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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