# 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 that returns a number given an input point, and vector fields will not be simply treated as a function that returns a vector given an input point.

## Basic structures[edit]

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

### Multi-points[edit]

A point is an arbitrary position. A "multi-point" is a set of point/weight pairs: where is the "weight" that is assigned to point . Given two point/weight pairs and that cover the same point , the weights add up to get which replaces and . Any pair is removed. can consist of infinitely many points, and each point may have an infinitesimal weight.

An arbitrary point can be described by the scalar field . This is the "Dirac delta function" centered on point . The is the inverse of an infinitely small volume that wraps point . To further explain this, let be a tiny volume with volume that wraps point . can be approximated by . A mass of 1 is being crammed into yielding an infinitely high density. Since is essentially a density function, it brings with it the units .

Multi-point can be described by the scalar field . If consists of infinitely many points with each point having infinitesimal weight, then 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 image below shows how a continuous scalar field can be generated as a collection of points. Consider position and the infinitesimal volume with volume . The total point weight contained by is . This weight of is then split up over an arbitrarily large number of points that are scattered over the volume .

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.

An important requirement that will be made of the scalar fields that denote multi-points is that the total point weight is finite. To ensure this, it will be assumed that there must always exist an such that is . In more rigorous terms, this means that there exists some and quantities and such that .

### Multi-paths[edit]

A simple path (also called a simple curve) is an oriented continuous curve that extends from a starting point to an ending point . Intermediate points are indexed by and are denoted by . 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: where is the weight that is assigned to path . Given two path/weight pairs and that cover the same path , the weights add up to get which replaces and . Any pair is removed. In addition given two path/weight pairs and with the same weight and , then and can be linked end-to-end to get the pair which replaces and . Assigning a path a negative weight effectively reverses its orientation: if denotes path with the opposite orientation, then is equivalent to . can consist of infinitely many paths, and each path may have an infinitesimal weight.

An arbitrary curve can be described by the vector field . This is the "Dirac delta function" for the curve . is the unit length tangent vector to path at point . if . If there are multiple tangent vectors due to intersecting itself, then is the sum of these tangent vectors. The is the inverse of the cross-sectional area of an infinitely thin tube that encloses . To further explain this, let be a thin tube with cross-sectional area that encloses . can be approximated by . is the generalization of to the tube . A path weight of 1 is being crammed into the cross-sectional area of yielding an infinitely high path density. Since is essentially a density over area, it brings with it the units .

The image to the right gives a depiction of the Dirac delta function for a simple curve. The vector field is 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 can be described by the vector field . If consists of infinitely many paths with each path having infinitesimal weight, then 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.

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.)

An important requirement that will be made of the vector fields that denote multi-paths is that the path weight that extends to infinity is 0. To ensure this, it will be assumed that . This condition ensures that the maximum path weight density passing through a sphere (in any direction) centered on the origin diminishes to 0 as the sphere's radius becomes infinitely large.

### Multi-surfaces[edit]

A simple surface 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: where is the weight that is assigned to surface . Given two surface/weight pairs and that cover the same surface , the weights add up to get which replaces and . Any pair is removed. In addition given two surface/weight pairs and with the same weight , then and can be combined to get the pair which replaces and . Assigning a surface a negative weight effectively reverses its orientation: if denotes surface with the opposite orientation, then is equivalent to . can consist of infinitely many surfaces, and each surface may have an infinitesimal weight.

An arbitrary surface can be described by the vector field . This is the "Dirac delta function" for the surface . is the unit length normal vector to surface at point . if . If there are multiple normal vectors due to intersecting itself, then is the sum of these normal vectors. The is the inverse of the thickness of an infinitely thin membrane that encloses . To further explain this, let be a thin membrane with thickness that encloses . can be approximated by . is the generalization of to the membrane . A surface weight of 1 is being sandwiched into the thickness of yielding an infinitely high surface density. Since is essentially a density over length, it brings with it the units .

Multi-surface can be described by the vector field . If consists of infinitely many surfaces with each surface having infinitesimal weight, then 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.

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.

An important requirement that will be made of the vector fields that denote multi-surfaces is that the surface weight that extends to infinity is 0. To ensure this, it will be assumed that . This condition ensures that the maximum surface weight density along a circle centered on the origin diminishes to 0 as the circle's radius becomes infinitely large.

### Multi-volumes[edit]

A volume is an arbitrary region of space. A "multi-volume" is a set of volume/weight pairs: where is the "weight" that is assigned to volume . Given two volume/weight pairs and that cover the same volume , the weights add up to get which replaces and . Any pair is removed. In addition given two volume/weight pairs and with the same weight and , then and can be combined to get the pair which replaces and . can consist of infinitely many volumes, and each volume may have an infinitesimal weight.

An arbitrary volume can be described by the scalar field . 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 or not, 1 being yes and 0 being no. Since is simply an indicator function, it brings with it no units (it is dimensionless).

Multi-volume can be described by the scalar field . If consists of infinitely many volumes with each volume having infinitesimal weight, then 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.

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.

An important requirement that will be made of the scalar fields that denote multi-volumes is that the volume weight at infinity is 0. To ensure this, it will be assumed that .

## Totals[edit]

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[edit]

Given a multi-point , the total point weight is clearly . Given a scalar field that denotes a multi-point, the total weight of is . Given a simple point , the total weight is 1 so .

### Total displacement[edit]

Given a simple path that starts at point and ends at point , the total displacement generated by is . This displacement is solely dependent on the endpoints as indicated by the top image to the right.

Given a multi-path , the total displacement generated by is .

Given a vector field that denotes a multi-path, the total displacement generated by is . Since the displacement generated by a simple path is , it is the case that .

One important observation from is that given a path integral over path , the differential is equal to in a volume integral: provided that function is linear in the second parameter. In the lower image to the right, the displacement differential is equated to the volume differential by diffusing the path over an infintely thin cross-sectional area . The integrand at points outside of the infinitely thin tube is 0: for all points , .

### Total surface vector[edit]

Given an arbitrary oriented surface , its "counter-clockwise boundary", denoted by , is the boundary of whose orientation is determined in the following manner: Looking at so that the preferred direction through is oriented towards the viewer, the boundary wraps 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 and is oriented to be perpendicular to unit-length normal vector . The "surface vector" of this surface is .

Given a non-flat surface , the total surface vector of is computed by summing the surface vectors of each infinitesimal portion of . The total surface vector is .

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:

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 , **so the sum of the unrotated surface vectors is also .**

The fact that the total surface vector of a closed ribbon is 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 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.

Given a multi-surface the total surface vector generated by is .

Given a vector field that denotes a multi-surface, the total surface vector generated by is . Since the surface vector generated by simple surface is , it is the case that . One important observation is that given a surface integral over , the differential is equal to in a volume integral: provided that function is a linear in the second parameter.

### Total volume[edit]

Consider a multi-volume , where the volumes of are respectively , then the total volume of is . Each volume can be computed by . The total volume of is .

If a multi-volume can be denoted by scalar field , then the volume of is .

Given an arbitrary volume , a volume integral over can be converted to a volume integral over by replacing the differential with :

provided that is linear in the second parameter.

## Intersections[edit]

The union of two multi-points denoted by scalar fields and is simply , 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.

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

### Point-Volume intersections[edit]

When a point with weight intersects a volume with weight , then the intersection is point with weight . 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.

Given a multi-point with scalar field , and multi-volume with scalar field , then the intersection is a multi-point with scalar field .

### Path-Surface intersections[edit]

When a path with weight intersects a surface with weight at point , then the intersection is point with weight . The weight is if passes through in the direction in which is oriented. The weight is if passes through opposite the direction in which 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.

In the image above to the far right, the multi-path is denoted by a vector field which has the value inside the blue tube, and is everywhere else. The multi-surface is denoted by a vector field which has the value among the red sheets, and is everywhere else. The total path weight in the blue tube is . The total surface weight in the red sheets is . The total weight of all the intersection points is . The volume that the intersection points are evenly spread out in is . The intersection point density is .

Given a multi-path with vector field , and a multi-surface with vector field , then the intersection is a multi-point with scalar field .

### Path-Volume intersections[edit]

When a path with weight intersects a volume with weight , then the intersection is path with weight . 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.

Given a multi-path with vector field , and a multi-volume with scalar field , then the intersection is a multi-path with vector field .

### Surface-Surface intersections[edit]

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

In the image above to the right, the first multi-surface is denoted by a vector field that has the value among the blue sheets, and is everywhere else. The second multi-surface is denoted by a vector field that has the value among the red sheets, and is everywhere else. The total surface weight in the blue sheets is , and the total surface weight in the red sheets is . The total weight of all the intersection paths is . The cross-sectional area that the intersection paths are evenly spread out over is . The intersection path density is . 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 , and a multi-surface with vector field , then the intersection is the multi-path with vector field .

### Surface-Volume intersections[edit]

When a surface with weight intersects a volume with weight , then the intersection is surface with weight . 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.

Given a multi-surface with vector field , and a multi-volume with scalar field , then the intersection is a multi-surface with vector field .

### Volume-Volume intersections[edit]

When a volume with weight intersects a volume with weight , then the intersection is the volume with weight . 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.

Given a multi-volume with scalar field , and a multi-volume with scalar field , then the intersection is a multi-volume with scalar field .

### Other intersections[edit]

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:

structure | multi-point | multi-path | multi-surface | multi-volume |
---|---|---|---|---|

multi-point | n/a | n/a | n/a | multi-point |

multi-path | n/a | n/a | multi-point | multi-path |

multi-surface | n/a | multi-point | multi-path | multi-surface |

multi-volume | multi-point | multi-path | multi-surface | multi-volume |

## Boundaries[edit]

### The endpoints of paths[edit]

Given a simple path that starts at point and ends at point , the "endpoints" of is the multi-point that consists of the starting point with a weight of +1, and the final point with a weight of -1. While is denoted by the vector field , the endpoints are denoted by the scalar field . The image below gives several examples of simple paths and their associated endpoints.

Given a multi-path , the endpoints of is the multi-point .

Given a vector field that denotes a multi-path, the multi-point that denotes the endpoints of is denoted by scalar field . The scalar field evaluated at point is denoted by , or .

The requirement that no path weight 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: .

The similarity of the notation to the intersection of multi-path with multi-surface , denoted by , makes sense if 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 is shown. On the right is a simple path , along with its endpoints . On the left is an extension of that is behind the "veil" of surface . pokes out of and into at points consistent with the endpoints of : i.e. .