This chapter will present an analog to vector calculus where space now consists of discrete "lumps". The purpose of this chapter is to provide an intuitive basis for vector calculus. The portrayal of vector calculus in this chapter will enable the generalization of vector calculus to non-Euclidean geometries.
The essence of Calculus[edit]
The key approach to calculus is to start with a model where each variable, using
as an example, takes on discrete quantities separated by small intervals of width
.
is a measure of the model's "coarseness" (when
is increasing) or "fineness" (when
is decreasing). When handling quantities at the scale of
, an approximation for which the error relative to
vanishes as
becomes infinitely small, becomes exact when
is treated as a continuous variable.
As an example, let
be small, and restrict
to the quantities
. Let
for any integer
.
The difference
approaches 0 as
, but it is not accurate to approximate
with 0, as the error relative to
does not vanish as
: in truth,
as
.
It is however, accurate to approximate
with
since the error relative to
does vanish as
: it is the case that
as
.
When we articulate continuous space into small but non-zero volumes for the purpose of approximating vector calculus, it is important to note that the only approximations that will be made, will be those that entail relative errors that vanish as the articulation of space becomes more and more fine.
In order to demonstrate the necessity of having the error relative to the differential
approach 0 as
as opposed to the absolute error, the following example will be used:
Consider the closed interval
divided into
intervals:
where
. From each
choose a representative point
, and let
. Let
and
be two functions whose first parameter is a representative point, and the second parameter is the interval width. Both
and
will be assumed to approach 0 as the interval width becomes infinitely small. Consider the two sums
and
. It will be assumed that
and
converge to
and
respectively as
and
.
The question of interest are circumstances under which
. If other words, as the interval
becomes more and more articulated into smaller and smaller intervals, what is required so that approximating
with
becomes exact?
If
as
, this is not sufficient to guarantee that
. In contrast, if
as
, this will force
. To prove this, the following derivation will be used:
For each
, it is assumed that
as
. Therefore
and hence
as
and
. Therefore
.
The left image shows the sums

and

. The white area of each strip depicts

, and the red area of each strip depicts

. The total area highlighted in red depicts the difference between

and

. As

increases, the center image shows how the total red area does not decrease if it is not the case that for each strip that the "ratio of red area to the strip width", which is the height of the red area, fails to approach 0 for all strips. This is despite the fact that the red area (absolute difference) in each strip is approaching 0. The right image shows how the total red area approaches 0 if the "red area to strip width ratio" for each strip is approaching 0.
Lumped space[edit]
Directed graphs[edit]
The lumped approximation of space will require the use of a "directed graph".
A "directed graph"
consists of a set
of nodes/vertices and a set
of directed edges. For each edge
,
has an origin node
that is the beginning of edge
, and a terminal node
that is the end of edge
. It will also be allowed that edges can form loops (
) and that multiple edges can have both the same beginning and end (
and
), though this will be irrelevant for the cases that will be considered.
An arbitrary function
over the set of nodes will be referred to as a "node based function", and is effectively a scalar field. The notation
will be used if the input parameter is ambiguous. A node based function can also be denoted by exhaustively listing the assignment to each node:
. If a node based function is denoted by a complex expression
, then the evaluation of
at a single node
will be denoted using the pipe-subscript notation:
.
An arbitrary function
over the set of edges will be referred to as an "edge based function", and is effectively a scalar or vector field. The notation
will be used if the input parameter is ambiguous. An edge based function can also be denoted by exhaustively listing the assignment to each edge:
. If an edge based function is denoted by a complex expression
, then the evaluation of
at a single edge
will be denoted using the pipe-subscript notation:
. When a real number assigned to an edge denotes a "vector", a positive value denotes an orientation along the edge's preferred direction, and a negative value denotes an orientation against the edge's preferred direction.
An example directed graph.
An example directed graph is shown to the right. The node set is
and the edge set is
. The origin and terminal nodes of each edge are:
;
;
;
;
; and
.
Interpreting nodes and edges[edit]
Two irregular volumes with an irregular boundary are characterized as two points linked by a curve. Each point is a node in a directed graph, and the curve denotes the directed edge that links the two points. The "length" of the directed edge is the distance between the two points, and the "cross-sectional area" of the directed edge is the area of the projection of the boundary surface onto a plane that is perpendicular to the displacement between the two points.
Each node
will correspond to both a point
in space, and a volume
that contains
.
will denote the volume of
, also referred to as the volume of node
. The volume of each node must be positive:
for all
. In the example directed graph, an example choice of volumes for each node is
.
Each edge
will correspond to both a curve
originating from
and terminating at
, and a surface
that separates
from
oriented from
to
.
will denote the distance
from
to
, which will be referred to as the length of edge
. The length of each edge must be positive:
for all
.
will denote the area of
projected onto a plane that is perpendicular to the displacement
, which will be referred to as the cross-sectional area, thickness, or width of edge
. The cross-sectional area of each edge must be positive:
for all
.
In the example directed graph, an example choice of lengths for each edge is
, and an example choice of areas for each edge is
.
In summary an arbitrary directed graph
will be characterized by:
- the set of nodes

- the set of edges

- the start
and end
of each edge 
- the volume
of each node 
- the length
and area
of each edge 
The point
and space
associated with each node
, and the curve
and surface
associated with each edge
, are not given consideration after the approximate directed graph model has been established. The parameters in the above list are all that is needed to utilize the directed graph model.
Points and multi-points[edit]
A "point" in space is quantified by a single node
, which corresponds to the space point
.
Point
can be denoted by the node based function
where for each node
it is the case that
. The term
arises from the fact that the point is "spread out" over the volume of
. The notation
denotes
with the parameter
not shown.
A superposition of points is a "multi-point". A multi-point is effectively a set of points with varying weights, which can be any real number. Let the set
of node/weight pairs denote a multi-point. For each
:
is the weight associated with node
. The node based function that denotes multi-point
is:
. Any node based function that returns real values can denote a multi-point.
Any node based function that denotes "density" is best interpreted as a multi-point.
Below is shown two examples of a multi-point
and the corresponding node based function
. On the left, there is a collection of points with a weight of +1 denoted by red dots, and points with a weight of -1 denoted by blue dots. For each volume
, the weights of all of the contained points are added together and the total weight is divided by
to spread it out over the volume
. On the right, shades of red denote positive values for
, while shades of blue denote negative values for
. White means that
for the current node
.
| Multi point example on the left, and function on the right.
|
| Multi point example on the left, and function on the right.
|
|
Volumes and multi-volumes[edit]
The example directed graph with the example multi-volume.
The discrete analog to volumes using a directed graph is that a volume
is a collection of nodes:
. The combined volume region denoted by
is
and the total volume amount is
.
Volume
can be denoted by a node based membership function:
where for each node
, it is the case that
. The notation
denotes
with the parameter
not shown.
A superposition of volumes forms a "multi-volume". Any node based function that returns real values can be the membership function of a multi-volume. For example, given the node based function
can be expressed as the following superpositions:
is the superposition of volumes
with the respective weights
.
is the superposition of volumes
with the respective weights
.
It is important to note that given an arbitrary node based function, decomposing the multi-volume denoted by the function into a superposition of volumes is not unique, as shown above where a node based function is decomposed into multiple superpositions of volumes.
Any node based function that denotes a "potential" is best interpreted as a multi-volume.
Paths and multi-paths[edit]
The example directed graph with the example multi-path.
The discrete analog to paths using a directed graph is that a path
is a series of edges and orientations:
where
is the
edge and
is an indicator that is
if
is traversed in the preferred direction, and is
if
is traversed in the opposite direction. The path must be unbroken: for each
it must be that
. The combined path denoted by
is
(path
is path
the orientation reversed) and the total length is
.
A path
can be denoted by the edge based function
where for each edge
, it is the case that
where
. The term
arises from the fact that the path is "spread out" over the thickness of
. The notation
denotes
with the parameter
not shown.
A superposition of paths forms a "multi-path". Any edge based function that returns real values can denote a multi-path. For example, assuming that
for all
, the edge based function
can be expressed as the following superpositions:
is the superposition of paths
, and
with the respective weights
.
is the superposition of paths
, and
with the respective weights
.
It is important to note that given an arbitrary edge based function, decomposing the multi-path denoted by the function into a superposition of paths is not unique, as shown above where an edge based function is decomposed into multiple superpositions of paths.
Any edge based function that denotes "flow density" (flow per unit cross-sectional area) is best interpreted as a multi-path.
Below is shown two examples of a multi-path
and the corresponding edge based function
. On the left a superposition of multiple paths is shown. For each surface
, the total "flow" through
is computed and is then divided by
to spread it out over the surface
. On the right, the flow enters a surface through the green side and leaves through the red side. The density of the flow is depicted by the strength of the shading. If the green to red orientation coincides with the orientation of
, then
is positive. If the green to red orientation is opposite the orientation of
, then
is negative.
| Multi path example on the left, and edge based function on the right.
|
| Multi path example on the left, and edge based function on the right.
|
|
Surfaces and multi-surfaces[edit]
The example directed graph with the example multi-surface.
The discrete analog to oriented surfaces using a directed graph is that an oriented surface
is a collection of edges and orientations:
where all of the edges
are distinct and
is an indicator that describes the surface's orientation. Each edge
is an edge that passes through the surface, and
is
if
passes through the surface in the preferred direction, and
is
if
passes through the surface in the opposite direction. The combined surface denoted by
is
(surface
is surface
with the orientation reversed) and the total area is
.
A surface
can be denoted by the edge based function
where for each edge
, it is the case that
. The term
arises from the fact that the surface is "spread out" over the length of
. The notation
denotes
with the parameter
not shown.
A superposition of surfaces forms a "multi-surface". Any edge based function that returns real values can denote a multi-surface. For example, assuming that
for all
, the edge based function
can be expressed as the following superpositions:
is the superposition of surfaces
and
with the respective weights
.
is the superposition of surfaces
and
with the respective weights
and
.
It is important to note that given an arbitrary edge based function, decomposing the multi-surface denoted by the function into a superposition of surfaces is not unique, as shown above where an edge based function is decomposed into multiple superpositions of surfaces.
Any edge based function that denotes the "rate of gain" (rate of gain per unit length) is best interpreted as a multi-surface.
Below is shown two examples of a multi-surface
and the corresponding edge based function
. On the left a superposition of multiple surfaces is shown. For each curve
, the net number of times edge
penetrates
in the preferred direction, referred to as the "gain" across
, is computed and is then divided by
to spread the gain out over the curve
. On the right, the net gain is positive if an edge is traversed from the green end to the red end. The amount of gain is depicted by the strength of the shading. If the green to red orientation coincides with the orientation of
, then
is positive. If the green to red orientation is opposite the orientation of
, then
is negative.
| Multi surface example on the left, and edge based function on the right.
|
| Multi surface example on the left, and edge based function on the right.
|
|
Intersections[edit]
Multi-point multi-volume intersections[edit]
A multi-volume and a multi-point is shown. The red volumes and points have a weight of +1, and the cyan volumes and points have a weight of -1. The intersection contributed by a point is its weight times the weight assigned to the volume that contains the point. The total intersection weight in this image is: N = (+1)x(+1+1+1-1-1) + 0x(+1+1-1-1-1) + (-1)x(+1-1) = +1.
Given an arbitrary point
and volume
, let
denote an indicator function that returns 1 if point
is contained by volume
and returns 0 if otherwise.
Node based function
denotes the point
, and node based function
denotes the volume
.
The indicator function
can be computed via the following sum:
.
The term
cancels out the
in
.
counts the number of times point
occurs in volume
which can only be 0 or 1 for simple points and volumes. This count can be generalized to sets of points. Let
denote a collection of points. The collection of points can be denoted by the node based function
. The number of points from
that are contained by volume
can again be computed by the sum:
.
In the general case, given a node based function
that denotes a multi-point, and a node based function
that denotes a multi-volume, then the total weight of all the occurrences of
being inside of
is
. The quantity
is called the net intersection or the total intersection weight of
with
.
It should also be noted that
is commutative:
Multi-path multi-surface intersections[edit]
A path is shown intersecting a surface multiple times. The boundary of the surface has a counter-clockwise orientation. The net number of times that the surface is crossed in the preferred direction is +1-1+1+1+1 = 3.
Given an arbitrary path
and surface
, let
denote the number of times path
passes through surface
in the preferred direction minus the number of times path
passes through surface
in the opposite direction. For an arbitrary
, and an arbitrary
, if
then the
link of path
coincides with the
segment of surface
. If
, then
and
passes through
in the preferred direction. If
, then
and
passes through
in the opposite direction.
can be computed via the following sum:
. The term
cancels out the
from
, and the term
cancels out the
from
.
In the general case, given an edge based function
that denotes a multi-path, and an edge based function
that denotes a multi-surface, then the total weight of all the intersections of
with
is
. The quantity
is called the net intersection or the total intersection weight of
with
.
It should also be noted that
is commutative:
Lumped coordinate system models[edit]
Model of Cartesian coordinates[edit]
A directed graph that models the Cartesian coordinate system. One dimension is suppressed for simplicity.
The discrete model used for Cartesian coordinates will consist of an infinite 3 dimensional grid of nodes that spans
. To establish the model, the following is needed:
- A spacing
for the x-coordinates.
- A spacing
for the y-coordinates.
- A spacing
for the z-coordinates.
For each triplet of integer indices
, there exists a node
and 3 edges
,
, and
.
The origin and terminal nodes of
are
and
The origin and terminal nodes of
are
and
The origin and terminal nodes of
are
and
The Cartesian coordinate associated with node
is
, and the space associated with node
is
.
The volume of node
is
.
The length of edge
is
.
The length of edge
is
.
The length of edge
is
.
The area of edge
is
.
The area of edge
is
.
The area of edge
is
.
Given a scalar field
over Cartesian coordinates,
will be approximated by the node based function
. Function
is defined by
for each
.
Given a vector field
over Cartesian coordinates,
will be approximated by the edge based function
. Function
is defined by
;
; and
.
Model of Cylindrical coordinates[edit]
A directed graph that models the cylindrical coordinate system. For simplicity, the vertical/z dimension is not shown.
The discrete model used for Cylindrical coordinates will consist of an infinite 3 dimensional cylindrical grid of nodes that spans
. To establish the model, the following is needed:
- A spacing
for the coordinate
.
- A large positive integer
that will yield a spacing
for the coordinate
.
- A spacing
for the coordinate
.
Each node
will correspond to a unique triplet
of integers, and will be denoted by
. The cylindrical coordinate denoted by
will be
.
Clearly, due to the nature of cylindrical coordinates (such as the fact that coordinate
is cyclical), not every triplet
will be allowed. The following three domains will be defined:
,
, and
.
There exists a node
for each
.
There exists an edge
for each
such that
and
There exists an edge
for each
such that
and
There exists an edge
for each
such that
and
Next, the volume of each node and the lengths and area of each edge will be calculated. To simplify notation, let
for any nonnegative real number
.
For each
the cylindrical coordinate that corresponds to
is
, and the space (described using cylindrical coordinates) associated with
is
For each
the volume of node
is
For each
the length and area of edge
is
and
respectively.
For each
the length and area of edge
is
and
respectively.
For each
the length and area of edge
is
and
respectively.
Given a scalar field
over cylindrical coordinates,
will be approximated by the node based function
. Function
is defined by
for each
.
Given a vector field
over cylindrical coordinates,
will be approximated by the edge based function
. Function
is defined by:
Model of Spherical coordinates[edit]
A directed graph that models the spherical coordinate system. The image is a cross-section that contains the vertical/z axis, which is "up" in the image.
The discrete model used for Spherical coordinates will consist of an infinite 3 dimensional spherical grid of nodes that spans
. To establish the model, the following is needed:
- A spacing
for the coordinate
.
- A large positive integer
that will yield a spacing
for the coordinate
.
- A large positive integer
that will yield a spacing
for the coordinate
.
Each node
will correspond to a unique triplet
of integers, and will be denoted by
. The spherical coordinate denoted by
will be
.
Clearly, due to the nature of spherical coordinates (such as the fact that coordinate
is cyclical), not every triplet
will be allowed. The following three domains will be defined:
,
,
, and
.
There exists a node
for each
.
There exists an edge
for each
such that
and
There exists an edge
for each
such that
and
There exists an edge
for each
such that
and
Next, the volume of each node and the lengths and area of each edge will be calculated. To simplify notation, let
for any nonnegative real number
, and let
for any nonnegative real number
and any real number
.
For each
the spherical coordinate that corresponds to
is
, and the space (described using spherical coordinates) associated with
is
For each
the volume of node
is
For each
the length and area of edge
is
and
respectively.
For each
the length and area of edge
is
and
respectively.
For each
the length and area of edge
is
and
respectively.
Given a scalar field
over spherical coordinates,
will be approximated by the node based function
. Function
is defined by
for each
.
Given a vector field
over spherical coordinates,
will be approximated by the edge based function
. Function
is defined by:
Model of Curvilinear coordinates[edit]
A curvilinear coordinate system is a coordinate system such as cylindrical or spherical coordinates where the position may not follow a straight line as one of the coordinates is changed. To model an arbitrary curvilinear coordinate system in 3 dimensional space, let the coordinates be denoted by
.
To avoid unnecessary repetition, the following notation will be used:
- Given any number
, and subset
, then
denotes the triple ![{\displaystyle [k]_{C}=\left(\left\{{\begin{array}{cc}k&(1\in C)\\0&(1\notin C)\end{array}}\right.,\left\{{\begin{array}{cc}k&(2\in C)\\0&(2\notin C)\end{array}}\right.,\left\{{\begin{array}{cc}k&(3\in C)\\0&(3\notin C)\end{array}}\right.\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dc5dbdd6edacf23ee05a0720f02c657b248fd499)
- Given any two triples of numbers
, then 
- Given any two triples of numbers
, then 
denotes an arbitrary curvilinear coordinate
.
denotes an arbitrary triplet of differences in the coordinates
.
Let
denote the position given by coordinate
.
For each
, the rate of change in the position
at coordinate
with respect to
is given by:
.
It will be assumed that
,
, and
are all mutually perpendicular, which is the case for Cartesian, cylindrical, and spherical coordinates.
In addition, for each
, vector
will also denote a unit length normalization of
.
For the purposes of simplicity, it will be assumed that the coordinates
can be any triple of real numbers. Singularities such as the z-axis in cylindrical and spherical coordinates will not be considered by this model. Since singularities are not considered, this model will be considerably simpler than the models given for cylindrical and spherical coordinates.
To model the curvilinear coordinate system, a lattice of of nodes
connected by directed edges
will be established:
Start by choosing "resolutions" for each coordinate
denoted by
. The quantities
should all be strictly positive, and ideally should be as small as possible.
- For each triplet of integer indices
, a node
will be created at position
.
- For each
and for each triplet of integer indices
, there will exist edge
such that
and
.
- For each
, the volume of node
is

- For each
and
, the length and area of edge
are
and
respectively.
Any node
for which
should be deleted along with all connecting edges.
For any edge
for which
and
, then nodes
and
should be fused into a single node, and edge
should be discarded. For any edge
for which
and
, then edge
should be discarded. Either action may be pursued when
.
Given a scalar field
over the curvilinear coordinate system,
will be approximated by the node based function
. Function
is defined by
for each
.
Given a vector field
over the curvilinear coordinates,
will be approximated by the edge based function
. Function
is defined by
for each
and for each
.
Discrete analog to Integrals[edit]
Volume Integrals[edit]
Let
denote an arbitrary volume. Given a node based function
, the "volume integral" of
over
is
(recall that
is the volume of node
, and
is the volume of space represented by
).
If node based function
is an approximation of a scalar field
, where
, then
is an approximation of the volume integral
.
Recall that
can be described via the node based function
. This allows the expression
. If node based function
denotes a multi-volume, then
.
most often denotes a density function, and
denotes the total "mass" contained by multi-volume
. The equality
means that if
is interpreted as a multi-point, then the total "mass" contained by
is the total intersection weight of
with
.
Path Integrals[edit]
Let
denote an arbitrary path. Given an edge based function
, the "path integral" of
along
is
(recall that
is the length of edge
, and
is the curve represented by
).
If edge based function
is an approximation of a vector field
, where
, then
is an approximation of the path integral
.
Recall that
can be described via the edge based function
. This allows the expression
. If edge based function
denotes a multi-path, then
.
most often denotes a rate of gain function, and
denotes the total "gain" caused by
along multi-path
. The equality
means that if
is interpreted as a multi-surface, then the total "gain" generated along
is the total intersection weight of