Posts Tagged ‘Stokes’ Theorem’

Monday Math 126

July 12, 2010

From the generalized Stokes’ theorem, which generalizes the fundamental theorem of calculus to higher dimensional differential forms on manifolds, one may derive a number of useful theorems of vector calculus, such as the gradient theorem, Kelvin-Stokes theorem (also frequently known as “Stokes’ theorem” or the “curl theorem”), the divergence theorem, and Green’s Theorem. One may also derive from it the formula for vector integration by parts: for a region Ω of with piecewise smooth boundary Γ, with outward surface normal , then for scalar function φ(r) and vector function v(r), then one has
,
or, rearranging,
,
or
.

Using the second form, and letting φ=1, we get
,
the divergence theorem.

Letting our vector field be the gradient of a scalar function, , in the first form, we obtain
,
which is Green’s first identity, often written as
,
and usually used in three dimensions:
,
Exchanging φ and ψ,
,
and subtracting this from the previous, the dot product of gradients terms cancel, giving Green’s second identity:
.
Taking Green’s first identity in the form
,
and setting ψ=φ, we get
.

Letting in the first form, we see
,
since the curl of a vector field always has zero divergence.

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Physics Friday 122

June 4, 2010

Part 2: The Vector Potential and Gauge Transformation.

In electrostatics, one often works not with the electric field, but the electric potential. Since the electric field is irrotational in electrostatics ( for all space), it must be the gradient of some scalar function. We choose the negative of this function as the potential, so that ; note that the potential is unique only up to a constant.

For magnetostatics, we have


,

Here, we use the first equation; since the magnetic field is divergence-free, it must be the curl of some vector field, which we call the vector potential:
. In fact, last week, we saw that , already showing that the magnetic field is the curl of a vector field, specifically .
However, just as the scalar electric potential may be transformed by addition of an arbitrary constant without changing the physics, the vector potential is not unique. Noting that the curl of the gradient of any scalar function is identically zero, adding the gradient of an arbitrary scalar field to the vector potential produces the same physics; this transformation is called a gauge transformation. Thus, our general form for the vector potential is
.

Substituting the definition of A into the equation for the curl of the field, we get
,
and using the identity for curl of a curl, we see
.
Thanks to the freedom of gauge transformation, we can choose a scalar φ so that . This is known as the Coulomb gauge. With this choice, we see the vector potential satisfies the (vector) Poisson equation .
Taking the divergence of
,
we see that the Coulomb gauge reduces to , since the integral term has zero divergence. For an unbounded space and the absence of sources at infinity, this reduces to a constant φ, which in turn gives us
.

Also of note, consider an orientable surface S bounded by a simple closed curve C, oriented via the right-hand rule. Then the magnetic flux through S is ; in terms of the vector potential, this is
;
and by Stokes’ theorem, the latter integral can be turned into an integral over the bounding curve C:
;
and the magnetic flux through a simple loop is equal to the line integral of the vector potential over the loop.