Challenge Problem Set # 5: Generalized Stokes’ Theorem

November 25, 2011

The object of this problem set is to tie together all of the “different” versions of the fundamental theorem of calculus in higher dimensions, e.g., Green’s Theorem, the Divergence Theorem, and (the book’s) Stokes’ Theorem. As we shall see, these are nothing more than special cases of the full Stokes’ Theorem. For our purposes it will suffice to stay in R2 or R3. Before we can state Stokes’ theorem in general we need an understanding of the exterior derivative d and 1, 2, and 3-forms.

Recall that the differential of a differentiable function f : R3 → R is the quantity: ∂f ∂f ∂f df = dx + dy + dz. (0.1) ∂x ∂y ∂z If f instead is a function f : R2 → R, then its differential is merely ∂f ∂f df = dx + dy (0.2) ∂x ∂y as f = f(x, y) does not depend on z. Now (0.1) and (0.2) are examples of (differential) 1-forms on R3, respectively R2. Definition 0.1. An expression of the form

P dx + Q dy + R dz, where P, Q, R : R3 → R is called a 1-form on R3.

1 Problem 1. What is a 1-form on R2? Now these forms can be generalized to 2 and 3 differentials, called 2 and 3-forms. Definition 0.2. An expression of the form P dxdy + Q dxdz + R dydz, where P, Q, R : R3 → R, is called a 2-form. An expression of the form P dxdydz where P : R3 → R is called a 3-form.

Problem 2. What is a 2-form on R2. Explain why there are no 3-forms on R2. Now we have left to define the differential, or exterior derivative, of a form. The way that we do this is by the following: Take, for example, the 1-form P dx + Q dy + R dz. Then the exterior derivative of this form is:

d(P dx + Q dy + R dz) = d(P ) dx + d(Q) dy + d(R) dz where d(P ), d(Q), d(R) are the plain-old differentials, as defined above. Therefore we have d(P dx + Q dy + R dz) = (Px dx + Py dy + Pz dz)dx + (Qx dx + Qy dy + Qz dz)dy

+(Rx dx + Ry dy + Rz dz) dz Using the algebra of differentials (dxdx = dydy = dzdz = 0, dydx = −dxdy, etc) , the above reduces to the 2-form

(Qx − Py)dxdy + (Rx − Pz)dxdz + (Ry − Qz)dydz

Problem 3. Compute the exterior derivative of the following forms: (a) P (x, y) dx + Q(x, y) dy. (b) P (x, y) dxdy (c) P (x, y, z) dxdy + Q(x, y, z) dzdx + R(x, y, z) dydz (d) P (x, y, z) dxdydz. We now are in position to state a generalized Stokes’ theorem.

2 Theorem 0.1 (Stokes’ Theorem (general)). Let U be a subset of R3 of di- mension k ≥ 2. Then Z Z dω = ω U ∂U for all k − 1 forms ω. Here ∂U is the boundary U and d denotes the external derivative.

Let us see, at least formally, the power of this theorem. Problem 4. Locate in your text and state the following theorems in terms of forms: (a) Green’s theorem. (b) Divergence theorem. (c) Stokes’ theorem. Problem 5. With a brief understanding of dimensions, e.g. solids have dimension 3, surfaces have dimension 2 and curves have dimension 1, prove (a), (b), (c) above using the generalized Stokes’ theorem.

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