Problem Set 21

1. Easy
Let $$S\subseteq \mathbb R^3$$ be the capped upper half of the unit sphere; that is, $S = \left\{ x^2+y^2+z^2 =1, z\geq 0\right\} \cup \left\{ x^2+y^2\leq 1, z=0\right\}.$ Let $$\mathbf F(x,y,z) = (2x,2y,2z)$$ be a given vector field.
1. Compute $$\displaystyle\iint_S \mathbf F \cdot \hat{\mathbf n}\, dA$$ using the Divergence Theorem.
2. Compute $$\displaystyle\iint_S \mathbf F \cdot \hat{\mathbf n}\, dA$$ by parameterizing the surface explicitly. Check that this agrees with your answer from part (1).

2. Easy
Let $$\mathbf F(x,y,z) = (xy,yz,xz)$$ and $$S = \left\{ x^2+y^2 \leq 1, 0\leq z \leq 1\right\}$$ be the solid cylinder.
1. Directly (without using any theorems) compute the flux of $$\mathbf F$$ through $$\partial S$$ if $$\partial S$$ has the Stokes' orientation relative to $$S$$.
2. Compute the flux of $$\mathbf F$$ through $$S$$ by using the Divergence theorem.

3. Easy
Let $$F(x,y,z)=x^2\mathbf{i} + y^2\mathbf{j} + z^2\mathbf{k}$$, and take $$S$$ to be the surface of the cube $$0\le x,y,z \le 10$$, oriented so that the positive normal points out of the region bounded by $$S$$.
Using the Divergence Theorem, or by any other means, evaluate the surface integral $\iint_S \mathbf{F}\cdot \hat{\mathbf{n}}\, \mathrm dA$.

4. Easy
Assume that $$\mathbf F: \mathbb R^3 \to \mathbb R^3$$ is a $$C^1$$-vector field and can be written as $$\mathbf F = \text{curl} \mathbf G$$ for some $$C^2$$-vector field $$G$$. Show that if $$S$$ is any piecwise smooth surface which bounds a regular region in $$\mathbb R^3$$, then the flux of $$\mathbf F$$ through $$S$$ is zero.

5. Medium
1. Let $$E: \mathbb R^3 \setminus \{(0,0,0)\} \to \mathbb R^3$$ be given by $E(x,y,z) = \frac{k}{(x^2+y^2+z^2)^{3/2}} (x,y,z)$ for some $$k>0$$. Show that $$\text{div }E = 0$$.
2. Let $$E$$ be the vector-field given in part (1), and let $$S_r = \{ x^2+y^2+z^2=r^2\}\subseteq \mathbb R^3$$ be the sphere of radius $$r$$. Compute the flux of $$E$$ through $$S_r$$.
3. You should have gotten different answers in part (1) and (2). Explain why this is not a contradiction to the Divergence Theorem applied with vector field $$E$$ and with region $$B$$ being the solid ball in $$\mathbb R^3$$ of radius r.

6. Medium
Consider the cube $C = \left\{ 0 \leq x \leq a, \ 0 \leq y \leq b, \ 0 \leq z \leq c\right\}, \qquad a,b,c > 0.$ and let $$\mathbf F(x,y,z) = \left[ (x-a)yz, x(y-b)z, xy(z-c) \right]$$.
1. Directly compute the flux of $$\mathbf F$$ through $$\partial C$$; that is, compute $$\iint_{\partial C} \mathbf F \cdot \hat{\mathbf n}\ dA$$. [Hint: Use symmetry to reduce this whole computation to a single integral.]
2. Directly compute the integral $$\iiint_C \text{div } \mathbf F dV$$.
3. These two quantities are equal by a theorem. State that theorem.

7. Medium
Evaluate $\iint_S \text{curl}(\mathbf{F})\cdot \hat{\mathbf{n}}\, \mathrm d\mathbf{A}$ where $$\mathbf{F}(x,y,z) = xyz\mathbf{i} + xy\mathbf{j} + x^2yz\mathbf{k}$$ and $$S$$ consists of the top and the four sides (but not the bottom) of the cube with vertices $$(\pm 1,\pm 1,\pm 1)$$, oriented outward.

8. Hard
Let $$\mathbf{F}(\mathbf x) = (F_1(\mathbf x), F_2(\mathbf x), F_3(\mathbf x))$$ be a vector field in $$\mathbb{R}^3$$.
1. For arbitrary $$h>0$$, let $$S_h = \left\{(x,y,z): x^2+y^2+z^2=h^2\right\}$$ be the sphere of radius $$h$$. Parameterize $$S_h$$ by a function $$\mathbf g:[a,b]\times[c,d] \to \mathbb{R}^3$$. Compute $$\frac{\partial {\mathbf g}}{\partial s} \times \frac{\partial{\mathbf g}}{\partial t}$$.
2. Under the assumption that $$h$$ is very small, we can use a first order approximation on the functions $$F_i$$. Write out the linear approximations for $$F_i(\mathbf x)$$ at $$(0,0,0)$$ and evaluate these on the parameterization.
3. Use parts (1) and (2) to determine $$\mathbf F(\mathbf g(t))\cdot \left( \frac{\partial {\mathbf g}}{\partial s} \times \frac{\partial{\mathbf g}}{\partial t}\right)$$. [Ignore terms in order $$h^4$$, or keep track of them by writing $$O(h^4)$$]
4. Compute $\lim_{h\rightarrow 0} \frac{1}{\frac{4}{3}\pi h^3} \iint_{S_h} \mathbf F \cdot \hat{\mathbf n} dS.$ Compare this to the divergence. Conclude that divergence is the infinitesimal flux. [Notice that $$\frac{4}{3} \pi h^3$$ is the volume of the sphere, so we are `normalizing' by the volume in our limit.]

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