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 $\mathbb{R}^n$ with piecewise smooth boundary Γ, with outward surface normal $\mathbf{\hat{n}}$ , then for scalar function φ(r) and vector function v(r), then one has
$\int_{\Omega}\mathbf{\nabla}\phi\cdot\mathbf{v}\,d\Omega=\int_{\Gamma}\phi\mathbf{v}\cdot\mathbf{\hat{n}}d\Gamma-\int_{\Omega}\phi\mathbf{\nabla}\cdot\mathbf{v}\,d\Omega$ ,
or, rearranging,
$\int_{\Omega}\phi\mathbf{\nabla}\cdot\mathbf{v}\,d\Omega=\int_{\Gamma}\phi\mathbf{v}\cdot\mathbf{\hat{n}}d\Gamma-\int_{\Omega}\mathbf{\nabla}\phi\cdot\mathbf{v}\,d\Omega$ ,
or
$\int_{\Gamma}\phi\mathbf{v}\cdot\mathbf{\hat{n}}d\Gamma=\int_{\Omega}\phi\mathbf{\nabla}\cdot\mathbf{v}\,d\Omega+\int_{\Omega}\mathbf{\nabla}\phi\cdot\mathbf{v}\,d\Omega$ .

Using the second form, and letting φ=1, we get
$\int_{\Omega}\mathbf{\nabla}\cdot\mathbf{v}\,d\Omega=\int_{\Gamma}\mathbf{v}\cdot\mathbf{\hat{n}}d\Gamma$ ,
the divergence theorem.

Letting our vector field be the gradient of a scalar function, $\mathbf{v}=\mathbf{\nabla}\psi$ , in the first form, we obtain
$\int_{\Omega}\mathbf{\nabla}\phi\cdot\mathbf{\nabla}\psi\,d\Omega=\int_{\Gamma}\phi\mathbf{\nabla}\psi\cdot\mathbf{\hat{n}}d\Gamma-\int_{\Omega}\phi\nabla^2\psi\,d\Omega$ ,
which is Green’s first identity, often written as
$\int_{\Omega}\mathbf{\nabla}\phi\cdot\mathbf{\nabla}\psi+\phi\nabla^2\psi\,d\Omega=\int_{\Gamma}\phi\mathbf{\nabla}\psi\cdot\mathbf{\hat{n}}d\Gamma$ ,
and usually used in three dimensions:
$\int_{V}\mathbf{\nabla}\phi\cdot\mathbf{\nabla}\psi+\phi\nabla^2\psi\,dV=\int_{\delta{V}}\phi\mathbf{\nabla}\psi\cdot\mathbf{\hat{n}}dA$ ,
Exchanging φ and ψ,
$\int_{V}\mathbf{\nabla}\psi\cdot\mathbf{\nabla}\phi+\psi\nabla^2\phi\,dV=\int_{\delta{V}}\psi\mathbf{\nabla}\phi\cdot\mathbf{\hat{n}}dA$ ,
and subtracting this from the previous, the dot product of gradients terms cancel, giving Green’s second identity:
$\int_{V}\phi\nabla^2\psi-\psi\nabla^2\phi\,dV=\int_{\delta{V}}\left[\phi\mathbf{\nabla}\psi-\psi\mathbf{\nabla}\phi\right]\cdot\mathbf{\hat{n}}dA$ .
Taking Green’s first identity in the form
$\int_{V}\phi\nabla^2\psi\,dV=\int_{\delta{V}}\phi\mathbf{\nabla}\psi\cdot\mathbf{\hat{n}}dA-\int_{V}\mathbf{\nabla}\phi\cdot\mathbf{\nabla}\psi\,dV$ ,
and setting ψ=φ, we get
$\int_{V}\phi\nabla^2\phi\,dV=\int_{\delta{V}}\phi\mathbf{\nabla}\phi\cdot{d}\mathbf{A}-\int_{V}\left|\mathbf{\nabla}\phi\right|^2\,dV$ .

Letting $\mathbf{v}=\mathbf{\nabla}\times\mathbf{a}$ in the first form, we see
$\begin{eqnarray}\int_{V}\mathbf{\nabla}\phi\cdot\left(\mathbf{\nabla}\times\mathbf{a})\,dV&=&\int_{\delta{V}}\phi\left(\mathbf{\nabla}\times\mathbf{a}\right)\cdot{d}\mathbf{A}-\int_{V}\phi\mathbf{\nabla}\cdot\left(\mathbf{\nabla}\times\mathbf{a}\right)\,dV\\\int_{V}\mathbf{\nabla}\phi\cdot\left(\mathbf{\nabla}\times\mathbf{a})\,dV&=&\int_{\delta{V}}\phi\left(\mathbf{\nabla}\times\mathbf{a}\right)\cdot{d}\mathbf{A}\end{eqnarray}$ ,
since the curl of a vector field always has zero divergence.

Like this:

Be the first to like this post.

Tags: Calculus, Divergence Theorem, Integration by Parts, Math, Monday Math, Stokes' Theorem

This entry was posted on July 12, 2010 at 12:23 am and is filed under Math/Science. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

5 Responses to “Monday Math 126”

Physics Friday 126 « Twisted One 151's Weblog Says:
July 12, 2010 at 12:25 am | Reply
[...] 151's Weblog Just another WordPress.com weblog « Should I even bother? A Poll Monday Math 126 [...]
Monday Math 127 « Twisted One 151's Weblog Says:
July 19, 2010 at 12:24 am | Reply
[...] 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 [...]
Monday Math 128 « Twisted One 151's Weblog Says:
July 26, 2010 at 12:14 am | Reply
[...] 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 [...]
Monday Math 129 « Twisted One 151's Weblog Says:
August 2, 2010 at 12:16 am | Reply
[...] 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 [...]
Monday Math 135 « Twisted One 151's Weblog Says:
September 13, 2010 at 12:17 am | Reply
[...] , 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 [...]

Twisted One 151's Weblog

Monday Math 126

Like this:

5 Responses to “Monday Math 126”

Leave a Reply Cancel reply

Follow “Twisted One 151's Weblog”