\displaystyle |\mathbf {F} |\Delta A}$. The total surface weight in the red sheets is ${\displaystyle |\mathbf {G} |\Delta t}$. The total weight of all the intersection points is ${\displaystyle (|\mathbf {F} |\Delta A)(|\mathbf {G} |\Delta t)=|\mathbf {F} ||\mathbf {G} |\Delta A\Delta t}$. The volume that the intersection points are evenly spread out in is ${\displaystyle \Delta A\Delta t/\cos \theta }$. The intersection point density is ${\displaystyle {\frac {|\mathbf {F} ||\mathbf {G} |\Delta A\Delta t}{\Delta A\Delta t/\cos \theta }}=|\mathbf {F} ||\mathbf {G} |\cos \theta =\mathbf {F} \cdot \mathbf {G} }$.

Given a multi-path with vector field ${\displaystyle \mathbf {J} }$, and a multi-surface with vector field ${\displaystyle \mathbf {F} }$, then the intersection is a multi-point with scalar field ${\displaystyle \mathbf {J} \cdot \mathbf {F} }$.

The total intersection between a multi-path ${\displaystyle \mathbf {J} }$ and a multi-surface ${\displaystyle \mathbf {F} }$ is ${\displaystyle \iiint _{\mathbf {q} \in \mathbb {R} ^{3}}(\mathbf {J} (\mathbf {q} )\cdot \mathbf {F} (\mathbf {q} ))dV}$.

If ${\displaystyle \mathbf {J} }$ is a simple path ${\displaystyle C}$, then the total intersection is ${\displaystyle \iiint _{\mathbf {q} \in \mathbb {R} ^{3}}(\delta _{1}(\mathbf {q} ;C)\cdot \mathbf {F} (\mathbf {q} ))dV=\int _{\mathbf {q} \in C}\mathbf {F} (\mathbf {q} )\cdot d\mathbf {q} }$.

If ${\displaystyle \mathbf {F} }$ is a simple surface ${\displaystyle \sigma }$, then the total intersection is ${\displaystyle \iiint _{\mathbf {q} \in \mathbb {R} ^{3}}(\mathbf {J} (\mathbf {q} )\cdot \delta _{2}(\mathbf {q} ;\sigma ))dV=\iint _{\mathbf {q} \in \sigma }\mathbf {J} (\mathbf {q} )\cdot d\mathbf {S} }$.

### Path-Volume intersections

When a path ${\displaystyle C}$ with weight ${\displaystyle w_{1}}$ intersects a volume ${\displaystyle \Omega }$ with weight ${\displaystyle w_{2}}$, then the intersection is path ${\displaystyle C\cap \Omega }$ with weight ${\displaystyle w_{1}w_{2}}$. Given a multi-path and a multi-volume, the intersection is the sum of the pair-wise intersections of each simple path with each simple volume. The image below gives an example of the intersection of a multi-path with a multi-volume.

The left panel depicts both a multi-path and a multi-volume. The right panel depicts the intersection between the multi-path and the multi-volume, which is itself a multi-path. Note that the path's orientation is reversed in the negatively weighted volumes. In addition, the path segment in the weight 2 volume region in the middle has a weight of 2 as indicated by the thicker line.

Given a multi-path with vector field ${\displaystyle \mathbf {J} }$, and a multi-volume with scalar field ${\displaystyle U}$, then the intersection is a multi-path with vector field ${\displaystyle \mathbf {J} U}$.

The total intersection between a multi-path ${\displaystyle \mathbf {J} }$ and a multi-volume ${\displaystyle U}$ is ${\displaystyle \iiint _{\mathbf {q} \in \mathbb {R} ^{3}}\mathbf {J} (\mathbf {q} )U(\mathbf {q} )dV}$.

If ${\displaystyle \mathbf {J} }$ denotes a simple path ${\displaystyle C}$, then the total intersection is ${\displaystyle \iiint _{\mathbf {q} \in \mathbb {R} ^{3}}\delta _{1}(\mathbf {q} ;C)U(\mathbf {q} )dV=\int _{\mathbf {q} \in C}U(\mathbf {q} )d\mathbf {q} }$.

If ${\displaystyle U}$ denotes a simple volume ${\displaystyle \Omega }$, then the total intersection is ${\displaystyle \iiint _{\mathbf {q} \in \mathbb {R} ^{3}}\mathbf {J} (\mathbf {q} )\delta _{3}(\mathbf {q} ;\Omega )dV=\iiint _{\mathbf {q} \in \Omega }\mathbf {J} (\mathbf {q} )dV}$.

### Surface-Surface intersections

When a surface ${\displaystyle \sigma _{1}}$ with weight ${\displaystyle w_{1}}$ intersects a surface ${\displaystyle \sigma _{2}}$ with weight ${\displaystyle w_{2}}$, then the intersection is the path ${\displaystyle \sigma _{1}\cap \sigma _{2}}$ with weight ${\displaystyle w_{1}w_{2}}$. The orientation given to path ${\displaystyle \sigma _{1}\cap \sigma _{2}}$ is defined as follows: viewing the intersection where the surface normal vectors of ${\displaystyle \sigma _{1}}$ and ${\displaystyle \sigma _{2}}$ are oriented towards the viewer, the intersection path has ${\displaystyle \sigma _{1}}$ to its right, and ${\displaystyle \sigma _{2}}$ to its left. Put another way, the intersection path is oriented according to the "right-hand rule" where the surface normals of ${\displaystyle \sigma _{1}}$ are the "x" direction, and the surface normals of ${\displaystyle \sigma _{2}}$ are the "y" direction. The images below give examples of the intersections of a multi-surface with a multi-surface.

 A 3D image that shows the intersection of 2 surfaces. Surface 1 is blue and the normal vectors are oriented upwards. Surface 2 is red and the normal vectors are oriented to the right. The intersection is the black curve. The orientation of the intersection curve is determined via the right-hand rule with the surface normals of surface 1 as the "x" direction, and the surface normals of surface 2 as the "y" direction. The intersection between two multi-surfaces. The first multi-surface is the layered blue sheets, and the second multi-surface is the layered red sheets. Vector F is the surface density in the blue sheets. Vector G is the surface density in the red sheets. The green parallelogram is a 2D cross-section of the prism that forms the intersection. The intersection paths become more dilute the further angle theta deviates from 90 degrees, so the intersection path density is the cross product of F and G. The intersection paths are also oriented out of the screen in this example.

In the image above to the right, the first multi-surface is denoted by a vector field that has the value ${\displaystyle \mathbf {F} }$ among the blue sheets, and is ${\displaystyle \mathbf {0} }$ everywhere else. The second multi-surface is denoted by a vector field that has the value ${\displaystyle \mathbf {G} }$ among the red sheets, and is ${\displaystyle \mathbf {0} }$ everywhere else. The total surface weight in the blue sheets is ${\displaystyle |\mathbf {F} |\Delta t_{1}}$, and the total surface weight in the red sheets is ${\displaystyle |\mathbf {G} |\Delta t_{2}}$. The total weight of all the intersection paths is ${\displaystyle (|\mathbf {F} |\Delta t_{1})(|\mathbf {G} |\Delta t_{2})=|\mathbf {F} ||\mathbf {G} |\Delta t_{1}\Delta t_{2}}$. The cross-sectional area that the intersection paths are evenly spread out over is ${\displaystyle \Delta t_{1}\Delta t_{2}/\sin \theta }$. The intersection path density is ${\displaystyle {\frac {|\mathbf {F} ||\mathbf {G} |\Delta t_{1}\Delta t_{2}}{\Delta t_{1}\Delta t_{2}/\sin \theta }}=|\mathbf {F} ||\mathbf {G} |\sin \theta =|\mathbf {F} \times \mathbf {G} |}$. Lastly, it should be noted that the intersection paths are oriented out of the screen as per the right-hand rule.

Given a multi-surface with vector field ${\displaystyle \mathbf {F} _{1}}$, and a multi-surface with vector field ${\displaystyle \mathbf {F} _{2}}$, then the intersection is the multi-path with vector field ${\displaystyle \mathbf {F} _{1}\times \mathbf {F} _{2}}$.

The total intersection between multi-surface ${\displaystyle \mathbf {F} _{1}}$ and multi-surface ${\displaystyle \mathbf {F} _{2}}$ is ${\displaystyle \iiint _{\mathbf {q} \in \mathbb {R} ^{3}}(\mathbf {F} _{1}(\mathbf {q} )\times \mathbf {F} _{2}(\mathbf {q} ))dV}$.

If ${\displaystyle \mathbf {F} _{2}}$ denotes a simple surface ${\displaystyle \sigma }$, then the total intersection is ${\displaystyle \iiint _{\mathbf {q} \in \mathbb {R} ^{3}}(\mathbf {F} _{1}(\mathbf {q} )\times \delta _{2}(\mathbf {q} ;\sigma ))dV=\iint _{\mathbf {q} \in \sigma }\mathbf {F} _{1}(\mathbf {q} )\times d\mathbf {S} }$.

### Surface-Volume intersections

When a surface ${\displaystyle \sigma }$ with weight ${\displaystyle w_{1}}$ intersects a volume ${\displaystyle \Omega }$ with weight ${\displaystyle w_{2}}$, then the intersection is surface ${\displaystyle \sigma \cap \Omega }$ with weight ${\displaystyle w_{1}w_{2}}$. Given a multi-surface and a multi-volume, the intersection is the sum of the pair-wise intersections of each simple surface with each simple volume. The image below gives an example of the intersection of a multi-surface with a multi-volume.

The left panel depicts both a multi-surface and a multi-volume. The right panel depicts the intersection between the multi-surface and the multi-volume, which is itself a multi-surface. Note that the surface's orientation is reversed in the negatively weighted volume. In addition, the surface segment in the weight 2 volume region in the upper-left has a weight of 2 as indicated by the thicker line.

Given a multi-surface with vector field ${\displaystyle \mathbf {F} }$, and a multi-volume with scalar field ${\displaystyle U}$, then the intersection is a multi-surface with vector field ${\displaystyle \mathbf {F} U}$.

The total intersection between a multi-surface ${\displaystyle \mathbf {F} }$ and a multi-volume ${\displaystyle U}$ is ${\displaystyle \iiint _{\mathbf {q} \in \mathbb {R} ^{3}}\mathbf {F} (\mathbf {q} )U(\mathbf {q} )dV}$.

If ${\displaystyle \mathbf {F} }$ denotes a simple surface ${\displaystyle \sigma }$, then the total intersection is ${\displaystyle \iiint _{\mathbf {q} \in \mathbb {R} ^{3}}\delta _{2}(\mathbf {q} ;\sigma )U(\mathbf {q} )dV=\iint _{\mathbf {q} \in \sigma }U(\mathbf {q} )d\mathbf {S} }$.

If ${\displaystyle U}$ denotes a simple volume ${\displaystyle \Omega }$, then the total intersection is ${\displaystyle \iiint _{\mathbf {q} \in \mathbb {R} ^{3}}\mathbf {F} (\mathbf {q} )\delta _{3}(\mathbf {q} ;\Omega )dV=\iiint _{\mathbf {q} \in \Omega }\mathbf {F} (\mathbf {q} )dV}$.

### Volume-Volume intersections

When a volume ${\displaystyle \Omega _{1}}$ with weight ${\displaystyle w_{1}}$ intersects a volume ${\displaystyle \Omega _{2}}$ with weight ${\displaystyle w_{2}}$, then the intersection is the volume ${\displaystyle \Omega _{1}\cap \Omega _{2}}$ with weight ${\displaystyle w_{1}w_{2}}$. Given two multi-volumes, the intersection is the sum of the pair-wise intersections of each simple volume from the first multi-volume with each simple volume from the second multi-volume. The image below gives an example of the intersection between two multi-volumes.

The left two panels each depict a multi-volume, and the rightmost panel depicts the intersection of the two multi-volumes. The weight of the intersection of two simple volumes is the product of the weight of the two volumes.

Given a multi-volume with scalar field ${\displaystyle U_{1}}$, and a multi-volume with scalar field ${\displaystyle U_{2}}$, then the intersection is a multi-volume with scalar field ${\displaystyle U_{1}U_{2}}$.

The total intersection between multi-volume ${\displaystyle U_{1}}$ and multi-volume ${\displaystyle U_{2}}$ is ${\displaystyle \iiint _{\mathbf {q} \in \mathbb {R} ^{3}}U_{1}(\mathbf {q} )U_{2}(\mathbf {q} )dV}$.

If ${\displaystyle U_{2}}$ denotes a simple volume ${\displaystyle \Omega }$, then the total intersection is ${\displaystyle \iiint _{\mathbf {q} \in \mathbb {R} ^{3}}U_{1}(\mathbf {q} )\delta _{3}(\mathbf {q} ;\Omega )dV=\iiint _{\mathbf {q} \in \Omega }U_{1}(\mathbf {q} )dV}$.

### Other intersections

Other types of intersections, such as Point-Point intersections, Point-Path intersections, Point-Surface intersections, and Path-Path intersections, are not considered since these kinds of intersections can occur only by design. For example, the probability that two randomly chosen points will intersect each other is 0, but if a point and a volume are randomly chosen, then the probability of the point landing in the volume is nonzero. Given two unrelated points, these two points will never land on each other, since a prior relationship has to exist for the points to coincide. Below is summarized the various types of intersections:

Intersections
structure multi-point ${\displaystyle \rho _{2}}$ multi-path ${\displaystyle \mathbf {J} _{2}}$ multi-surface ${\displaystyle \mathbf {F} _{2}}$ multi-volume ${\displaystyle U_{2}}$
multi-point ${\displaystyle \rho _{1}}$ n/a n/a n/a multi-point ${\displaystyle \rho _{1}U_{2}}$
multi-path ${\displaystyle \mathbf {J} _{1}}$ n/a n/a multi-point ${\displaystyle \mathbf {J} _{1}\cdot \mathbf {F} _{2}}$ multi-path ${\displaystyle \mathbf {J} _{1}U_{2}}$
multi-surface ${\displaystyle \mathbf {F} _{1}}$ n/a multi-point ${\displaystyle \mathbf {F} _{1}\cdot \mathbf {J} _{2}}$ multi-path ${\displaystyle \mathbf {F} _{1}\times \mathbf {F} _{2}}$ multi-surface ${\displaystyle \mathbf {F} _{1}U_{2}}$
multi-volume ${\displaystyle U_{1}}$ multi-point ${\displaystyle U_{1}\rho _{2}}$ multi-path ${\displaystyle U_{1}\mathbf {J} _{2}}$ multi-surface ${\displaystyle U_{1}\mathbf {F} _{2}}$ multi-volume ${\displaystyle U_{1}U_{2}}$

## Boundaries

### The endpoints of paths

Given a simple path ${\displaystyle C}$ that starts at point ${\displaystyle C(0)}$ and ends at point ${\displaystyle C(1)}$, the "endpoints" of ${\displaystyle C}$ is the multi-point ${\displaystyle \{(C(0),+1),(C(1),-1)\}}$ that consists of the starting point with a weight of +1, and the final point with a weight of -1. While ${\displaystyle C}$ is denoted by the vector field ${\displaystyle \delta _{1}(\mathbf {q} ;C)}$, the endpoints are denoted by the scalar field ${\displaystyle \delta _{0}(\mathbf {q} ;C(0))-\delta _{0}(\mathbf {q} ;C(1))}$. The image below gives several examples of simple paths and their associated endpoints.

A series of panels, each depicting a directed path and its endpoints. The endpoints of a path consists of a positively weighted point at the start and a negatively weighted point at the end.

Given a multi-path ${\displaystyle \mathbf {C} =\{(C_{1},w_{1}),(C_{2},w_{2}),...,(C_{k},w_{k})\}}$, the endpoints of ${\displaystyle \mathbf {C} }$ is the multi-point ${\displaystyle \{(C_{1}(0),+1),(C_{1}(1),-1),(C_{2}(0),+1),(C_{2}(1),-1),...,(C_{k}(0),+1),(C_{k}(1),-1)\}}$.

Given a vector field ${\displaystyle \mathbf {J} }$ that denotes a multi-path, the multi-point that denotes the endpoints of ${\displaystyle \mathbf {J} }$ is denoted by scalar field ${\displaystyle \nabla \cdot \mathbf {J} }$. The scalar field ${\displaystyle \nabla \cdot \mathbf {J} }$ evaluated at point ${\displaystyle \mathbf {q} }$ is denoted by ${\displaystyle \nabla \cdot \mathbf {J} (\mathbf {q} )}$, ${\displaystyle (\nabla \cdot \mathbf {J} )(\mathbf {q} )}$ or ${\displaystyle \nabla \cdot \mathbf {J} |_{\mathbf {q} }}$.

The requirement that no path extends to infinity means that every starting point is paired with a final point, and therefore the total weight of all of the endpoints together is 0: ${\displaystyle \iiint _{\mathbf {q} \in \mathbb {R} ^{3}}(\nabla \cdot \mathbf {J} (\mathbf {q} ))dV=0}$.

The path endpoints are the intersections of the path with the "surface of reality".

The similarity of the notation ${\displaystyle \nabla \cdot \mathbf {J} }$ to the intersection of multi-path ${\displaystyle \mathbf {J} }$ with multi-surface ${\displaystyle \mathbf {F} }$, denoted by ${\displaystyle \mathbf {F} \cdot \mathbf {J} }$, makes sense if ${\displaystyle \nabla }$ is interpreted as the "surface of reality". A starting point forms when a path pokes into reality, and a final point forms when a path pokes out of reality.

In the image to the right, a depiction of the "surface of reality" interpretation of ${\displaystyle \nabla }$ is shown. On the right is a simple path ${\displaystyle \mathbf {F} }$, along with its endpoints ${\displaystyle \nabla \cdot \mathbf {F} }$. On the left ${\displaystyle \mathbf {F} _{\text{ext}}}$ is an extension of ${\displaystyle \mathbf {F} }$ that is behind the "veil" of surface ${\displaystyle \mathbf {G} _{\nabla }}$. ${\displaystyle \mathbf {F} _{\text{ext}}}$ pokes out of and into ${\displaystyle \mathbf {G} _{\nabla }}$ at points consistent with the endpoints of ${\displaystyle \mathbf {F} }$: i.e. ${\displaystyle \mathbf {G} _{\nabla }\cdot \mathbf {F} _{\text{ext}}=\nabla \cdot \mathbf {F} }$.

### The counter-clockwise oriented boundaries of surfaces

Given an oriented surface ${\displaystyle \sigma }$, the "counter-clockwise oriented boundary" of ${\displaystyle \sigma }$ is a path ${\displaystyle \partial \sigma }$ that traces the boundary of ${\displaystyle \sigma }$ in a counter-clockwise direction. The counter-clockwise direction is better described as follows: While located on the boundary, the counter-clockwise direction is the "forwards" direction when the surface normal vectors point "up" and the surface itself is on the "left". The image below gives several examples of oriented surfaces and their counter-clockwise boundaries. Note in particular the 4th panel that shows that the orientation around a hole in the surface appears to be clockwise.

A series of panels, each depicting an oriented surface and its counterclockwise oriented boundary. The surface normal vectors are depicted by the red arrows.

Given a multi-surface ${\displaystyle \mathbf {S} =\{(\sigma _{1},w_{1}),(\sigma _{2},w_{2}),...,(\sigma _{k},w_{k})\}}$, the counter-clockwise boundary of ${\displaystyle \mathbf {S} }$ is the multi-path ${\displaystyle \{(\partial \sigma _{1},w_{1}),(\partial \sigma _{2},w_{2}),...,(\partial \sigma _{k},w_{k})\}}$.

Given a vector field ${\displaystyle \mathbf {F} }$ that denotes a multi-surface, the multi-path that denotes the counter-clockwise boundary of ${\displaystyle \mathbf {F} }$ is denoted by vector field ${\displaystyle \nabla \times \mathbf {F} }$. The vector field ${\displaystyle \nabla \times \mathbf {F} }$ evaluated at point ${\displaystyle \mathbf {q} }$ is denoted by ${\displaystyle \nabla \times \mathbf {F} (\mathbf {q} )}$, ${\displaystyle (\nabla \times \mathbf {F} )(\mathbf {q} )}$, or ${\displaystyle \nabla \times \mathbf {F} |_{\mathbf {q} }}$.

The requirement that no surface weight extends to infinity means that all counter-clockwise boundaries form closed loops, and therefore the total displacement of the total counter-clockwise boundary is ${\displaystyle \mathbf {0} }$: ${\displaystyle \iiint _{\mathbf {q} \in \mathbb {R} ^{3}}(\nabla \times \mathbf {F} (\mathbf {q} ))dV=\mathbf {0} }$.

It is also important to note that the counter-clockwise boundary has no endpoints: ${\displaystyle \nabla \cdot (\nabla \times \mathbf {F} )=0}$.

The boundary of a surface is analogous to the intersection of the surface with the "surface of reality".

The similarity of the notation ${\displaystyle \nabla \times \mathbf {F} _{2}}$ to the intersection of multi-surface ${\displaystyle \mathbf {F} _{1}}$ with multi-surface ${\displaystyle \mathbf {F} _{2}}$, denoted by ${\displaystyle \mathbf {F} _{1}\times \mathbf {F} _{2}}$, again makes sense if ${\displaystyle \nabla }$ is interpreted as the "surface of reality". An edge is formed when a surface "slices" into reality.

In the image to the right, a depiction of the "surface of reality" interpretation of ${\displaystyle \nabla }$ is shown. On the right is a simple surface ${\displaystyle \mathbf {F} }$, along with its counter-clockwise boundary ${\displaystyle \nabla \times \mathbf {F} }$. On the left ${\displaystyle \mathbf {F} _{\text{ext}}}$ is an extension of ${\displaystyle \mathbf {F} }$ that is behind the "veil" of surface ${\displaystyle \mathbf {G} _{\nabla }}$. ${\displaystyle \mathbf {F} _{\text{ext}}}$ slices into ${\displaystyle \mathbf {G} _{\nabla }}$ at curves consistent with the boundary of ${\displaystyle \mathbf {F} }$: i.e. ${\displaystyle \mathbf {G} _{\nabla }\times \mathbf {F} _{\text{ext}}=\nabla \times \mathbf {F} }$.

### The inwards-oriented surfaces of volumes

Given a volume ${\displaystyle \Omega }$, the "inwards oriented surface" of ${\displaystyle \Omega }$ is a surface ${\displaystyle \partial \Omega }$ that wraps the volume ${\displaystyle \Omega }$ with the surface normals pointing inwards. The image below gives several examples of volumes and their inwards oriented surfaces.

A series of panels, each depicting a volume and its inwards oriented surface. The inwards orientation of the surface is indicated by the red arrows pointing inwards.

Given a multi-volume ${\displaystyle \mathbf {U} =\{(\Omega _{1},w_{1}),(\Omega _{2},w_{2}),...,(\Omega _{k},w_{k})\}}$, the inwards oriented surface of ${\displaystyle \mathbf {U} }$ is the multi-surface ${\displaystyle \{(\partial \Omega _{1},w_{1}),(\partial \Omega _{2},w_{2}),...,(\partial \Omega _{k},w_{k})\}}$.

Given a scalar field ${\displaystyle U}$ that denotes a multi-volume, the multi-surface that denotes the inwards oriented surface of ${\displaystyle U}$ is denoted by vector field ${\displaystyle \nabla U}$. The vector field ${\displaystyle \nabla U}$ evaluated at point ${\displaystyle \mathbf {q} }$ is denoted by ${\displaystyle \nabla U(\mathbf {q} )}$, ${\displaystyle (\nabla U)(\mathbf {q} )}$, or ${\displaystyle \nabla U|_{\mathbf {q} }}$.

The requirement that no volume weight extends to infinity means that all inwards oriented surfaces form closed surfaces, and therefore the total surface vector of the total inwards oriented surface is ${\displaystyle \mathbf {0} }$: ${\displaystyle \iiint _{\mathbf {q} \in \mathbb {R} ^{3}}(\nabla U(\mathbf {q} ))dV=\mathbf {0} }$.

It is also important to note that the inwards oriented surface has no boundary: ${\displaystyle \nabla \times (\nabla U)=\mathbf {0} }$.

In this 2D cross-section, the surface of a volume is analogous to the intersection of the volume with the "surface of reality".

The similarity of the notation ${\displaystyle \nabla U}$ to the intersection of multi-surface ${\displaystyle \mathbf {F} }$ with multi-volume ${\displaystyle U}$, denoted by ${\displaystyle \mathbf {F} U}$, again makes sense if ${\displaystyle \nabla }$ is interpreted as the "surface of reality". A surface is formed from the surface of reality when the volume "pushes" into reality.

In the image to the right, a depiction of the "surface of reality" interpretation of ${\displaystyle \nabla }$ is shown. The image is a 2D cross-section for simplicity. On the right is a simple volume ${\displaystyle U}$, along with its inwards oriented surface ${\displaystyle \nabla U}$. On the left ${\displaystyle U_{\text{ext}}}$ is an extension of ${\displaystyle U}$ that is behind the "veil" of surface ${\displaystyle \mathbf {G} _{\nabla }}$. ${\displaystyle U_{\text{ext}}}$ pushes through ${\displaystyle \mathbf {G} _{\nabla }}$ at surfaces consistent with the surface of ${\displaystyle U}$: i.e. ${\displaystyle \mathbf {G} _{\nabla }U_{\text{ext}}=\nabla U}$.

### Closed loops and closed surfaces

A simple path is "closed" or a "loop" if its starting and final points are the same, so the total endpoints is 0 since the weights of the starting and final points cancel out. More generally, a multi-path ${\displaystyle \mathbf {J} }$ is "closed" or a "multi-loop" if ${\displaystyle \nabla \cdot \mathbf {J} =0}$. As previously noted, the counter-clockwise boundary of a surface is closed.

A simple surface is "closed" or a "bubble" if it has no boundary. More generally, a multi-surface ${\displaystyle \mathbf {F} }$ is "closed" or a "multi-bubble" if ${\displaystyle \nabla \times \mathbf {F} =\mathbf {0} }$. As previously noted, the inwards oriented surface of a volume is closed.

It is clear that the total displacement present in a closed multi-path is ${\displaystyle \mathbf {0} }$: ${\displaystyle \nabla \cdot \mathbf {J} =0\implies \iiint _{\mathbf {q} \in \mathbb {R} ^{3}}\mathbf {J} dV=\mathbf {0} }$, and it is also clear that the total surface vector of a closed multi-surface is also ${\displaystyle \mathbf {0} }$: ${\displaystyle \nabla \times \mathbf {F} =\mathbf {0} \implies \iiint _{\mathbf {q} \in \mathbb {R} ^{3}}\mathbf {F} dV=\mathbf {0} }$.

Given a simple loop and simple bubble, the number of times that the loop enters the bubble is equal to the number of times that the loop leaves the bubble.

Given both a simple loop and a simple bubble, the total point weight of all intersection points is 0: every time the loop enters the bubble, it must also leave the bubble, and the weights of these two intersection points cancel out. More generally, given a closed multi-path ${\displaystyle \mathbf {J} }$ and a closed multi-surface ${\displaystyle \mathbf {F} }$, then the total intersection point weight is 0: ${\displaystyle (\nabla \cdot \mathbf {J} =0\;{\text{and}}\;\nabla \times \mathbf {F} =\mathbf {0} )\implies \iiint _{\mathbf {q} \in \mathbb {R} ^{3}}(\mathbf {J} \cdot \mathbf {F} )dV=0}$.

The above identity gives rise to the following observations:

• The total intersection point weight of a multi-loop and a multi-surface is purely a function of the multi-loop and the multi-surface's counter-clockwise boundary: the interior of the multi-surface does not matter. If ${\displaystyle \nabla \cdot \mathbf {J} =0}$ and ${\displaystyle \nabla \times \mathbf {F} _{1}=\nabla \times \mathbf {F} _{2}}$, then ${\displaystyle \iiint _{\mathbf {q} \in \mathbb {R} ^{3}}(\mathbf {J} \cdot \mathbf {F} _{1})dV=\iiint _{\mathbf {q} \in \mathbb {R} ^{3}}(\mathbf {J} \cdot \mathbf {F} _{2})dV}$.
• The total intersection point weight of a multi-path and a multi-bubble is purely a function of the multi-bubble and the multi-path's endpoints: the interior of the multi-path does not matter. If ${\displaystyle \nabla \times \mathbf {F} =\mathbf {0} }$ and ${\displaystyle \nabla \cdot \mathbf {J} _{1}=\nabla \cdot \mathbf {J} _{2}}$, then ${\displaystyle \iiint _{\mathbf {q} \in \mathbb {R} ^{3}}(\mathbf {J} _{1}\cdot \mathbf {F} )dV=\iiint _{\mathbf {q} \in \mathbb {R} ^{3}}(\mathbf {J} _{2}\cdot \mathbf {F} )dV}$.

The inwards oriented surface of a volume is closed. Conversely, given a closed surface, there exists a volume that "fills" the surface. More generally, given a multi-bubble ${\displaystyle \mathbf {F} }$, there exists a multi-volume ${\displaystyle U}$ for which ${\displaystyle \mathbf {F} }$ is the inwards oriented multi-surface of ${\displaystyle U}$: ${\displaystyle \nabla \times \mathbf {F} =\mathbf {0} \implies \exists U:\nabla U=\mathbf {F} }$. This multi-volume is referred to as the "scalar potential" of ${\displaystyle \mathbf {F} }$. The requirement that volumes cannot extend to infinity means that ${\displaystyle U}$ is unique.

The counter-clockwise oriented boundary of a surface is closed. Conversely, given a loop, there exists a surface that "fills" the loop. More generally, given a multi-loop ${\displaystyle \mathbf {J} }$, there exists a multi-surface ${\displaystyle \mathbf {F} }$ for which ${\displaystyle \mathbf {J} }$ is the counter-clockwise boundary of ${\displaystyle \mathbf {F} }$: ${\displaystyle \nabla \cdot \mathbf {J} =0\implies \exists \mathbf {F} :\nabla \times \mathbf {F} =\mathbf {J} }$. This multi-surface is referred to as the "vector potential" of ${\displaystyle \mathbf {J} }$. Even with the requirement that surfaces cannot extend to infinity, ${\displaystyle \mathbf {F} }$ is not unique without additional restrictions.

## Coordinate Systems

This image depicts a generalized coordinate lattice at the top. At the bottom of the image is a single volume element with the basis displacement (contravariant) vectors, alongside the basis surface (covariant) vectors.

This section will describe how to compute various quantities such as intersections, endpoints, boundaries, and surfaces given a curvilinear coordinate system.

Let the curvilinear coordinate system be arbitrary. Let the 3 coordinates that index all points be ${\displaystyle c_{1},c_{2},c_{3}}$. Coordinates will be denoted by the triple ${\displaystyle (c_{1},c_{2},c_{3})}$.

The following notation will be used in the following discussions:

• Given an arbitrary expression ${\displaystyle f:\{1,2,3\}\to \mathbb {R} }$ that assigns a real number to each index ${\displaystyle i=1,2,3}$, then ${\displaystyle (i;f(i))}$ will denote the triple ${\displaystyle (f(1),f(2),f(3))}$.
• Given index variables ${\displaystyle i,j\in \{1,2,3\}}$, the expression ${\displaystyle \mathbf {1} (i=j)}$ equals 1 if ${\displaystyle i=j}$ and 0 if otherwise.
• Given an arbitrary expression ${\displaystyle f:\{1,2,3\}\to \mathbb {R} }$ that assigns a real number to each index ${\displaystyle i=1,2,3}$, then ${\displaystyle \sum _{i}f(i)}$ will denote the sum ${\displaystyle f(1)+f(2)+f(3)}$.
• Given an index variable ${\displaystyle i\in \{1,2,3\}}$, ${\displaystyle i+1}$ will rotate ${\displaystyle i}$ forwards by 1, and ${\displaystyle i+2}$ will rotate ${\displaystyle i}$ forwards by 2. In essence, ${\displaystyle i+1=\left\{{\begin{array}{cc}i+1&(i=1,2)\\1&(i=3)\end{array}}\right.}$ and ${\displaystyle i+2=\left\{{\begin{array}{cc}3&(i=1)\\i-1&(i=2,3)\end{array}}\right.}$.

Start with an arbitrary coordinate ${\displaystyle (c'_{1},c'_{2},c'_{3})=(j;c'_{j})}$ and infinitesimal differences ${\displaystyle \Delta c_{1}}$, ${\displaystyle \Delta c_{2}}$, and ${\displaystyle \Delta c_{3}}$. The following 3 paths, 3 surfaces, and volume will be associated with point ${\displaystyle (j;c'_{j})}$:

• For each ${\displaystyle i\in \{1,2,3\}}$ there exists an infinitely short path ${\displaystyle C_{i}((j;c'_{j}))}$ starting from point ${\displaystyle (j;c'_{j})}$ and ending on point ${\displaystyle (j;c'_{j}+\Delta c_{i}\mathbf {1} (j=i))}$ along the curve defined by ${\displaystyle c'_{i}\leq c_{i}, ${\displaystyle c_{i+1}=c'_{i+1}}$ and ${\displaystyle c_{i+2}=c'_{i+2}}$. The displacement covered by ${\displaystyle C_{i}((j;c'_{j}))}$ is approximately ${\displaystyle \Delta c_{i}\cdot l_{i}((j;c'_{j}))\cdot {\hat {\mathbf {a} }}_{i}((j;c'_{j}))}$ where ${\displaystyle {\hat {\mathbf {a} }}_{i}((j;c'_{j}))}$ is a unit length vector that is parallel to the displacement between points ${\displaystyle (j;c'_{j})}$ and ${\displaystyle (j;c'_{j}+\Delta c_{i}\mathbf {1} (j=i))}$, and ${\displaystyle \Delta c_{i}\cdot l_{i}((j;c'_{j}))}$ is the length of the displacement. Note that the length of the displacement is proportional to ${\displaystyle \Delta c_{i}}$, with ${\displaystyle l_{i}((j;c'_{j}))}$ being the constant of proportionality. The set of vectors ${\displaystyle \{{\hat {\mathbf {a} }}_{1}((j;c'_{j})),{\hat {\mathbf {a} }}_{2}((j;c'_{j})),{\hat {\mathbf {a} }}_{3}((j;c'_{j}))\}}$ is the set of displacement basis vectors.
• For each ${\displaystyle i\in \{1,2,3\}}$ there exists an infinitely small surface ${\displaystyle \sigma _{i}((j;c'_{j}))}$ that is defined by the following: ${\displaystyle c_{i}=c'_{i}}$, ${\displaystyle c'_{i+1}\leq c_{i+1}, and ${\displaystyle c'_{i+2}\leq c_{i+2}. The orientation of ${\displaystyle \sigma _{i}((j;c'_{j}))}$ is in the direction of increasing ${\displaystyle c_{i}}$. The surface vector of ${\displaystyle \sigma _{i}((j;c'_{j}))}$ is approximately ${\displaystyle \Delta c_{i+1}\Delta c_{i+2}\cdot A_{i}((j;c'_{j}))\cdot {\hat {\mathbf {a} }}^{i}((j;c'_{j}))}$ where ${\displaystyle {\hat {\mathbf {a} }}^{i}((j;c'_{j}))}$ is a unit length vector that is perpendicular to ${\displaystyle \sigma _{i}((j;c'_{j}))}$, and ${\displaystyle \Delta c_{i+1}\Delta c_{i+2}\cdot A_{i}((j;c'_{j}))}$ is the area of ${\displaystyle \sigma _{i}((j;c'_{j}))}$. Note that the area of ${\displaystyle \sigma _{i}((j;c'_{j}))}$ is proportional to ${\displaystyle \Delta c_{i+1}\Delta c_{i+2}}$, with ${\displaystyle A_{i}((j;c'_{j}))}$ being the constant of proportionality. The set of vectors ${\displaystyle \{{\hat {\mathbf {a} }}^{1}((j;c'_{j})),{\hat {\mathbf {a} }}^{2}((j;c'_{j})),{\hat {\mathbf {a} }}^{3}((j;c'_{j}))\}}$ is the set of surface basis vectors.
• There is an infinitely small volume ${\displaystyle \Omega ((j;c'_{j}))}$ defined by ${\displaystyle c'_{1}\leq c_{1}, ${\displaystyle c'_{2}\leq c_{2}, and ${\displaystyle c'_{3}\leq c_{3}. ${\displaystyle \Omega ((j;c'_{j}))}$ has a shape that is approximately that of a parallelepiped. The volume of ${\displaystyle \Omega ((j;c'_{j}))}$ is approximately ${\displaystyle \Delta c_{1}\Delta c_{2}\Delta c_{3}\cdot V((j;c'_{j}))}$. Note that the volume of ${\displaystyle \Omega ((j;c'_{j}))}$ is proportional to ${\displaystyle \Delta c_{1}\Delta c_{2}\Delta c_{3}}$, with ${\displaystyle V((j;c'_{j}))}$ being the constant of proportionality.

It is important to note that:

• ${\displaystyle (i;c_{i})\in \Omega ((j;c'_{j}))}$ if and only if ${\displaystyle c'_{1}\leq c_{1}, ${\displaystyle c'_{2}\leq c_{2}, and ${\displaystyle c'_{3}\leq c_{3} (note the strictness of the upper bounds).
• For all ${\displaystyle i\in \{1,2,3\}}$, ${\displaystyle C_{i}((j;c_{j}))\subseteq \Omega ((j;c'_{j}))}$ if and only if ${\displaystyle c_{i}=c'_{i}}$, ${\displaystyle c'_{i+1}\leq c_{i+1}, and ${\displaystyle c'_{i+2}\leq c_{i+2} (note the strictness of the upper bounds).
• For all ${\displaystyle i\in \{1,2,3\}}$, ${\displaystyle \sigma _{i}((j;c_{j}))\subseteq \Omega ((j;c'_{j}))}$ if and only if ${\displaystyle c'_{i}\leq c_{i} (note the strictness of the upper bound), ${\displaystyle c_{i+1}=c'_{i+1}}$, and ${\displaystyle c_{i+2}=c'_{i+2}}$.

Converting between multi-points, multi-paths, multi-surfaces, and multi-volumes and their respective scalar fields and vector fields proceeds as follows:

This conversion is performed by subdividing space into discrete volumes or cells. Infinitesimal differences ${\displaystyle \Delta c_{1}}$, ${\displaystyle \Delta c_{2}}$, and ${\displaystyle \Delta c_{3}}$ are chosen, and a lattice consisting of the points ${\displaystyle (j;k_{j}\Delta c_{j})}$ where ${\displaystyle (j;k_{j})}$ is an arbitrary triple of integers is generated. The cell indexed by ${\displaystyle (j;k_{j})}$ consists of the point ${\displaystyle (j;k_{j}\Delta c_{j})}$, the paths ${\displaystyle C_{i}((j;k_{j}\Delta c_{j}))}$ for each ${\displaystyle i\in \{1,2,3\}}$, the surfaces ${\displaystyle \sigma _{i}((j;k_{j}\Delta c_{j}))}$ for each ${\displaystyle i\in \{1,2,3\}}$, and the volume ${\displaystyle \Omega ((j;k_{j}\Delta c_{j}))}$. All points ${\displaystyle (j;c_{j})}$ where ${\displaystyle k_{i}\Delta c_{i}\leq c_{i}<(k_{i}+1)\Delta c_{i}}$ for all ${\displaystyle i\in \{1,2,3\}}$ "belong" to the cell indexed by ${\displaystyle (j;k_{j})}$ (note that the upper bounds are excluded). Given an arbitrary point ${\displaystyle (j;c_{j})}$, the cell that contains ${\displaystyle (j;c_{j})}$ is indexed by ${\displaystyle (j;k_{j})=\left(j;\left\lfloor {\frac {c_{j}}{\Delta c_{j}}}\right\rfloor \right)}$. The point ${\displaystyle (j;c'_{j})=(j;k_{j}\Delta c_{j})}$ is the vertex that the cell is associated with.

A multi-point, multi-path, multi-surface, or multi-volume is converted to a scalar field or vector field by computing the total point weight, displacement, surface vector, or volume contained by each cell and then averaging over the cell's volume.

A scalar-field ${\displaystyle \rho }$ is converted to a multi-point by doing the following for each cell ${\displaystyle (j;k_{j})}$. First compute the total point weight contained inside the cell: ${\displaystyle \iiint _{\mathbf {q} \in \Omega ((j;k_{j}\Delta c_{j}))}\rho (\mathbf {q} )dV\approx \rho ((j;k_{j}\Delta c_{j}))V((j;k_{j}\Delta c_{j}))\Delta c_{1}\Delta c_{2}\Delta c_{3}}$. Next assign this weight to the point ${\displaystyle (j;k_{j}\Delta c_{j})}$.

A vector-field ${\displaystyle \mathbf {J} =\sum _{i}J_{i}{\hat {\mathbf {a} }}_{i}}$ is converted to a multi-path by doing the following for each cell ${\displaystyle (j;k_{j})}$. First compute the total displacement contained inside the cell: ${\displaystyle \iiint _{\mathbf {q} \in \Omega ((j;k_{j}\Delta c_{j}))}\mathbf {J} (\mathbf {q} )dV\approx \left(\sum _{i}J_{i}((j;k_{j}\Delta c_{j})){\hat {\mathbf {a} }}_{i}((j;k_{j}\Delta c_{j}))\right)V((j;k_{j}\Delta c_{j}))\Delta c_{1}\Delta c_{2}\Delta c_{3}}$. Next separate this total displacement into components according to the basis ${\displaystyle {\hat {\mathbf {a} }}_{1}}$, ${\displaystyle {\hat {\mathbf {a} }}_{2}}$, and ${\displaystyle {\hat {\mathbf {a} }}_{3}}$: for each ${\displaystyle i\in \{1,2,3\}}$ the coefficient of ${\displaystyle {\hat {\mathbf {a} }}_{i}}$ is ${\displaystyle \iiint _{\mathbf {q} \in \Omega ((j;k_{j}\Delta c_{j}))}J_{i}(\mathbf {q} )dV\approx J_{i}((j;k_{j}\Delta c_{j}))V((j;k_{j}\Delta c_{j}))\Delta c_{1}\Delta c_{2}\Delta c_{3}}$. Next for each ${\displaystyle i\in \{1,2,3\}}$, divide the coefficient of ${\displaystyle {\hat {\mathbf {a} }}_{i}}$ by the length of ${\displaystyle C_{i}((j;k_{j}\Delta c_{j}))}$, which results in approximately ${\displaystyle J_{i}((j;k_{j}\Delta c_{j})){\frac {V((j;k_{j}\Delta c_{j}))}{l_{i}((j;k_{j}\Delta c_{j}))}}\Delta c_{i+1}\Delta c_{i+2}}$, and assign this weight to ${\displaystyle C_{i}((j;k_{j}\Delta c_{j}))}$.

A vector-field ${\displaystyle \mathbf {F} =\sum _{i}F_{i}{\hat {\mathbf {a} }}^{i}}$ is converted to a multi-surface by doing the following for each cell ${\displaystyle (j;k_{j})}$. First compute the total surface vector contained inside the cell: ${\displaystyle \iiint _{\mathbf {q} \in \Omega ((j;k_{j}\Delta c_{j}))}\mathbf {F} (\mathbf {q} )dV\approx \left(\sum _{i}F_{i}((j;k_{j}\Delta c_{j})){\hat {\mathbf {a} }}^{i}((j;k_{j}\Delta c_{j}))\right)V((j;k_{j}\Delta c_{j}))\Delta c_{1}\Delta c_{2}\Delta c_{3}}$. Next separate this total surface vector into components according to the basis ${\displaystyle {\hat {\mathbf {a} }}^{1}}$, ${\displaystyle {\hat {\mathbf {a} }}^{2}}$, and ${\displaystyle {\hat {\mathbf {a} }}^{3}}$: for each ${\displaystyle i\in \{1,2,3\}}$ the coefficient of ${\displaystyle {\hat {\mathbf {a} }}^{i}}$ is ${\displaystyle \iiint _{\mathbf {q} \in \Omega ((j;k_{j}\Delta c_{j}))}F_{i}(\mathbf {q} )dV\approx F_{i}((j;k_{j}\Delta c_{j}))V((j;k_{j}\Delta c_{j}))\Delta c_{1}\Delta c_{2}\Delta c_{3}}$. Next for each ${\displaystyle i\in \{1,2,3\}}$, divide the coefficient of ${\displaystyle {\hat {\mathbf {a} }}^{i}}$ by the area of ${\displaystyle \sigma _{i}((j;k_{j}\Delta c_{j}))}$, which results in approximately ${\displaystyle F_{i}((j;k_{j}\Delta c_{j})){\frac {V((j;k_{j}\Delta c_{j}))}{A_{i}((j;k_{j}\Delta c_{j}))}}\Delta c_{i}}$, and assign this weight to ${\displaystyle \sigma _{i}((j;k_{j}\Delta c_{j}))}$.

A scalar-field ${\displaystyle U}$ is converted to a multi-volume by doing the following for each cell ${\displaystyle (j;k_{j})}$. First compute the total volume contained inside the cell: ${\displaystyle \iiint _{\mathbf {q} \in \Omega ((j;k_{j}\Delta c_{j}))}U(\mathbf {q} )dV\approx U((j;k_{j}\Delta c_{j}))V((j;k_{j}\Delta c_{j}))\Delta c_{1}\Delta c_{2}\Delta c_{3}}$. Next divide this weight by the volume of ${\displaystyle \Omega ((j;k_{j}\Delta c_{j}))}$, which results in approximately ${\displaystyle U((j;k_{j}\Delta c_{j}))}$, and assign this weight to ${\displaystyle \Omega ((j;k_{j}\Delta c_{j}))}$.

### Computing various intersections

Computing the intersection of any structure with a multi-volume is trivial matter: Simply multiply the scalar of vector field by the scalar field that denotes the multi-volume. When both structures are denoted by vector fields however, computing the intersection is far less trivial.

#### Computing path-surface intersections

To save space, the notation ${\displaystyle (j;c_{j})}$ and ${\displaystyle (j;k_{j}\Delta c_{j})}$ will be omitted from the various terms.

Given a multi-path ${\displaystyle \mathbf {C} }$ denoted by vector field ${\displaystyle \mathbf {J} =\sum _{i}J_{i}{\hat {\mathbf {a} }}_{i}}$, and a multi-surface ${\displaystyle \mathbf {S} }$ denoted by vector field ${\displaystyle \mathbf {F} =\sum _{i}F_{i}{\hat {\mathbf {a} }}^{i}}$, the scalar field that denotes the intersection can be computed as follows:

The following computations applies to each cell:

For each ${\displaystyle i\in \{1,2,3\}}$, the weight assigned to ${\displaystyle C_{i}}$ by ${\displaystyle \mathbf {C} }$ is computed as follows: ${\displaystyle \Delta c_{1}\Delta c_{2}\Delta c_{3}\cdot V\cdot J_{i}}$ is the ${\displaystyle {\hat {\mathbf {a} }}_{i}}$ component of the total displacement contained by the current cell. Computing the weight assigned to ${\displaystyle C_{i}}$ requires that this displacement be spread over the length of ${\displaystyle C_{i}}$: ${\displaystyle {\frac {\Delta c_{1}\Delta c_{2}\Delta c_{3}\cdot V\cdot J_{i}}{\Delta c_{i}\cdot l_{i}}}={\frac {V}{l_{i}}}\cdot \Delta c_{i+1}\Delta c_{i+2}\cdot J_{i}}$.

For each ${\displaystyle i\in \{1,2,3\}}$, the weight assigned to ${\displaystyle \sigma _{i}}$ by ${\displaystyle \mathbf {S} }$ is computed as follows: ${\displaystyle \Delta c_{1}\Delta c_{2}\Delta c_{3}\cdot V\cdot F_{i}}$ is the ${\displaystyle {\hat {\mathbf {a} }}^{i}}$ component of the total surface vector contained by the current cell. Computing the weight assigned to ${\displaystyle \sigma _{i}}$ requires that this surface vector be spread over the area of ${\displaystyle \sigma _{i}}$: ${\displaystyle {\frac {\Delta c_{1}\Delta c_{2}\Delta c_{3}\cdot V\cdot F_{i}}{\Delta c_{i+1}\Delta c_{i+2}\cdot A_{i}}}={\frac {V}{A_{i}}}\cdot \Delta c_{i}\cdot F_{i}}$.

The intersection between ${\displaystyle C_{i}}$ and ${\displaystyle \sigma _{i}}$ is the current lattice point with weight ${\displaystyle \left({\frac {V}{l_{i}}}\cdot \Delta c_{i+1}\Delta c_{i+2}\cdot J_{i}\right)\left({\frac {V}{A_{i}}}\cdot \Delta c_{i}\cdot F_{i}\right)}$${\displaystyle ={\frac {V^{2}}{l_{i}A_{i}}}\cdot \Delta c_{1}\Delta c_{2}\Delta c_{3}\cdot J_{i}F_{i}}$.

Aside from the intersections between ${\displaystyle C_{i}}$ and ${\displaystyle \sigma _{i}}$ for each cell and ${\displaystyle i\in \{1,2,3\}}$, no other intersections occur. The total weight of the intersection at the vertex of the current cell is ${\displaystyle \sum _{i}{\frac {V^{2}}{l_{i}A_{i}}}\cdot \Delta c_{1}\Delta c_{2}\Delta c_{3}\cdot J_{i}F_{i}}$${\displaystyle =V^{2}\cdot \Delta c_{1}\Delta c_{2}\Delta c_{3}\sum _{i}{\frac {1}{l_{i}A_{i}}}\cdot J_{i}F_{i}}$.

The value of ${\displaystyle \mathbf {J} \cdot \mathbf {F} }$ at the current cell is approximately