Monday Math 126

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.

5 Responses to “Monday Math 126”

1. Physics Friday 126 « Twisted One 151's Weblog Says:

[…] 151's Weblog Just another WordPress.com weblog « Should I even bother? A Poll Monday Math 126 […]

2. Monday Math 127 « Twisted One 151's Weblog Says:

[…] Suppose their are two solutions, φ1 and φ2. We can then define . Hence, . Now, recall that Green’s first identity states that for scalar fields f and g, , where S is the surface bounding the volume V, with outward […]

3. Monday Math 128 « Twisted One 151's Weblog Says:

[…] one-dimensional function f(x), with , the derivative has Fourier transform: . Similarly, we can use vector integration by parts for our multi-dimensional transforms. Working in three dimensions from here: First, one form of the […]

4. Monday Math 129 « Twisted One 151's Weblog Says:

[…] now consider the Fourier transform of the divergence of a vector field in three dimensions. First, vector integration by parts tells us that for vector field v and scalar field ψ . Letting , this means , and since , as […]

5. Monday Math 135 « Twisted One 151's Weblog Says:

[…] , where is the del operator with respect to the primed coordinates. Then, we see . Now, recall vector integration by parts; specifically, for scalar function φ(r) and vector function v(r), then one has on volume V with […]