| 15-a-1 |
a. Line integrals - (i) By parameterizing curves
|
0.475
|
0.699
|
10.968
|
15
|
0.937
|
0.462
|
For $y>0$, let \begin{align*} \textbf{F}(x,y,z)&=(e^{-x}\ln y-z)\textbf{i}+\left(2yz-e^{-x}/y\right)\textbf{j}+(y^2-x)\textbf{k}\mbox{ and}\\ \textbf{G}(x,y,z)&=e^{-x}\ln y\,\textbf{i}+\left(2yz-e^{-x}/y\right)\textbf{j}-x~\textbf{k}. \end{align*}
- Show that the vector function $\textbf{F}$ is a gradient on $\{(x,y,z)|~y>0\}$ by finding an $f$ such that $\nabla f=\textbf{F}$.
- Evaluate the line integral $\displaystyle\int_C\textbf{G}(\textbf{r})\cdot \textbf{dr}$, where $C$ is the curve given by $\textbf{r}(u)=(1+u^2)\textbf{i}+e^u\textbf{j}+(1+u)\textbf{k}$, $u\in[0,1]$.
|
| 15-a-2 |
a. Line integrals - (i) By parameterizing curves
|
0.548
|
0.355
|
4.185
|
12
|
0.629
|
0.081
|
Find the line integral \[ \int_{{\cal C}}\,(2 x \sin(\pi y) - e^{z})\,dx + (\pi x^2 \cos(\pi y) - 3 e^{z})\,dy - x e^z\,dz\] $\mbox{along the curve}\;\;{\cal C} = \left\{(x, y, z) | z = \ln \sqrt{1 + x^2},\; y = x,\; 0 \le x \le 1 \right\}. $
|
| 15-a-3 |
a. Line integrals - (i) By parameterizing curves
|
0.398
|
0.417
|
4.170
|
10
|
0.616
|
0.218
|
Let $C$ be the upper half of the curve $(x^2+y^2)^2=x^2-y^2$. Evaluate $\displaystyle\int_{C}y\,ds$.
圖
|
| 15-a-4 |
a. Line integrals - (i) By parameterizing curves
|
0.432
|
0.746
|
11.570
|
15
|
0.962
|
0.531
|
- Find a scalar function $f(x,y,z)$ such that $\nabla f=\sin y\,{\bf i}+x\cos y\,{\bf j}-\sin z\,{\bf k}$.
- Find the line integral $\displaystyle\int_C\sin y\,dx+x\cos y\,dy+(y-\sin z)dz$, where $C:\ {\bf r}(t)=\left\langle t,\frac{\pi}{2}\cos t,\frac{\pi}{2}\sin t\right\rangle$, $0\leq t\leq \pi$.
|
| 15-a-5 |
a. Line integrals - (i) By parameterizing curves
|
0.605
|
0.439
|
6.280
|
16
|
0.741
|
0.136
|
Let $S$ be a toroidal shell (not a solid torus!) obtained by rotating about the $z$-axis the circle $C$ in the plane $y=0$ of radius $a$ centred at $(b,0,0)$, where $0 < a < b\:$. It can be parametrized by
$ \textbf r(\theta, \alpha) = \left\langle\:(b +a\cos\alpha) \cos\theta , (b +a\cos\alpha) \sin\theta , a\sin\alpha \right\rangle$,
$0 \leq \theta, \alpha \leq 2\pi$.
Suppose that the temperature at each point of the surface is proportional to its distance from the plane $z=0\:$, i.e., the temperature $T(x,y,z)$ for every point $(x,y,z) \in S$ is given by $T(x,y,z) = \lambda |z|$ for some constant $\lambda >0$.
圖
- Find the average temperature along the circle $C$ which is $\frac{\displaystyle\int_C T \ ds}{\displaystyle\int_C 1\ ds}$.
- Find the area of $S$, $A(S)$. Then evaluate the average temperature of the toroidal shell $S$ which is $\displaystyle\frac{\iint_S T\ dS}{A(S)}$.
|
| 15-a-6 |
a. Line integrals - (i) By parameterizing curves
|
0.502
|
0.361
|
3.720
|
10
|
0.612
|
0.109
|
Let $C$ be the curve of intersection of $x^2+y^2+z^2=4$, $x^2+y^2=2x$, $z\geq 0$, oriented $C$ to be counterclockwise when viewed from above. Evaluate $\displaystyle\int_Cy^2dx+z^2dy+x^2dz$.
|
| 15-a-7 |
a. Line integrals - (i) By parameterizing curves
|
0.642
|
0.573
|
7.540
|
13
|
0.894
|
0.252
|
Compute the line integral \[ \int_Cz^2\,\mathrm{d}x-x^2\,\mathrm{d}y+2yz\,\mathrm{d}z, \] where $C$ is the curve of intersection of the upper half sphere $z=\sqrt{4-x^2-y^2}$ and the circular cylinder $x^2+y^2=2y$, orientated counterclockwise viewed from the above, as Figure.
圖
|
| 15-a-8 |
a. Line integrals - (i) By parameterizing curves
|
0.654
|
0.442
|
4.720
|
12
|
0.769
|
0.115
|
Let $C$ be the polar curve defined by $r^2=\cos 2\theta$ in the first quadrant. Evaluate $\displaystyle{\int_Cy\ ds}$.
圖
|
| 15-a-9 |
a. Line integrals - (i) By parameterizing curves
|
0.543
|
0.451
|
4.250
|
9
|
0.722
|
0.179
|
Let $C$ be the closed curve formed by $y=x^2$, where $0\leq x\leq 1$, and $x=y^2$, where $0\leq y\leq 1$. Given $C$ the counterclockwise orientation, evaluate $\displaystyle\int_C xy\,ds$ and $\displaystyle\oint_C xy \,dx$.
|
| 15-a-10 |
a. Line integrals - (ii) By the fundamental theorem for line integrals (conservative vector fields)
|
0.475
|
0.699
|
10.968
|
15
|
0.937
|
0.462
|
For $y>0$, let \begin{align*} \textbf{F}(x,y,z)&=(e^{-x}\ln y-z)\textbf{i}+\left(2yz-e^{-x}/y\right)\textbf{j}+(y^2-x)\textbf{k}\mbox{ and}\\ \textbf{G}(x,y,z)&=e^{-x}\ln y\,\textbf{i}+\left(2yz-e^{-x}/y\right)\textbf{j}-x~\textbf{k}. \end{align*}
- Show that the vector function $\textbf{F}$ is a gradient on $\{(x,y,z)|~y>0\}$ by finding an $f$ such that $\nabla f=\textbf{F}$.
- Evaluate the line integral $\int_C\textbf{G}(\textbf{r})\cdot \textbf{dr}$, where $C$ is the curve given by $\textbf{r}(u)=(1+u^2)\textbf{i}+e^u\textbf{j}+(1+u)\textbf{k}$, $u\in[0,1]$.
|
| 15-a-11 |
a. Line integrals - (ii) By the fundamental theorem for line integrals (conservative vector fields)
|
0.548
|
0.355
|
4.185
|
12
|
0.629
|
0.081
|
Find the line integral \[ \int_{{\cal C}}\,(2 x \sin(\pi y) - e^{z})\,dx + (\pi x^2 \cos(\pi y) - 3 e^{z})\,dy - x e^z\,dz\] $\mbox{along the curve}\;\;{\cal C} = \left\{(x, y, z) | z = \ln \sqrt{1 + x^2},\; y = x,\; 0 \le x \le 1 \right\}. $
|
| 15-a-12 |
a. Line integrals - (ii) By the fundamental theorem for line integrals (conservative vector fields)
|
0.454
|
0.750
|
8.110
|
10
|
0.976
|
0.523
|
Evaluate $\displaystyle\int_C(yze^{xyz}+x)\,dx+xze^{xyz}\,dy+xye^{xyz}\,dz$, where $C$ is the curve ${\bf r}(t)=\left\langle t,\,\cos(\pi t),\,\tan^{-1}t\right\rangle$, $0\leq t\leq 1$.
|
| 15-a-13 |
a. Line integrals - (ii) By the fundamental theorem for line integrals (conservative vector fields)
|
0.490
|
0.584
|
8.450
|
14
|
0.829
|
0.338
|
Consider the vector field
$\textbf F (x,y)=P(x,y)\,\textbf i +Q(x,y)\,\textbf j=\frac{x+3y}{x^2+y^2}\,\textbf i +\frac{-3x+y}{x^2+y^2}\,\textbf j $.
- Show that $\textbf F$ is conservative on the half plane $D=\{(x,y)|x<0\}$.
- Compute $\displaystyle\int_{C_0}\textbf F \cdot d\textbf r$, where $C_0$ is the unit circle $x^2+y^2=1$ oriented counterclockwise. Is $\textbf F$ conservative on $\mathbb{R}^2\backslash \{(0,0)\}$?
- Compute $\displaystyle\int_C \textbf F\cdot d\textbf r$, where $C$ is a piecewise smooth path consisting of the ellipse $\frac{x^2}{4}+y^2=1$ and the triangle formed by the lines $x=-3$, $x+y=-2$, and $y-x=2$. The orientation of $C$ is shown in the figure.
圖
|
| 15-a-14 |
a. Line integrals - (ii) By the fundamental theorem for line integrals (conservative vector fields)
|
0.295
|
0.238
|
4.740
|
20
|
0.385
|
0.091
|
Let $\textbf{F}=\frac{(x-y)^2y}{(x^2+y^2)^2}\textbf{i} +\frac{-(x-y)^2x}{(x^2+y^2)^2}\textbf{j}$.
- Verify that $\textbf F$ is conservative on the right half plane $x>0$. Find a potential function of $\textbf F$ on the right half plane.
- Evaluate $\displaystyle\oint_{C_1}\textbf{F}\cdot d\textbf{r}$ where $C_1$ is the ellipse $\frac{x^2}{4}+(y-2)^2=1$.
- Evaluate $\displaystyle\int_{C_2}\textbf{F}\cdot d\textbf{r}$ where $C_2$ is the curve with polar equation $r=e^{|\theta|}$, $-\frac{9\pi}{4}\leq\theta\leq\frac{9\pi}{4}$.
圖
|
| 15-a-15 |
a. Line integrals - (ii) By the fundamental theorem for line integrals (conservative vector fields)
|
0.454
|
0.752
|
9.800
|
12
|
0.979
|
0.524
|
- Find the potential function of the vector field \[ \mathbf{F}(x,y,z)=\frac{2x(1-\mathrm{e}^y)}{(1+x^2)^2}\,\mathbf{i}+\left(\frac{\mathrm{e}^y}{1+x^2}+(y+1)\mathrm{e}^y\right)\,\mathbf{j}+\mathbf{k}. \]
- Evaluate $\displaystyle\int_C\mathbf{F}\cdot d\mathbf{r}$, where $C$ is any curve from $(0,0,0)$ to $(1,1,1)$.
|
| 15-a-16 |
a. Line integrals - (ii) By the fundamental theorem for line integrals (conservative vector fields)
|
0.577
|
0.560
|
10.180
|
18
|
0.848
|
0.271
|
Consider the vector field defined by $\textbf{G}(x,y)=(3x^2+y)\textbf{i}+(2x^2y-x)\textbf{j}$, $(x,y)\in\mathbb{R}^2$.
- Is $\textbf{G}(x,y)$ conservative?
- Find a function $\mu(x)$ with $\mu(1)=1$ such that $\mu(x)\textbf{G}(x,y)$ is conservative.
- Set $\textbf{F}(x,y)$ to be the conservative vector field in (b). Find the potential function $f(x,y)$ of $\textbf{F}$ with $f(1,0)=3$.
- Let $C$ be the curve with defining equation in polar coordinate given by \[r=\sec\theta+\frac{\sqrt{2}}{\pi}\theta,\ \theta\in\left[0,\frac{\pi}{4}\right].\] Evaluate the integral $\displaystyle\int_C\textbf{F}\cdot d\textbf{r}$.
|
| 15-a-17 |
a. Line integrals - (ii) By the fundamental theorem for line integrals (conservative vector fields)
|
0.456
|
0.548
|
6.870
|
12
|
0.776
|
0.320
|
Let $\textbf{F}(x,y)=\frac{y^3}{(x^2+y^2)^2}\textbf{i}-\frac{xy^2}{(x^2+y^2)^2}\textbf{j}$.
- Show that $\textbf{F}$ is conservative on the domain $D=R^2-\{(0,y)| y\leq 0\}$.
- Compute $\displaystyle\int_C\textbf{F}\cdot d\textbf{r}$, where $C$ is the part of the polar curve $r=1+\sin\theta$, $0\leq \theta\leq \pi$.
|
| 15-a-18 |
a. Line integrals - (ii) By the fundamental theorem for line integrals (conservative vector fields)
|
0.324
|
0.828
|
11.390
|
13
|
0.990
|
0.666
|
- Find the value $\lambda$ such that the vector field
$\textbf{F}=(x^2+4xy^\lambda)\textbf{i}+(6x^{\lambda-1}y^2-2y)\textbf{j}$ is conservative.
- For this $\lambda$, find a potential function of $\textbf{F}$.
- For $\lambda$ in (a), evaluate $\displaystyle\int_C\textbf{F}\cdot \mathrm{d}\textbf{r}$, where $C$ is the path described by $\frac{x^2}{9}+(y-1)^2=1$ counterclockwise from $(0,0)$ to $(3,1)$.
|
| 15-a-19 |
a. Line integrals - (ii) By the fundamental theorem for line integrals (conservative vector fields)
|
0.468
|
0.673
|
8.220
|
12
|
0.907
|
0.438
|
Let $\displaystyle\mathbf{F}(x,y)=\frac{x+y}{x^2+y^2}\,\mathbf{i}+\frac{-x+y}{x^2+y^2}\,\mathbf{j}$.
- Is $\mathbf{F}(x,y)$ conservative on the half plane $D=\{(x,y)|x>0\}$?
- Evaluate the line integral $\displaystyle\int_{C_1}\mathbf{F}\cdot\mathrm{d}\mathbf{r}$, where $C_1$ is the part of the parabola $y=(x-2)^2$ from $(2,0)$ to $(4,4)$.
- Evaluate the line integral $\displaystyle\int_{C_2}\mathbf{F}\cdot\mathrm{d}\mathbf{r}$, where $C_2$ is the unit circle $x^2+y^2=1$ oriented counterclockwise. Is $\mathbf{F}(x,y)$ conservative on $\mathbb{R}^2-\{(0,0)\}$?
|
| 15-a-20 |
a. Line integrals - (ii) By the fundamental theorem for line integrals (conservative vector fields)
|
0.163
|
0.889
|
10.120
|
11
|
0.970
|
0.807
|
Let $\textbf{F}=z\cos(xz)\textbf{i}+ze^{yz}\textbf{j} +(x\cos(xz)+ye^{yz})\textbf{k}$.
- Find a scalar function $\varphi(x,y,z)$ such that $\nabla \varphi=\textbf{F}$.
- Evaluate $\displaystyle\int_C\textbf{F}\cdot d\textbf{r}$, where $C$ is the curve ${\bf r}(t)=(\cos (\pi t^2), \ln (t+1) ,\tan^{-1}(t)), 0\leq t\leq 1.$
|
| 15-a-21 |
a. Line integrals - (ii) By the fundamental theorem for line integrals (conservative vector fields)
|
0.573
|
0.539
|
8.800
|
16
|
0.825
|
0.252
|
Let $\textbf F(x,y)=P(x,y)\ \textbf i+Q(x,y)\ \textbf j$, where $P(x,y)=\frac{xy}{\sqrt{x^2+y^2}}$, $Q(x,y)=\frac{x^2+2y^2}{\sqrt{x^2+y^2}}$.
- Compute $\frac{\partial P}{\partial y}$ and $\frac{\partial Q}{\partial x}$. Is $\textbf F$ conservative on the right half plane $D=\{(x,y)|x>0\}$? Justify your answer.
- Compute $\displaystyle\int_C\textbf F\cdot d\textbf r $, where $C$ is any curve in the right half plane $D$ from $(1,1)$ to $(2,2)$.
- Compute $\displaystyle\oint_C\textbf F\cdot d\textbf r$, where $C$ is a positively oriented circle centered at $(0,0)$ with radius $r>0$.
- Compute $\displaystyle\oint_C \textbf F{\bf\cdot} d\textbf r $, where $C$ is any positively oriented simple closed curve, $C\subset \mathbb{R}^2\backslash\{(0,0)\}$.
(Hint: You need to discuss whether $C$ encloses $(0,0)$ or not.)
- Is $\textbf F$ conservative on $\mathbb{R}^2\backslash\{(0,0)\}$? Justify your answer.
|
| 15-a-22 |
a. Line integrals - (iii) By Green's theorem
|
0.521
|
0.718
|
11.836
|
15
|
0.978
|
0.458
|
Let $C$ be a piecewise-smooth Jordan curve that does not pass through the origin.
Evaluate $\displaystyle\oint_C \frac{-y^5}{(x^2+y^2)^3} dx + \frac{xy^4}{(x^2+y^2)^3} dy $ for the following two cases, where $C$ is traversed in the counterclockwise direction.
- $C$ does not enclose the origin.
- $C$ does enclose the origin.
|
| 15-a--23 |
a. Line integrals - (iii) By Green's theorem
|
0.449
|
0.676
|
7.173
|
10
|
0.901
|
0.452
|
Evaluate $\displaystyle\oint_{r=1-\cos\theta}(x^2y+y)dx-(xy^2-x)dy$ with the curve oriented counterclockwise.
|
| 15-a-24 |
a. Line integrals - (iii) By Green's theorem
|
0.672
|
0.502
|
5.510
|
10
|
0.838
|
0.167
|
Let the vector field ${\bf F}(x,y)=\frac{x^2y}{(x^2+y^2)^2}\,{\bf i}-\frac{x^3}{(x^2+y^2)^2}\,{\bf j}$, $C_1$ be the curve $|x|+|y|=1$ and $C_2$ be the curve $x^2+(y-2)^2=1$. Find $\displaystyle\oint_{C_1}{\bf F}\cdot d{\bf r}$ and $\displaystyle\oint_{C_2}{\bf F}\cdot d{\bf r}$.
|
| 15-a-25 |
a. Line integrals - (iii) By Green's theorem
|
0.272
|
0.860
|
13.650
|
15
|
0.996
|
0.724
|
Let $D$ be the bounded region in the first quadrant enclosed by $y=0$, $x=1$, and $y=\sqrt{x}$ with positively oriented boundary $C$ (i.e. counter clockwise.). Evaluate
\begin{align*}\oint_{C}\left[9x^2y(x^3+1)^{\frac{1}{2}}-xy^2(x^3+1)^{\frac{3}{2}}\right]dx \\ +\left[2(x^3+1)^{\frac{3}{2}}+2(y^3+1)^{\frac{3}{2}}\right]dy.\end{align*}
|
| 15-a-26 |
a. Line integrals - (iii) By Green's theorem
|
0.586
|
0.602
|
8.790
|
14
|
0.894
|
0.309
|
- Evaluate the line integral $\displaystyle I_1=\int_{C_1}(-\sin x+\mathrm{e}^y)\,\mathrm{d}x+(2x+x\mathrm{e}^y)\,\mathrm{d}y$, where $C_1$ is the line segment from $(0,0)$ to $(2,2)$.
圖
- Find the area of the region between $C_1$ and $C_2$, where $C_2$ is a curve from $(0,0)$ to $(2,2)$ parameterized by \[ \mathbf{r}(t)=\frac2\pi(t-\sin t)\,\mathbf{i}+(1-\cos t)\,\mathbf{j},\quad 0\leq t\leq \pi. \] (c) Evaluate the line integral $\displaystyle I_2=\int_{C_2}(-\sin x+\mathrm{e}^y)\,\mathrm{d}x+(2x+x\mathrm{e}^y)\,\mathrm{d}y$.
|
| 15-a-27 |
a. Line integrals - (iii) By Green's theorem
|
0.430
|
0.544
|
6.420
|
12
|
0.759
|
0.329
|
Determine whether the statement is true of false. Fill $\textbf T$ (true) or $\textbf F$ (false) in the blanks. If the statement is false, write down a reason, or give a correct statement, or find a counterexample.
- Let $\mathbf{F}(x,y)=P(x,y)\,\mathbf{i}+Q(x,y)\,\mathbf{j}$. If $\displaystyle\frac{\partial Q}{\partial x}=\frac{\partial P}{\partial y}$ throughout the domain of $\mathbf{F}(x,y)$, then the line integrals of $\mathbf{F}(x,y)$ is independent of path on the domain.
- Let $f(x,y)$ be a smooth function. Suppose that a smooth curve $C$ gives an orientation from initial point $p$ to terminal point $q$. If $-C$ denotes the curve consisting of the same points as $C$ but with the opposite orientation (from initial point $q$ to terminal point $p$), then $\displaystyle\int_{-C}f(x,y)\,\mathrm{d}s=-\int_Cf(x,y)\,\mathrm{d}s$.
- For a unit circle $C: x^2+y^2=1$, we have $\displaystyle\oint_Cx\,\mathrm{d}y=0$ by symmetry.
- Any smooth function $f(x,y,z)$ satisfies $\mathrm{div}(\nabla f)=0$.
|
| 15-a-28 |
a. Line integrals - (iii) By Green's theorem |
0.623
|
0.620
|
7.700
|
12
|
0.931
|
0.308
|
Find the value $k\in\mathbb{R}$ such that the line integral \[ I(k)=\int_{C_k}(1+y^2+y\,\mathrm{e}^{xy})\,\mathrm{d}x+(2x+y+x\,\mathrm{e}^{xy})\,\mathrm{d}y \] achieves the minimum value, where $C_k$ is the curve $y=k\sin x$ from $(0,0)$ to $(\pi,0)$.
|
| 15-a-29 |
a. Line integrals - (iii) By Green's theorem
|
0.392
|
0.738
|
9.350
|
12
|
0.934
|
0.542
|
Evaluate $\displaystyle\oint_C(x^2y^2+y)dx-(2xy^3-3x)dy$, where $C$ is described by the polar equation $r=1-\cos\theta$ oriented counterclockwise.
|
| 15-a-30 |
a. Line integrals - (iii) By Green's theorem
|
0.506
|
0.626
|
7.820
|
12
|
0.879
|
0.373
|
Evaluate the line integral $\displaystyle\int_C\sin \pi x\ dx+(e^{y^2}+x^2)dy$ along the following choices of the curve $C$.
圖
- $C=C_0$ is the line segment from $(-1,0)$ to $(0,0)$.
- $C=C_1\cup C_2$, where $C_1$ is the polar curve $r=2\sin\theta$, $0\leq\theta\leq\frac{\pi}{2}$ and $C_2$ is the cardioid $r=1+\sin\theta$, $\frac{\pi}{2}\leq\theta\leq \pi$.
|
| 15-a-31 |
a. Line integrals - (iii) By Green's theorem
|
0.548
|
0.567
|
8.640
|
15
|
0.841
|
0.293
|
Evaluate the line integral $\displaystyle\int_C\left(-x-y+\frac{y^2}{2}\right)dx+(x+2xy+3)dy$, where $C$ consists of the arc $C_1$ of the quarter circle $x^2+y^2=1, x\geq 0, y\leq 0$, from $(0,-1)$ to $(1,0)$ followed by the arc $C_2$ of the quarter ellipse $4x^2+y^2=4$, $x\geq 0$, $y\geq 0$, from $(1,0)$ to $(0,2)$. (Hint: You may use Green's Theorem, but note that $C$ is not closed.)
|
| 15-b-1 |
b. Surface integrals
|
0.660
|
0.544
|
8.670
|
15
|
0.874
|
0.214
|
- Find the area of the part of the sphere $x^2+y^2+z^2=2$ that lies above the plane $z=1$.
- Let the canopy be the part of the upper hemisphere $x^2+y^2+z^2=2$ that lies above the square $-1\leq x\leq 1$, $-1\leq y\leq 1$, and let $C$ be the boundary of canopy oriented counterclockwise when viewed from above. Evaluate $\displaystyle\oint_C{\bf F}\cdot d{\bf r}$, where ${\bf F}(x,y,z)=(xz+\tan x^2)\,{\bf i}+(\sin x\cos y+e^{y^2})\,{\bf j}+\left(-\frac{y^2}{2}+\sin\sqrt{z}\right)\,{\bf k}$.
圖
|
| 15-b-2 |
b. Surface integrals - (i) By parameterizing surfaces
|
0.616
|
0.538
|
8.250
|
15
|
0.846
|
0.230
|
Let $S$ be the triangular region with vertices $(0,0,0)$, $(a,0,0)$, and $(a,a,a)$, $a>0$, with upward unit normal $\textbf{n}$, and $C$ be the positively oriented boundary of $S$. Let \[\textbf{F}=\left(y-z\cos(x^2)\right)\textbf{i} +\left(2x-\sin(z^2)\right)\textbf{j} +\left(3z-\tan(y^2)\right)\textbf{k}.\]
- Find a parametrization of $S$ and find the upward unit normal $\textbf{n}$. (Hint. Consider the projection of $S$ to $xy$-plane.)
- Evaluate $\nabla\times\textbf{F}$.
- Evaluate $\displaystyle\oint_{C}\textbf{F}\cdot \textbf{dr}$.
|
| 15-b-3 |
b. Surface integrals - (i) By parameterizing surfaces
|
0.726
|
0.512
|
7.181
|
15
|
0.875
|
0.149
|
Let $S_1$ be the surface $\{(x,y,z)|~z=x^2+y^2,~z\leq y\}$, $S_2$ be the surface $\{(x,y,z)|~z=y,~x^2+y^2\leq z\}$, and $\textbf{V}(x,y,z)=-y\textbf{i}+x\textbf{j}+z\textbf{k}$.
- Compute directly the downward flux of $\textbf{V}$ across $S_1$.
- Use the divergence theorem to compute the upward flux of $\textbf{V}$ across $S_2$.
|
| 15-b-4 |
b. Surface integrals - (i) By parameterizing surfaces
|
0.592
|
0.663
|
8.576
|
12
|
0.959
|
0.367
|
Evaluate the surface integral $\displaystyle\iint\limits_{S}(x^2+y^2)zd\sigma$, where $S$ is the part of the plane $z=4+x+y$ that lies inside the cylinder $x^2+y^2=4$.
|
| 15-b-5 |
b. Surface integrals - (i) By parameterizing surfaces
|
0.492
|
0.315
|
4.070
|
15
|
0.561
|
0.069
|
Let ${\bf F}(x,y,z)=\frac{1}{\left(x^2+y^2+z^2\right)^{3/2}}(x\,{\bf i}+y\,{\bf j}+z\,{\bf k})$.
- Let $S_1$ be the part of the sphere $x^2+y^2+z^2=1$ in the first octant bounded by the planes $y=0$, $y=\sqrt{3}x$ and $z=0$, oriented upward. Find the flux of ${\bf F}$ across $S_1$.
- Let $S_2$ be the part of the surface $\frac{x^2}{4}+\frac{y^2}{9}+\frac{z^2}{16}=1$ in the first octant bounded by the planes $y=0$, $y=\sqrt{3}x$ and $z=0$, oriented upward. Find the flux of ${\bf F}$ across $S_2$.
圖
|
| 15-b-6 |
b. Surface integrals - (i) By parameterizing surfaces
|
0.493
|
0.720
|
7.620
|
10
|
0.966
|
0.473
|
Find the area of the surface $\{ x^2 + y^2 + z^2 = 4,\,1 \le x^2+y^2 \le 3,\,z\geq 0 \}$.
|
| 15-b-7 |
b. Surface integrals - (i) By parameterizing surfaces
|
0.605
|
0.439
|
6.280
|
16
|
0.741
|
0.136
|
Let $S$ be a toroidal shell (not a solid torus!) obtained by rotating about the $z$-axis the circle $C$ in the plane $y=0$ of radius $a$ centred at $(b,0,0)$, where $0 < a < b\:$. It can be parametrized by
$ \textbf r(\theta, \alpha) = \left\langle\:(b +a\cos\alpha) \cos\theta , (b +a\cos\alpha) \sin\theta , a\sin\alpha \right\rangle$,
$0 \leq \theta, \alpha \leq 2\pi$.
Suppose that the temperature at each point of the surface is proportional to its distance from the plane $z=0\:$, i.e., the temperature $T(x,y,z)$ for every point $(x,y,z) \in S$ is given by $T(x,y,z) = \lambda |z|$ for some constant $\lambda >0$.
圖
- Find the average temperature along the circle $C$ which is $\frac{\displaystyle\int_C T \ ds}{\displaystyle\int_C 1\ ds}$.
- Find the area of $S$, $A(S)$. Then evaluate the average temperature of the toroidal shell $S$ which is $\displaystyle\frac{\iint_S T\ dS}{A(S)}$.
|
| 15-b-8 |
b. Surface integrals - (i) By parameterizing surfaces
|
0.726
|
0.403
|
3.670
|
10
|
0.766
|
0.040
|
Let $S$ be the surface $x^2+y^2+z^2=a^2$, $x\geq 0$, $y\geq 0$, $z\geq 0$ ($a>0$), and let $C$ be the boundary of $S$. Find the centroid of $C$.
|
| 15-b-9 |
b. Surface integrals - (i) By parameterizing surfaces
|
0.625
|
0.451
|
4.190
|
10
|
0.763
|
0.138
|
Evaluate $\displaystyle\iint_SxdS$ where $S$ is the part of the cone $z=\sqrt{2(x^2+y^2)}$ that lies below the plane $z=1+x$.
|
| 15-b-10 |
b. Surface integrals - (i) By parameterizing surfaces
|
0.621
|
0.637
|
6.980
|
10
|
0.947
|
0.326
|
Evaluate the surface integral $\displaystyle\iint\limits_S(x^2+y^2)dS$, where $S$ is the surface $z=\sqrt{x^2+y^2}$ with $0\leq z\leq 1$.
|
| 15-b-11 |
b. Surface integrals - (i) By parameterizing surfaces
|
0.677
|
0.393
|
3.470
|
10
|
0.732
|
0.054
|
Let $S$ be a cone has radius $a$ and height $h$ without base. Evaluate the integral of the distance of the points to its axis over $S$.
|
| 15-b-12 |
b. Surface integrals - (i) By parameterizing surfaces
|
0.583
|
0.499
|
4.890
|
10
|
0.790
|
0.207
|
Compute the surface integral \[ \iint_S xz\,\mathrm{d}S, \] where $S$ is the part of the cone $z=\sqrt{x^2+y^2}$ inside the circular cylinder $x^2+y^2=2x$.
|
| 15-b-13 |
b. Surface integrals - (i) By parameterizing surfaces
|
0.623
|
0.646
|
8.190
|
12
|
0.958
|
0.335
|
Evaluate the surface integral $\displaystyle\iint_S\sqrt{x^2+y^2}\,\mathrm{d}S$, where $S$ is the part of the surface $\displaystyle z=\tan^{-1}\left(\frac{y}x\right)$ inside the circular cylinder $x^2+y^2=1$ and in the first octant.
|
| 15-b-14 |
b. Surface integrals - (i) By parameterizing surfaces
|
0.573
|
0.455
|
5.600
|
12
|
0.741
|
0.169
|
Find the area of the sphere $x^2+y^2+z^2=4$ lying inside the cylinder $(x-1)^2+y^2=1$.
|
| 15-b-15 |
b. Surface integrals - (i) By parameterizing surfaces
|
0.480
|
0.563
|
6.060
|
10
|
0.803
|
0.323
|
Find the area of the part of the surface $x^2+y^2+z^2=1$ that lies within the cylinder $x^2+y^2+x=0$ and above $z=0$.
|
| 15-b-16 |
b. Surface integrals - (ii) By Stokes' theorem
|
0.616
|
0.538
|
8.250
|
15
|
0.846
|
0.230
|
Let $S$ be the triangular region with vertices $(0,0,0)$, $(a,0,0)$, and $(a,a,a)$, $a>0$, with upward unit normal $\textbf{n}$, and $C$ be the positively oriented boundary of $S$. Let \[\textbf{F}=\left(y-z\cos(x^2)\right)\textbf{i} +\left(2x-\sin(z^2)\right)\textbf{j} +\left(3z-\tan(y^2)\right)\textbf{k}.\]
- Find a parametrization of $S$ and find the upward unit normal $\textbf{n}$. (Hint. Consider the projection of $S$ to $xy$-plane.)
- Evaluate $\nabla\times\textbf{F}$.
- Evaluate $\displaystyle\oint_{C}\textbf{F}\cdot \textbf{dr}$.
|
| 15-b-17 |
b. Surface integrals - (ii) By Stokes' theorem
|
0.654
|
0.606
|
7.711
|
12
|
0.933
|
0.279
|
Let $\textbf{V}=(2x-y)\textbf{i}+(2y+z)\textbf{j}+x^2y^2z^2\textbf{k}$ and let $S$ be the upper half of the ellipsoid $\frac{x^2}{4}+\frac{y^2}{9}+\frac{z^2}{25}=1$. Find the flux of curl$\textbf{V}$ in the direction of the upper unit normal $\textbf{n}$ (pointing away from the origin.).
|
| 15-b-18 |
b. Surface integrals - (ii) By Stokes' theorem
|
0.727
|
0.603
|
9.790
|
15
|
0.967
|
0.239
|
Let $S$ be the part of the sphere $x^2+y^2+(z-2)^2=8$ that lies above the $xy$-plane and that has outward normal (i.e. with ${\bf k}$-component $\geq 0$). Let ${\bf F}(x,y,z)=\left\langle -y^3\cos xz,\,x^3e^{yz},\,-e^{xyz}\right\rangle$. Find $\displaystyle\iint_S\mbox{curl}\,{\bf F}\cdot d{\bf S}$.
|
| 15-b-19 |
b. Surface integrals - (ii) By Stokes' theorem
|
0.613
|
0.554
|
6.810
|
12
|
0.860
|
0.247
|
Let $f(x,y,z)=x + x y + y z + z x$, and $g(x,y,z)=x+2y+3z$.
- Show by direct calculation that $\mbox{curl}\,(f\,\nabla g)\,=\,\nabla f\,\times\,\nabla g$.
- Find $\displaystyle\iint_{S}\,(\nabla f\,\times\,\nabla g)\cdot\,\textbf n\,dS$ where $S$ is the surface $z=\sqrt{4 - x^2 - y^2}$, and $\textbf n$ is the unit normal on $S$ pointing upwards.
|
| 15-b-20 |
b. Surface integrals - (ii) By Stokes' theorem
|
0.074
|
0.424
|
2.050
|
5
|
0.461
|
0.387
|
Determine the statement is true ($\bigcirc$) or false ($\times$).
- If $f(x,y)$ is continuous on the rectangle $R=\{(x,y)|\ a\leq x\leq b,\ c\leq y\leq d\}$ except for finitely many points, then $f(x,y)$ is integrable on $R$ and \[\iint_Rf(x,y)dA=\int_c^d\int_a^bf(x,y)dxdy =\int_a^b\int_c^df(x,y)dydx.\]
- If $\textbf{F}(x,y)=P(x,y)\textbf{i}+Q(x,y)\textbf{j}$ and $\frac{\partial P}{\partial y}=\frac{\partial Q}{\partial x}$ on an open connected region $D$, then $\textbf{F}$ is conservative on $D$.
- If curl$\textbf{F}=$curl$\textbf{G}$ on $\mathbb{R}^3$, then $\displaystyle\int_C\textbf{F}\cdot d\textbf{r}=\int_C\textbf{G}\cdot d\textbf{r}$ for all closed path $C$.
- If $\textbf{F}$ and $\textbf{G}$ are vector fields and curl$\textbf{F}=$curl$\textbf{G}$, div$\textbf{F}=$div$\textbf{G}$, then $\textbf{F}-\textbf{G}$ is a constant vector field.
- Let $B$ be a rigid body rotating about the $z$-axis with constant angular speed $\omega$. If $\textbf{v}(x,y,z)$ is the velocity at point $(x,y,z)\in B$, then curl$\textbf{v}$ is parallel to $\textbf{k}$.
|
| 15-b-21 |
b. Surface integrals - (ii) By Stokes' theorem
|
0.585
|
0.372
|
3.750
|
10
|
0.664
|
0.079
|
Evaluate $\displaystyle\int_C(y+\sin^3x)dx+(z^2+\cos^4y)dy +(x^3+\tan^5z)dz$ where $C$ is the curve $\textbf{r}(t)=\sin t\ \textbf{i}+\cos t\ \textbf{j}+\sin 2t\ \textbf{k}$, $0\leq t\leq 2\pi$. (Hint: $C$ lies on the surface $z=2xy$.)
|
| 15-b-22 |
b. Surface integrals - (ii) By Stokes' theorem
|
0.443
|
0.411
|
4.250
|
10
|
0.633
|
0.190
|
Let $C$ be the curve formed by the intersection of the plane $z=x$ and the surface $z=x^2+y^2$. $C$ is oriented counterclockwise when viewed from above. Evaluate $\displaystyle\oint_C(xyz+\tan^{-1}x)dx+(x^2+\sinh y)dy +(xz+\ln z)dz$.
|
| 15-b-23 |
b. Surface integrals - (ii) By Stokes' theorem
|
0.469
|
0.722
|
11.680
|
15
|
0.957
|
0.488
|
- Find curl$\textbf{F}$, where $\textbf{F}(x,y,z)=(y^2-z^2)\textbf{i} +(z^2-x^2)\textbf{j}+(x^2-y^2)\textbf{k}$.
- Compute the line integral \[ \oint_C(y^2-z^2)\,\mathrm{d}x+(z^2-x^2)\,\mathrm{d}y+(x^2-y^2)\,\mathrm{d}z, \] where $C$ is the hexagon which is the boundary of the intersection of the plane $x+y+z=\frac32$ and the unit cube $B=\{(x,y,z)|0\leq x\leq 1,0\leq y\leq 1,0\leq z\leq 1\}$, oriented as pictured.
圖
|
| 15-b-24 |
b. Surface integrals - (ii) By Stokes' theorem
|
0.642
|
0.573
|
7.540
|
13
|
0.894
|
0.252
|
Compute the line integral \[ \int_Cz^2\,\mathrm{d}x-x^2\,\mathrm{d}y+2yz\,\mathrm{d}z, \] where $C$ is the curve of intersection of the upper half sphere $z=\sqrt{4-x^2-y^2}$ and the circular cylinder $x^2+y^2=2y$, orientated counterclockwise viewed from the above, as Figure.
圖
|
| 15-b-25 |
b. Surface integrals - (ii) By Stokes' theorem
|
0.435
|
0.530
|
7.290
|
14
|
0.747
|
0.312
|
Compute $\displaystyle\iint_S\text{curl}\textbf{F}\cdot\text{d}\textbf{S}$, where ${\bf F}=e^{xz} {\bf i}+(x^2+z^2){\bf j}+(y+\cos z){\bf k}$ and where $S=\left\{(x,y,z)\Big|\frac{x^2}{4}+\frac{y^2}{9}+z^2=1\ \text{and }x+2z\geq 0\right\}$ oriented so that the boundary is counterclockwise when viewed from above.
圖
|
| 15-b-26 |
b. Surface integrals - (ii) By Stokes' theorem
|
0.468
|
0.444
|
6.260
|
14
|
0.678
|
0.210
|
Let $\textbf F = (x-y) \textbf i + (y-z) \textbf j + (z-x) \textbf k$ be a vector field on $\mathbb{R}^3$.
- Compute $\text{curl} \textbf{F}$ and div$\textbf{F}$ on $\mathbb{R}^3$.
- Let $S_1$ be the part of paraboloid $z=x^2+(y-1)^2$ that is below the plane $z=5-2y$ with downward orientation. Find the flux of $\textbf{F}$ across $S_1$, $\displaystyle\iint_{S_1} \textbf{F} \cdot d\textbf S$.
- Let $S_2$ be a surface defined by the equation $\frac{x^2}{9}+\frac{y^2}{36}-z^2=1$ for $z \in [0,1]$ and endowed with the orientation given by the downward normal vector. Find the flux of $\text{curl}\textbf{F}$ across $S_2$, $\displaystyle\iint_{S_2} \text{curl}\textbf{F} \cdot d\textbf S$.
圖 1 圖 2
|
| 15-b-27 |
b. Surface integrals - (ii) By Stokes' theorem
|
0.528
|
0.543
|
7.650
|
14
|
0.808
|
0.279
|
Let $\textbf F = (x-y) \textbf i + (y-z) \textbf j + (z-x) \textbf k$ be a vector field on $\mathbb{R}^3$.
- Compute $\text{curl} \textbf{F}$ on $\mathbb{R}^3$.
- Let $S_1$ be a parametric surface given by $\textbf r(r,\theta) = r \cos\theta \textbf i + 2r \sin\theta \textbf j + (9-r^2) \textbf k$ for $r\in [0,3]$ and $\theta \in [0,2\pi]$, which comes with the standard orientation given by the normal vector $\textbf{r}_r \times \textbf{r}_\theta$. Find the flux of $\text{curl} \textbf{F}$ across $S_1$.
- Let $S_2$ be a surface defined by the equation $\frac{x^2}{9}+\frac{y^2}{36}-z^2=1$ for $z \in [0,1]$ and endowed with the orientation given by the downward normal vector. Find the flux of $\text{curl}\textbf{F}$ across $S_2$.
圖 1 圖 2
|
| 15-b-28 |
b. Surface integrals - (ii) By Stokes' theorem
|
0.630
|
0.513
|
7.490
|
15
|
0.828
|
0.198
|
Evaluate $\displaystyle\iint\limits_S \nabla\times\textbf{F}\cdot d\textbf{S}$, where $\textbf{F}(x,y,z)=(y+\sin x)\,\textbf{i}+(z^2+\cos y)\,\textbf{j}+x^3\,\textbf{k}$ and where $S$ is the surface $z=2xy$ inside the cylinder $x^2+y^2=1$ and with the normal pointing in the positive $z$-direction.
|
| 15-b-29 |
b. Surface integrals - (iii) By the divergence theorem
|
0.726
|
0.512
|
7.181
|
15
|
0.875
|
0.149
|
Let $S_1$ be the surface $\{(x,y,z)|~z=x^2+y^2,~z\leq y\}$, $S_2$ be the surface $\{(x,y,z)|~z=y,~x^2+y^2\leq z\}$, and $\textbf{V}(x,y,z)=-y\textbf{i}+x\textbf{j}+z\textbf{k}$.
- Compute directly the downward flux of $\textbf{V}$ across $S_1$.
- Use the divergence theorem to compute the upward flux of $\textbf{V}$ across $S_2$.
|
| 15-b-30 |
b. Surface integrals - (iii) By the divergence theorem
|
0.649
|
0.436
|
4.711
|
12
|
0.760
|
0.111
|
Evaluate the flux of \[\textbf{V}(x,y,z)=(z^2x+y^2z)\textbf{i} +\left(\frac{1}{3}y^3+z\tan x\right)\textbf{j}+(x^2z+2y^2+1)\textbf{k}\] across $S$: the upper half sphere $x^2+y^2+z^2=1$, $z\geq 0$ with normal pointing away from the origin.
|
| 15-b-31 |
b. Surface integrals - (iii) By the divergence theorem
|
0.492
|
0.315
|
4.070
|
15
|
0.561
|
0.069
|
Let ${\bf F}(x,y,z)=\frac{1}{\left(x^2+y^2+z^2\right)^{3/2}}(x\,{\bf i}+y\,{\bf j}+z\,{\bf k})$.
- Let $S_1$ be the part of the sphere $x^2+y^2+z^2=1$ in the first octant bounded by the planes $y=0$, $y=\sqrt{3}x$ and $z=0$, oriented upward. Find the flux of ${\bf F}$ across $S_1$.
- Let $S_2$ be the part of the surface $\frac{x^2}{4}+\frac{y^2}{9}+\frac{z^2}{16}=1$ in the first octant bounded by the planes $y=0$, $y=\sqrt{3}x$ and $z=0$, oriented upward. Find the flux of ${\bf F}$ across $S_2$.
圖
|
| 15-b-32 |
b. Surface integrals - (iii) By the divergence theorem
|
0.642
|
0.642
|
10.720
|
15
|
0.963
|
0.321
|
Let ${\bf F}=\left\langle 3xy^2,\,y^3,\,e^{x^2+y^2}\right\rangle$. Let $S$ be the part of the surface $z=1-x^2-y^2$ that lies above $xy$-plane oriented upwards (that is, with normal having ${\bf k}$-component $\geq 0$). Calculate the flux $\displaystyle\int_{S}{\bf F}\cdot d{\bf S}$ of ${\bf F}$ across $S$. Note that $S$ is not closed.
|
| 15-b-33 |
b. Surface integrals - (iii) By the divergence theorem
|
0.427
|
0.420
|
6.980
|
16
|
0.633
|
0.206
|
Consider the unit disk \[ S_1 = \left\{(x,y,z) \in \mathbb{R}^3| x^2 + y^2 \leq 1 , z = 0\right\} \] and the half cone \[ S_2 = \left\{(x,y,z) \in \mathbb{R}^3| 2z = 1-\sqrt{x^2+y^2} , z \geq 0\right\} \; , \] and let $S = S_1 \cup S_2$ be the closed surface of a cone with the positive (outward) orientation. Both $S_1$ and $S_2$ are endowed with the induced orientation from $S$.
- Let $\textbf F = \left\langle 0 , y^2 , z-2yz\right\rangle\:$. Find $\displaystyle\iint_{S} \textbf F \cdot d\textbf S \:$.
- Let $\textbf G =\textbf F + \textbf E$, where \[\textbf E = \left\langle\frac x{(x^2 +y^2 +z^2)^{\frac 32}} , \frac y{(x^2 +y^2 +z^2)^{\frac 32}} , \frac z{(x^2 +y^2 +z^2)^{\frac 32}} \right\rangle\:\] defined on $\mathbb{R}^3\backslash \{(0 , 0 , 0)\}$. Find $\displaystyle\iint_{S_2} \textbf G \cdot d\textbf S \:$.
(Note that the integral is only over $S_2$.)
|
| 15-b-34 |
b. Surface integrals - (iii) By the divergence theorem
|
0.621
|
0.602
|
9.430
|
15
|
0.913
|
0.292
|
Evaluate the flux integral $\displaystyle\iint\limits_S\mathbf{F}\cdot\mathbf{n}\,dS$, where \[ \mathbf{F}(x,y,z)=(x+z)\,\mathbf{i}-(z+y)\,\mathbf{j}+(y+z^3)\,\mathbf{k}, \] and $S$ is the sphere $(x-2)^2+y^2+z^2=4$ with outward normal.
|
| 15-b-35 |
b. Surface integrals - (iii) By the divergence theorem
|
0.430
|
0.580
|
6.250
|
10
|
0.795
|
0.365
|
Let $S$ be the surface $x^2+y^2+z^2=1$, $x,y,z\geq 0$, an eighth of a sphere, and $\textbf{F}=x^2\textbf{i}+y^2\textbf{j}+z^2\textbf{k}$. Find the outward flux of $\textbf{F}$ across $S$.
|
| 15-b-36 |
b. Surface integrals - (iii) By the divergence theorem
|
0.451
|
0.725
|
11.290
|
15
|
0.950
|
0.499
|
- Find div$\textbf{F}$, where $\textbf{F}(x,y,z)=(y^2x+\sin z)\textbf{i}+(x^2y-\cos x)\textbf{j}+\left(\frac{1}{3}z^3+y^2\right)\textbf{k}$.
- Evaluate $\displaystyle\iint_S\textbf{F}\cdot d\textbf{S}$, where $S$ is the top half of the sphere $\begin{cases} x^2+y^2+z^2=1\\ z\geq 0 \end{cases}$ oriented upward.
|
| 15-b-37 |
b. Surface integrals - (iii) By the divergence theorem
|
0.741
|
0.449
|
6.400
|
15
|
0.819
|
0.078
|
Consider the vector field $\mathbf{F}(\mathbf{x})=\frac{\mathbf{x}}{|\mathbf{x}|^3}$, that is,
$ \mathbf{F}(x,y,z)=\frac{x}{(x^2+y^2+z^2)^{\frac32}}\,\mathbf{i}+\frac{y}{(x^2+y^2+z^2)^{\frac32}}\,\mathbf{j}+\frac{z}{(x^2+y^2+z^2)^{\frac32}}\,\mathbf{k}.$
- Evaluate $\displaystyle\iint_{S_1}\mathbf{F}\cdot\mathrm{d}\mathbf{S}$, where $S_1$ is the part of the sphere $x^2+y^2+z^2=1$ inside the cone $z=\sqrt{\frac{x^2+y^2}{3}}$ with upward orientation.
- Evaluate $\displaystyle\iint_{S_2}\mathbf{F}\cdot\mathrm{d}\mathbf{S}$, where $S_2$ is the part of the cone $z=\sqrt{\frac{x^2+y^2}{3}}$ between planes $z=\frac12$ and $z=\frac2{\sqrt{3}}$ with outward orientation.
- Use the Divergence Theorem to evaluate $\displaystyle\iint_{S_3}\mathbf{F}\cdot\mathrm{d}\mathbf{S}$, where $S_3$ is the part of the paraboloid $z=\frac{6-x^2-y^2}{\sqrt{3}}$ inside the cone $z=\sqrt{\frac{x^2+y^2}{3}}$ with upward orientation.
|
| 15-b-38 |
b. Surface integrals - (iii) By the divergence theorem
|
0.491
|
0.544
|
9.280
|
16
|
0.789
|
0.299
|
Let $E$ be the space region bounded by the surfaces \begin{align*} S_1 &:= \{(x, y, z)|\ z = -1+\sqrt{x^2+y^2},\ z\leq0\},\\ S_2 &:= \{(x, y, z)|\ z = 1-x^2-y^2,\ z\geq0\},\\ \end{align*} and ${\bf V}(x, y, z) = -y{\bf i} + x{\bf j} + z{\bf k}.$
圖
- Evaluate $\displaystyle\iiint_{\Omega}\text{div}\textbf{V}\, dV$, where $\Omega$ is the solid region enclosed by $S_1\cup S_2$.
- State the Divergence Theorem and evaluate the total outward flux $\displaystyle\iint_{S_1\cup S_2}\bf{V}\cdot d\bf{S}$.
- Compute the upward flux of $\bf{V}$ across $S_2$.
|
| 15-b-39 |
b. Surface integrals - (iii) By the divergence theorem
|
0.211
|
0.155
|
1.400
|
10
|
0.260
|
0.049
|
Suppose that $f(x,y,z)$ is a scalar function with continuous second partial derivatives. Fix a point $P_0=(x_0,y_0,z_0)$. Consider spheres $S_\rho$ centered at $P_0$ with radius $\rho>0$.
- Parametrize $S_\rho$ with spherical coordinates $\textbf r(\varphi,\theta)=(x_0+\rho\sin\varphi\cos\theta, y_0+\rho\sin\varphi\sin\theta, z_0+\rho\cos\varphi)$, $0\leq\varphi\leq\pi$, and $0\leq \theta\leq 2\pi$. Write down the double integral in $\varphi$ and $\theta$ that represents the average value of $f$ on $S_\rho$.
- Let function $A(\rho)$ be the average value of $f$ on $S_\rho$, for $\rho>0$. Evaluate $A'(\rho)$ in terms of $\displaystyle\iint_{S_\rho}\nabla f\cdot d \textbf S$.
- If $\nabla^2f=f_{xx}+f_{yy}+f_{zz}$ is always positive, show that $A(\rho)$ is increasing. If $\nabla^2f(x,y,z)=0$ for all $(x,y,z)$, compute $A(\rho)$.
|
| 15-b-40 |
b. Surface integrals - (iii) By the divergence theorem
|
0.501
|
0.434
|
5.020
|
12
|
0.685
|
0.184
|
Let $S$ be the boundary surface of the union of the balls $x^2 + y^2 + z^2 \le 1$ and $x^2 + y^2 +(z-1)^2 \le 1$.
圖 1
- Use spherical coordinates to parametrize $S$.
- Find $\displaystyle\iint_S \textbf F \cdot d\textbf S$ where $\textbf F = \textbf i + \textbf j + z^2\,\textbf k$ and $S$ is given the outward orientation.
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