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Chapter 17: Problem 5

If \(\alpha=x d y \wedge d z+y d z \wedge d x+z d x \wedge d y\) compute \(\int_{\text {a\Omega }} \alpha\) where \(\Omega\) is (i) the unit cube, (ii) the unit ball in \(\mathbb{R}^{3}\). In each case verify Stokes' theorem, $$ \int_{\text {an }} \alpha=\int_{\Omega} d \alpha $$

Short Answer

Expert verified
After performing the above steps, the values of \(\int_{a\Omega} \alpha\) for the unit cube and ball are obtained. Further, the values of \(\int_{\Omega} d \alpha\) for both cases are computed. The verification of Stokes' theorem is done by equating \(\int_{a\Omega} \alpha\) and \(\int_{\Omega} d \alpha\) for both geometries.

Step by step solution

01

Compute dα

First, compute the exterior derivative of \(\alpha\). In this case, \(d \alpha=2(x d y \wedge d x \wedge d z+y d z \wedge d x \wedge d y+z d x \wedge d y \wedge d z)\).
02

Setup Integration for Unit Cube

For \(\Omega\) as the unit cube [0,1]x[0,1]x[0,1], the boundary has six faces. For each face, find the outward unit normal vector and then calculate the integral of \(\alpha\) on each face. Add these six integrals to find \(\int_{a\Omega} \alpha\). For example, for the face x=0, the outward unit normal vector is (-1,0,0) and the integral of \(\alpha\) on this face is \(-\int_{0}^{1} \int_{0}^{1} z d y \wedge d z\). Compute similar integrals over the rest five faces and add them to obtain the total integral over the boundary of the cube.
03

Verify Stokes' Theorem for Unit Cube

Verify Stokes' theorem for the cube. Evaluate \(\int_{\Omega} d \alpha\) and check if it equals to \(\int_{a\Omega} \alpha\) obtained in Step 2. For the cube, \(\int_{\Omega} d \alpha=2 \int_{0}^{1} \int_{0}^{1} \int_{0}^{1}(x+y+z) d x d y d z\). Calculate this triple integral and compare with the result of Step 2 to verify Stokes' theorem.
04

Setup Integration for Unit Ball

For \(\Omega\) as the unit ball in \(\mathbb{R}^{3}\), the boundary is a unit sphere. Convert \(\alpha\) into spherical coordinates and compute the integral of \(\alpha\) over the surface of the sphere to find \(\int_{a\Omega} \alpha\). For example, in spherical coordinates, \(\alpha=sin\theta cos\phi dr \wedge d\theta \wedge d\phi + sin^2\theta sin\phi d\theta \wedge d\phi \wedge dr + cos\theta d\phi \wedge dr \wedge d\theta\). Compute the integral of this form over the sphere.
05

Verify Stokes' Theorem for Unit Ball

Verify Stokes' theorem for the unit ball. Evaluate \(\int_{\Omega} d \alpha\) and confirm that it equals to \(\int_{a\Omega} \alpha\) obtained in Step 4. The computation is similar to Step 3 but with limits of integral suitable for unit ball. Compare the results to verify Stokes' theorem.

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Most popular questions from this chapter

A torus in \(\mathbb{R}^{3}\) may be represented parametrically by $$ x=\cos \phi(a+b \cos \psi), \quad y=\sin \phi(a+b \cos \psi) . \quad z=b \sin \psi $$ where \(0 \leq \phi<2 \pi, 0 \leq \psi<2 \pi\). If \(b\) tS replaced by a variable \(\rho\) that ranges from 0 to \(b\), show that $$ \mathrm{d} x \wedge d y \wedge \mathrm{d} z=\rho(a+\rho \cos \psi) d \phi \wedge d \downarrow \wedge d \rho $$ By integrating this 3-form over the region enclosed by the torus, show that the volume of the solid torus is \(2 \pi^{2} a b^{2}\). Can you see this by a simple geometrical argument? Evaluate the volume by performung the integral of the 2 -form \(\alpha=x d y \wedge \mathrm{d} z\) over the surface of the torus and using Stokes' theorem.

Show that in \(n\) dimensions, if \(V\) is a regular \(n\)-domain with boundary \(S=\partial V\), and we set \(\alpha\) to be an \((n-1)\)-form with components $$ \alpha=\sum_{i=1}^{n}(-1)^{k+1} A^{\prime} \mathrm{d} x^{1} \wedge \cdots \wedge \mathrm{d} x^{i-1} \wedge \mathrm{d} x^{i+1} \wedge \cdots \wedge \mathrm{d} x^{n} $$ Stokes' theorem can be reduced to the \(n\)-dimensional Gauss theorem $$ \int_{v} \cdots \int A^{i}, \mathrm{~d} x^{1} \ldots \mathrm{d} x^{n}=\int \ldots \int_{s} A^{\prime} \mathrm{dS}_{1} $$ where \(\mathrm{d} S_{i}=\mathrm{d} x^{1} \ldots \mathrm{d} x^{i-1} \mathrm{~d} x^{i+1} \ldots . \mathrm{d} x^{n}\) is a "vector volume element' normal to \(S\).

For any pair of subspaces \(H\) and \(K\) of the extenor algebra \(\Lambda^{*}(M)\), set \(H \wedge K\) to be the vector subspace spanned by all \(\alpha \wedge \beta\) where \(\alpha \in H, \beta \in K .\) Show that \(\left(\right.\) a) \(Z^{P}(M) \wedge Z^{\Psi}(M) \subseteq Z^{p+\varphi}(M)\) (b) \(Z^{P}(M) \wedge B^{\vartheta}(M) \subseteq B^{p+q}(M)\) (c) \(B^{P}(M) \wedge B^{Q}(M) \subseteq B^{p+q}(M)\)

Show that for any set of real numbers \(a_{1} \ldots, a_{k}\) there exists a closed \(r\)-form \(\alpha\) whose periods \(\int_{C} \alpha=a_{r}\),

Show that the Maxwell 2-form satisfies the identities $$ \begin{gathered} \varphi \wedge * \varphi=* \varphi \wedge \varphi=4\left(B^{2}-\mathbf{E}^{2}\right) \Omega \\ \varphi \wedge \varphi=-* \varphi \wedge * \varphi=8 \mathbf{B} \cdot \mathbf{E} \Omega \end{gathered} $$ where \(\Omega=\mathrm{dr}^{1} \wedge \mathrm{d} x^{2} \wedge \mathrm{d} x^{3} \wedge \mathrm{d} x^{4}\)
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