Chaitanya's Random Pages

August 26, 2010

My Six Favourite Formulas – #2

Filed under: mathematics — ckrao @ 2:21 pm

In this post I would like to say more about the generalised Stokes Theorem:

\displaystyle \int_M d\omega = \int_{\partial M}\omega

Here:

  • M is an oriented smooth n-manifold (a space whose local neighbourhoods look like n-dimensional Euclidean space)
  • \omega is an (n-1)-form with compact support on M (further explanation will follow)
  • \partial M is the boundary of M with induced orientation (sign/orientation matter!)
  • d\omega is the exterior derivative of \omega (further explanation will follow)

This gem of a formula shows a duality between the boundary operator and the exterior derivative, where the two are linked by the integration operation. It relates the algebraic structure of differential forms (integrands) to the topology of manifolds. This duality is understood better by studying homology (of chains) and (de Rham) cohomology.

Apart from this duality, another reason this formula appeals is that through the language of differential forms it generalises many integration formulas:

1. The fundamental theorem of calculus (FTOC):

\displaystyle \int_a^b f(x)\ dx = F(b)-F(a) where F' = f

2. The gradient theorem (FTOC for line integrals):

\displaystyle \int_p^q \mathbf{\nabla} \Phi . d\mathbf{r} = \Phi(q) - \Phi(p)

3. Green’s theorem in the plane:

\displaystyle \oint_C M dx + N dy = \int \int_R \left(\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y}\right)\ dx dy

4. Cauchy’s integral theorem (can be proved from Green’s theorem and Cauchy-Riemann conditions if we assume continuous partial derivatives):

\displaystyle \oint_\gamma f(z)\ dz = 0

5. The curl theorem (Stokes’s theorem), a more general form of the previous two equations:

\displaystyle \int_S \mathbf{\nabla}\times\mathbf{F}. d\mathbf{S} = \oint_{\partial S} \mathbf{F}.d\mathbf{r}

6. The divergence theorem:

\displaystyle \int_V \mathbf{\nabla}.\mathbf{F}\ dV = \int_S \mathbf{F}.d\mathbf{S}

One only needs to recall the general form and 1-6 all drop out as special cases.

About Differential Forms

Élie Cartan was the genius who recognised that differential forms were the right objects to study when generalising the fundamental theorem of calculus. These can be viewed simply as things that are integrated and combine the ideas of smooth functions and multilinear forms (i.e. many-variable real-valued functions that are linear in each variable).

A 0-form is simply a smooth function on a manifold M. For k > 0 a differential k-form assigns to each point p of M an alternating multilinear map on k tangent vectors in the tangent space at p. Think of the tangent space at p as the vector space of directions in which one could pass through p locally. A map is alternating if it evaluates to 0 when two of its variables are the same. The wedge product of k vectors is such an example and it turns out by linearity that a k-form can be written as a linear combination of wedge products of k 1-forms. Alternating forms generalise the calculation of the signed volume of a parallelepiped spanned by k vectors.

For example the 1-form dx associates to any point p a map which assigns to any tangent vector at p its x-coordinate. A 1-form multiplied by a smooth function is also a 1-form.

Higher k-forms can be formed by taking wedge product of two forms. The 2-form dx\wedge dy (also written dxdy) evaluated at a point p maps a pair of tangent vectors at p to the signed area of the projection onto the x-y plane of the parallelogram spanned by the tangent vectors.

In general a k-form may be written as

\displaystyle \alpha = \sum_I f_I\ dx_I

where the multi-index I represents a k-tuple of increasing integers from 1 to n. The notation dx_I is short for

\displaystyle dx_{i_1} \wedge dx_{i_2} \wedge \ldots \wedge dx_{i_k}.

Here are some examples of differential forms on the manifold \mathbb{R}^3. Note that one cannot add a k-form and an l-form if k and l are not equal.

0-form: x - \cos y + xyz

1-form: xdx - yzdy + xdz

2-form: (z-xy)dxdy + z dydz + dzdx

3-form: (xy + 3)dx dy dz

Exterior derivative

The exterior derivative unifies the grad/div/curl operations of vector calculus and generalises the notion of a differential of a function. In fact the exterior derivative of a 0-form (smooth function) f is defined as its differential:

\displaystyle df = \sum_{i=1}^n \frac{\partial f}{\partial x_i}\ dx_i

This is seen as a linear combination of 1-forms dx_i and so df is a 1-form. With a general differential k-form \alpha = \sum_I f_I\ dx_I as given above, its exterior derivative d\alpha is a (k+1)-form that may be defined as

\displaystyle d\alpha = \sum_I df_I \wedge dx_I

The differentiation operator d obeys the product rule and commutativity of partial derivatives implies that d^2 = 0. This is dual to the result that the boundary of a boundary is 0. The wedge product operation is alternating and therefore skew-symmetric, meaning \alpha \wedge \beta = -\beta \wedge \alpha whenever \alpha and \beta are 1-forms.

Integration

Differential forms are integrated by mapping the region of integration on the manifold to open sets in \mathbb{R}^n (via a parametrisation), then performing the integration there as a multi-dimensional integral on chains (linear combinations of simplices). Intuitively, to find \int_M \alpha:

(i) divide M into infinitesimal parallelepipeds (which reduce to segments or parallelograms in lower dimensions)

(ii) for each vertex of the subdivision evaluate \alpha at the k tangent vectors spanning the parallelepiped to obtain an infinitesimal scalar

(iii) sum the infinitesimal scalars over all infinitesimal parallelepipeds in the entire region of integration to obtain a scalar. Take the limit as the largest infinitesimal parallelepiped tends to 0 in volume.

Example: Green’s Theorem

We will show some of the manipulations of differential forms required in establishing Green’s theorem (3., above) from the generalised Stokes theorem. Here \omega is the 1-form Mdx + Ndy in a subset of the plane and

\begin{array}{lcl}d\omega & = & d(Mdx + Ndy)\\ & = & d(Mdx) + d(Ndy)\\& = & dM \wedge dx + dN \wedge dy\\ & = & \left(\frac{\partial M}{\partial x}dx + \frac{\partial M}{\partial y}dy \right) \wedge dx + \left(\frac{\partial N}{\partial x}dx + \frac{\partial N}{\partial y}dy \right) \wedge dy\\ & = & \frac{\partial M}{\partial x}dx\wedge dx + \frac{\partial M}{\partial y}dy \wedge dx + \frac{\partial N}{\partial x}dx\wedge dy + \frac{\partial N}{\partial y}dy\wedge dy\\& = & \frac{\partial M}{\partial y}dy \wedge dx + \frac{\partial N}{\partial x}dx\wedge dy\\ & = & \left(\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} \right) dx \wedge dy\\ & = & \left(\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} \right) dx dy \end{array}

Substituting this into the generalised Stokes theorem gives Green’s theorem.

Apart from Stokes’s formula, the language of differential forms is useful for making the change of variable formula for integration straightforward. More about that in a future post perhaps!

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1 Comment »

  1. […] the Jacobian determinant when changing variables, also used in a previous post here and alluded to here when first explaining differential forms in the generalised Stokes […]

    Pingback by My Six Favourite Formulas – #5 « Chaitanya's Random Pages — October 14, 2010 @ 10:08 am | Reply


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