In the chapter on vector calculus, the differential operator of the gradient (
), the divergence (
), and the curl (
) were introduced. This chapter will focus on inverting these differential operators.
The gradient, divergence, and curl operators are all "linear", meaning that given arbitrary scalar fields
, vector fields
, and scalars
, that:
More generally given a family of scalar fields
, vector fields
, and coefficients
(each
corresponds to a scalar field, a vector field, and a scalar coefficient),
Inverting Linear OperatorsEdit
Recall from linear algebra that when given a bijective linear operator
, that an inverse can be created by computing a solution to each of
where
is the
elementary basis vector for each
. When attempting to find an
that solves
for an arbitrary
, the solutions to
can be stacked in a linear manner to get
as a possible solution. This yields an approach to inverting
.
This same approach will be used to compute "Green's functions" for each vector calculus differential operator.
The Dirac delta functionEdit
The Dirac delta function is a hypothetical function
that returns
for all
and
for
. The Dirac delta is not meant to be evaluated at
, but instead assumed to satisfy the following integral property:
. More generally, given any interval
that strictly contains 0:
, then
.
is effectively a density function that describes an infinitely dense total mass of 1 at
. For an arbitrary
,
describes a density function that describes an infinitely dense total mass of 1 at
.
Even though
, it is not the case that
, or
. The integrals are:
and
. In general, for any
,
.
Addressing non-differentiabilityEdit
Consider the piece-wise function
defined by:
. It is common to accept that
is not differentiable at
and that
. With the Dirac delta function, the derivative of
can be expressed as
. With this derivative, part II of the fundamental theorem of calculus holds even for intervals that contain
.
In this chapter, it will generally be assumed that all scalar fields
and vector fields
are continuous and differentiable everywhere. However if this is not the case, the Dirac delta function will be used to model the derivative operators at points of non-differentiability.
Inverting linear differential operatorsEdit
Given an arbitrary function
,
can be expressed as the linear combination
of Dirac delta functions. Let
be a linear differential operator that takes a real valued single variable function
, and returns another real valued single variable function
. If a solution
exists to
for all
, then given an arbitrary
, a solution
exists to
which is
. The family of functions
are referred to as "Green's functions".
Dirac delta function variantsEdit
In this chapter, given an arbitrary point
, the function
will denote a 3-dimensional variant of the Dirac delta function. The key integral property is that for any volume
, that
. Note that
. It should also be noted that
carries with it the dimensions
.
Given an arbitrary oriented curve
,
will denote another variant of the Dirac delta function which returns vectors.
for all
, and is infinite in the direction of
for all points from
. The key integral property is that for any oriented surface
, that
where
is the net number of times
passes through
in the preferred direction (
passing through
in the reverse direction reduces
by 1). It should also be noted that
carries with it the dimensions
.
Given an arbitrary oriented surface
,
will denote another variant of the Dirac delta function which returns vectors.
for all
, and is infinite in the direction of the oriented normals of
for all points from
. The key integral property is that for any oriented curve
, that
where
is the net number of times
passes through
in the preferred direction (
passing through
in the reverse direction reduces
by 1). It should also be noted that
carries with it the dimensions
.
Multi-paths and multi-surfacesEdit
Given an oriented curve
, then
can be denoted by the Dirac delta vector field
. If
is continuous and starts at
and ends at
, then it can be proven using Gauss's Divergence Theorem that
.
Given a collection
of oriented paths, then the vector field
effectively denotes the "superposition" of
. This superposition is referred to as a "multi-path". Not all paths have to have a weight of 1. With multi-path
, the weights on
and
are both
. This multi-path is an even 50%/50% superposition between
and
.
Any vector field
can be envisioned as a superposition of a possibly infinite number of paths. Each path may have an infinitesimal weight. When a vector field is envisioned as a multi-path, the decomposition into individual paths is not unique. When vector field
denotes a multi-path,
is the net density of path origin points minus the density of destination points: the path starting point density.
When
everywhere,
can be envisioned as a superposition of a possibly infinite number of paths that are either closed or extend to infinity. If it is also the case that
is
for some
, then all of the paths have to close and
is effectively a "multi-loop". (
is
if and only if there exists some threshold
and factor
such that
)
A demonstration of how a divergence free vector field can be envisioned as the superposition of several simple loops.
In the image to the right, a divergence free vector field that denotes flow density is decomposed into the superposition of multiple simple loops. 2 dimensional space is depicted as a lattice of infinitely small squares. The vector field is the top-left section. The flow along each horizontal edge, denoted by the direction and number of arrows, is the horizontal component of the vector field at the current edge (or neighboring vertex). The flow along each vertical edge, denoted by the direction and number of arrows, is the vertical component of the vector field at the current edge (or neighboring vertex). The remaining 3 sections show 3 simple loops whose superposition forms the vector field.
Given an oriented surface
, then
can be denoted by the Dirac delta vector field
. If
has the counter-clockwise oriented boundary
, then it can be proven using Stokes' Theorem that
.
Given a collection
of oriented surfaces, then the vector field
effectively denotes the "superposition" of
. This superposition is referred to as a "multi-surface". Not all surfaces have to have a weight of 1. With multi-surface
, the weights on
and
are both
. This multi-surface is an even 50%/50% superposition between
and
.
Any vector field
can be envisioned as a superposition of a possibly infinite number of surfaces. Each surface may have an infinitesimal weight. When a vector field is envisioned as a multi-surface, the decomposition into individual surfaces is not unique. When vector field
denotes a multi-surface,
is the multi-loop that is the counter-clockwise oriented boundary of the multi-surface denoted by
.
When
everywhere,
can be envisioned as a superposition of a possibly infinite number of surfaces that are either closed with no boundaries or extend to infinity. If it is also the case that
is
for some
, then all of the surfaces have to close without extending to infinity and
is effectively a "multi closed surface".
Given an oriented curve
and a vector field
, then the path integral
is equivalent to the volume integral
. This statement, while intuitive to a certain degree given the definition of
, is proven in the box below:
In a general sense, the infinitesimal displacement
from a path integral along
can be replaced by
when the integral is converted to a volume integral over
. For example, given a continuous curve
which begins at
and ends at
, it can be derived that:
.
Given an oriented surface
and a vector field
, then the surface integral
is equivalent to the volume integral
. This statement, while intuitive to a certain degree given the definition of
, is proven in the box below:
In a general sense, the infinitesimal surface vector
from a surface integral over
can be replaced by
when the integral is converted to a volume integral over
. For example, given an oriented surface
which has a total surface vector of
, it can be derived that:
.
Given a curve
and a surface
, the net number of times
passes through
in the preferred direction, denoted by
, is given by:
Given vector fields
and
where
denotes a multi-path and
denotes a multi-surface, then
denotes the total "flux" of multi-path
through multi-surface
.
If
is closed loop and
is a closed surface then
since every time
passes through
,
must pass through
in the opposite direction in order to close itself. More generally, if vector field
denotes a multi-loop, which means that
and
is
for some
, and vector field
denotes a multi closed surface, which means that
and
is
for some
, then
. An algebraic proof (which is not truly necessary) is given in the box below:
Proof
To start, let
and
.
where
and
where
It is important to note that:
- As
,
, or
, that
.
- As
,
, or
, that
.
- Since
as
, then
as
.
Evaluating the integral gives:
An example application of multi-paths and multi-surfaces is given in the box below:
Electromagnetic Induction
Given a closed loop
that is carrying an electric current of
, the magnetic field
generated by this closed loop obeys the following equations:
(Ampere's Law in magnetostatics)
where
is magnetic permeability of free space [1]. It is also required that
as
.
Let
and
be two closed loops which are the counterclockwise boundaries of surfaces
and
. Let
denote the magnetic field generated by running an electric current of
around
. Note that
is proportional to
. Now let
denote the total flux of
through
in the preferred direction:
. Note that since
, that
is a function of
instead of
. It is the case that
is proportional to
, and the constant of proportionality
is the "mutual inductance" from
to
.
The mutual inductance
is purely a function of
and
. It is also the case that the mutual inductance is symmetric:
. The symmetry of the mutual inductance is not obvious, and while the symmetry is apparent from explicit formulas for the mutual inductance such as the "Neumann formula" [1], the symmetry can be made clear by interpreting the magnetic field as both a multi-loop and a multi-surface.
The vector field
satisfies both
and
. While not proven here, the magnetic field generated by a closed loop of current that does not extend to infinity is
so
is both a multi-loop and a multi-surface with counterclockwise boundary
. Doing the same for loop
gives
where
is the magnetic field generated solely by running a current of
around
.
is both a multi-loop and a multi-surface with counterclockwise boundary
.
The flux of
through
is:
Since
, the vector field
is a multi closed surface, and the total flux of multi-loop
through this multi closed surface is 0:
.
Therefore
so the mutual inductance is
. The symmetry of the mutual inductance is now apparent from this formula:
.
Electromagnetic Induction Generalized
Given a closed loop
that is carrying an electric current of
, the magnetic field
and the auxiliary field
generated by this closed loop obeys the following equations:
(Ampere's Law in a magnetic material)
(Definition of the auxiliary field
)
where
is the magnetic permeability of the magnetic material.
may vary over space. It is also required that
as
.
Let
and
be two closed loops which are the counterclockwise boundaries of surfaces
and
. Let
and
respectively denote the magnetic field and auxiliary field generated by running an electric current of
around
. Note that both
and
are proportional to
. Now let
denote the total flux of
through
in the preferred direction:
. Note that since
(
is a multi-loop), that
is a function of
instead of
. It is the case that
is proportional to
, and the constant of proportionality
is the "mutual inductance" from
to
.
The mutual inductance
is purely a function of
and
. It is also the case that the mutual inductance is symmetric:
. The symmetry of the mutual inductance is not obvious, and can be made clear by interpreting the magnetic field as a multi-loop and the auxiliary field as a multi-surface.
The vector fields
and
satisfies
,
, and
. While not proven here, the magnetic field generated by a closed loop of current that does not extend to infinity is
so
is a multi-loop and
is a multi-surface with counterclockwise boundary
. Doing the same for loop
gives
and
.
is the magnetic field and
is the auxiliary field generated solely by running a current of
around
.
is a multi-loop and
is a multi-surface with counterclockwise boundary
.
The flux of
through
is:
Since
, the vector field
is a multi-closed surface, and the total flux of multi-loop
through this multi-closed surface is 0:
.
Therefore
so the mutual inductance is
. The symmetry of the mutual inductance is now apparent from this formula:
.
Given a continuous oriented curve
that originates from
and terminates at
, it has been noted that
. It can be derived that:
.
Generalizing to a multi-path vector field
, it is the case that:
. Recall that
is a measure of the "starting point density" of
.
Given an oriented surface
with counterclockwise boundary
, it has been noted that
. It can be derived that:
.
Generalizing to a multi-surface vector field
, it is the case that:
. Recall that
is the counterclockwise boundary density of
.
Inverting the divergence operator (the inverse square law)Edit
Given an arbitrary scalar field
, the problem of interest is that of finding a vector field
that satisfies
and
. In other words, we want to find a vector field whose divergence is given by
and is irrotational. For reasons that will soon become apparent, it will be assumed that
is
as
for some
. The "big O" means that
.
can be expressed as a linear combination of Dirac delta functions:
.
If an irrotational vector field
can be determined such that
(that is divergence-free everywhere except the origin where the divergence is infinite), then for all
:
. Since the divergence operator distributes over linear combinations,
is an irrotational vector field that satisfies
.
An intuitive candidate for an irrotational vector field that is divergence free everywhere except the origin is a radially symmetric vector field
where
is the distance from the origin, and
is the unit vector that points away from the origin (see spherical coordinates).
is unknown at this point. The inverse square indicates that the flow diffuses out over a larger area as the distance from the origin increases. It can easily be checked that
is irrotational everywhere (including the origin), and is divergence free everywhere except the origin. All that remains is to determine
such that
. Consider a sphere of radius
centered on the origin. The total outwards flow/flux through the surface of this sphere is
, so the total flow generated inside the sphere is
. Since
, the total flow generated inside the sphere is
. This gives
. Therefore
.
In total,
is an irrotational vector field that satisfies
. The assumption that
is
as
for some
implies that the volume integral does not diverge to infinity. Also note that the apparent singularity at
does not impact the integral (to be discussed below).
UniquenessEdit
An important question is if the vector field
is the only irrotational vector field that satisfies
. If
is another possible solution, then
is a vector field that is both irrotational and divergence free at all points:
and
. There is then the following theorem:
is a constant vector
, and therefore an irrotational vector field
that satisfies
is unique up to the addition of a constant vector field.
About Improper IntegralsEdit
The volume integral
has a pole/singularity when
. Its range also extends to infinity. Both of these irregularities have the potential to result in a divergent (infinite) integral.
To analyse whether or not the integral diverges due to the pole/singularity or infinite range, the volume integral will be expressed as the integral of concentric spherical shells centered on
:
where
denotes the outwards oriented surface of a sphere centered on
with radius
.
denotes a solid sphere centered on
with radius
.
The inner surface integral does not present any irregularities. The lower bound of
denotes the pole, and the upper bound of
denotes the infinite range.
The integral
where the inner radius
is small and the outer radius
is large is regular. The goal is to analyse the integral's behavior as
and
.
Assume that
for all
for each radius
. The magnitude of the inner surface integral is now bounded from above by:
As can be seen, the surface area of
cancels out
.
The magnitude of
is bounded from above by:
Assuming that
is bounded everywhere,
is bounded everywhere, so it is clear that
does not approach infinity as
. This settles the pole/singularity at
. For the integral to not diverge as
, the condition that
is
as
for some
is sufficient. This condition implies that
is less than a multiple of
when
becomes sufficiently large. The integral of
converges at infinity provided that
.
Inverting the gradient operatorEdit
Given a vector field
, the problem of interest is finding a scalar field
such that
. It should be noted that
does not exist for most vector fields. The gradient theorem implies that for all closed continuous curves
, that
. It must be the case that
for all closed continuous curves
:
must be conservative. Equivalently,
everywhere:
should be irrotational.
Given a conservative vector field
,
can be determined by choosing an origin point
, and a constant
. For each
, a continuous oriented curve
that starts at
and ends at
should be generated.
is assigned:
. The choice of curve
is irrelevant since
is conservative/irrotational.
It can be confirmed that
satisfies
for each point
by evaluating the directional derivative along arbitrary curves that pass through
. For an arbitrary curve
parameterized by
that passes through
at
, the directional derivative of
at
is
. This confirms that
.
Spherical volume integral solutionEdit
Assume that
is
as
for some
. This means that for some
, that there exists
such that
.
This property implies that given a sufficiently large
, that any path integral between any two points outside of the sphere
that remains outside of the sphere is arbitrarily small. Therefore the origin point
can freely shift between points at infinity.
Choose infinity as the origin point and let
. Given an arbitrary point
, choose a direction quantified by unit length vector
. The path
travels backwards along a ray that starts at
and points in the direction given by
.
. Since this integral does not depend on the direction
, the average over all directions
is:
where
is an infinitesimal surface portion of the unit sphere that constrains
.
Letting the position vector variable
be
, the new volume differential is
.
Lastly, letting
gives:
This new formula
is a volume integral that expresses the potential
as a linear combination of functions that exhibit a degree of spherical symmetry. This formula is similar to the inverse square law for the inverse of the divergence.
Next, the above formula will be derived using a Green's function approach in a manner similar to the inverse square law for the divergence operator.
Green's function solutionEdit
This section will derive a formula identical to the formula above using a Green's function approach. While the derivation will be complicated and the result will not be new, the derivation itself will yield many interesting intermediate results. Again, it will be assumed that
is
as
for some
. It will also be assumed that
is continuous.
Vector field
can be expressed as the following linear combination of Dirac delta functions:
.
This linear combination however does not facilitate the inversion of the gradient since for each
, the vector fields
,
, and
are all not conservative, so there do not exist any scalar fields
,
, and
such that
,
, and
.
This prevents a simple solution of
.
It is first necessary to express
as a linear combination of vector fields that are conservative, so that for each component/basis vector field, a scalar field exists where the gradient is the component vector field. In mathematical terms
should be decomposed into the following linear combination:
where for each
, vector fields
,
, and
are all conservative. Specifically a vector valued function
is required that is linear with respect to
:
. In essence,
is effectively a linear combination of
,
, and
where the coefficients are the components of
. The linear combination that
must be decomposed into is:
.
A candidate function that exhibits a degree of spherical symmetry has the form: