면적분과 플럭스

We are to find the surface area of part of a hemisphere. On the hemisphere

$$\begin{aligned} z=f(x,y)=\sqrt{a^2-x^2-y^2}\qquad f_x(x,y)=-\frac{x}{\sqrt{a^2-x^2-y^2}}\qquad f_y(x,y)=-\frac{y}{\sqrt{a^2-x^2-y^2}} \end{aligned}$$

so that

$$\begin{aligned} \mathrm{d}S&=\sqrt{1+f_x(x,y)^2+f_y(x,y)^2}\,\mathrm{d}x\,\mathrm{d}y =\sqrt{1+\frac{x^2}{a^2-x^2-y^2}+\frac{y^2}{a^2-x^2-y^2}}\,\mathrm{d}x\,\mathrm{d}y \\ &=\sqrt{\frac{a^2}{a^2-x^2-y^2}}\,\mathrm{d}x\,\mathrm{d}y \end{aligned}$$

In polar coordinates, this is $\mathrm{d}S=\frac{a}{\sqrt{a^2-r^2}}\,r\,\mathrm{d}r\,\mathrm{d}\theta$. We are to find the surface area of the part of the hemisphere that is inside the cylinder, $x^2-ax+y^2=0$, which in polar coordinates becomes $r^2-ar\cos\theta=0$ or $r=a\cos\theta$. The top half of the domain of integration is sketched below.

So the

$$\begin{aligned} {\rm Surface\ Area} &= 2\int_0^{\pi/2}\mathrm{d}\theta\int_0^{a\cos\theta}\mathrm{d}r\ r \frac{a}{\sqrt{a^2-r^2}} = 2a\int_0^{\pi/2}\mathrm{d}\theta\ \Big[-\sqrt{a^2-r^2}\,\Big]_0^{a\cos\theta} \\ &= 2a\int_0^{\pi/2}\mathrm{d}\theta\ \big[a-a\sin\theta\big] \\ &= 2a^2\Big[\theta+\cos\theta\Big]_0^{\pi/2} =a^2[\pi-2] \end{aligned}$$

By symmetry, the centroid $(\bar x,\bar y,\bar z)$ obeys $\bar x=\bar y=\bar z$. Parametrize the sphere using spherical coordinates.

$$\mathbf{r}(\theta,\varphi) = a\sin\varphi\cos\theta\,\hat{\boldsymbol{\imath}} +a\sin\varphi\sin\theta\,\hat{\boldsymbol{\jmath}} +a\cos\varphi\,\hat{\mathbf{k}}$$

Then

$$\begin{aligned} \frac{\partial \mathbf{r}}{\partial \theta} =-a\sin\varphi\sin\theta\,\hat{\boldsymbol{\imath}}+a\sin\varphi\cos\theta\,\hat{\boldsymbol{\jmath}}\qquad \frac{\partial \mathbf{r}}{\partial \varphi} =a\cos\varphi\cos\theta\,\hat{\boldsymbol{\imath}}+a\cos\varphi\sin\theta\,\hat{\boldsymbol{\jmath}} -a\sin\varphi\,\hat{\mathbf{k}} \end{aligned}$$

so that

$$\begin{aligned} \frac{\partial \mathbf{r}}{\partial \theta}\times\frac{\partial \mathbf{r}}{\partial \varphi} &=\det\left[\begin{matrix} \hat{\boldsymbol{\imath}} & \hat{\boldsymbol{\jmath}} & \hat{\mathbf{k}} \\ -a\sin\varphi\sin\theta & a\sin\varphi\cos\theta & 0 \\ a\cos\varphi\cos\theta & a\cos\varphi\sin\theta & -a\sin\varphi \end{matrix}\right] \\ &=-a^2\sin^2\varphi\cos\theta\,\hat{\boldsymbol{\imath}}-a^2\sin^2\varphi\sin\theta\,\hat{\boldsymbol{\jmath}} -a^2\sin\varphi\cos\varphi\,\hat{\mathbf{k}}\\ \implies \mathrm{d}S&=\left|\frac{\partial \mathbf{r}}{\partial \theta}\times\frac{\partial \mathbf{r}}{\partial \varphi}\right| \,\mathrm{d}\theta\,\mathrm{d}\varphi =a^2\sin\varphi\,\mathrm{d}\theta\,\mathrm{d}\varphi \end{aligned}$$

As the surface area of the part of the sphere in the first octant is $\frac{1}{8}4\pi a^2=\frac{\pi a^2}{2}$

$$\begin{aligned} \bar x=\bar y=\bar z &=\frac{\iint _S z\,\mathrm{d}S}{\iint _S\,\mathrm{d}S} =\frac{1}{\pi a^2/2}\int_0^{\pi/2}\mathrm{d}\theta\int_0^{\pi/2}\mathrm{d}\varphi \ (a^2\sin\varphi)(\overbrace{a\cos\varphi}^{z}) \\ &=\frac{2a}{\pi}\frac{\pi}{2}\int_0^{\pi/2}\mathrm{d}\varphi \ \sin\varphi\cos\varphi =a\bigg[\frac{1}{2}\sin^2\varphi\bigg]_0^{\pi/2} =\frac{a}{2} \end{aligned}$$

In cylindrical coordinates

$$x=r\cos\theta\qquad y=r\sin\theta\qquad z=z$$

In these coordinates the equation, $x^2+y^2=2ay$, of the cylinder becomes

$$r^2=2ar\sin\theta\qquad\text{or}\qquad r=2a\sin\theta$$

That is, $r=f(\theta)$ with $f(\theta) =2a\sin\theta$. Parametrize the cylinder by $\mathbf{r}(\theta,z) = x(\theta,z)\,\hat{\boldsymbol{\imath}} + y(\theta,z)\,\hat{\boldsymbol{\jmath}} + z(\theta,z)\,\hat{\mathbf{k}}$ with

$$\begin{aligned} x(\theta,z)&=f(\theta)\cos\theta=2a\sin\theta\cos\theta=a\sin2\theta\\ y(\theta,z)&=f(\theta)\sin\theta=2a\sin\theta\sin\theta=a(1-\cos2\theta)\\ z(\theta,z)&=z \end{aligned}$$

Under this parametrization,

$$\begin{aligned} \frac{\partial \mathbf{r}}{\partial \theta}=2a\cos2\theta\,\hat{\boldsymbol{\imath}}+2a\sin2\theta\,\hat{\boldsymbol{\jmath}}\qquad \frac{\partial \mathbf{r}}{\partial z}=\hat{\mathbf{k}} &\implies \frac{\partial \mathbf{r}}{\partial \theta}\times\frac{\partial \mathbf{r}}{\partial z} =-2a\cos2\theta\,\hat{\boldsymbol{\jmath}}+2a\sin2\theta\,\hat{\boldsymbol{\imath}} \\ &\implies \mathrm{d}S=\left|\frac{\partial \mathbf{r}}{\partial \theta}\times\frac{\partial \mathbf{r}}{\partial z}\right|\, \mathrm{d}\theta\,\mathrm{d}z =2a\,\mathrm{d}\theta\,\mathrm{d}z \end{aligned}$$

We still have to determine the limits of integration. The figure on the left below provides a top view of the cylinder.

From it we see that $0\le\theta\le\pi$. The cone $z^2=x^2+y^2=r^2$ (i.e. $z=\pm r$) and the cylinder $r=2a\sin\theta$ intersect at $z^2=r^2=4a^2\sin^2\theta$. So, for each fixed $\theta$, $z$ runs from $-2a\sin\theta$ to $z=+2a\sin\theta$. (See the figure on the right above. It shows a constant $\theta$ cross-section.) Finally,

$$\begin{aligned} {\rm Area} &= \int_{|z|\le 2a\sin\theta}2a\,\mathrm{d}\theta\,\mathrm{d}z =2a\int_0^\pi \!\!\mathrm{d}\theta\int_{-2a\sin\theta}^{2a\sin\theta} \!\!\!\mathrm{d}z =8a^2\int_0^\pi \!\!\mathrm{d}\theta\ \sin\theta =8a^2\Big[-\cos\theta\Big]_0^\pi \\ &=16a^2 \end{aligned}$$

(a) Each (horizontal) constant $z$ cross-section is a circle centred on the $z$-axis. The radius varies linearly from $2$, when $z=0$ to $0$, when $z=3$. So the radius at height $z$ is $\frac{2}{3}(3-z)$ and we can use

$$\begin{aligned} \mathbf{r}(\theta,z) = \frac{2}{3}(3-z)\cos\theta\ \hat{\boldsymbol{\imath}} +\frac{2}{3}(3-z)\sin\theta\ \hat{\boldsymbol{\jmath}} +z\,\hat{\mathbf{k}}\qquad 0\le\theta<2\pi,\quad 0\le z\le 3 \end{aligned}$$

as the parametrization.

(b) By symmetry the centre of mass will lie on the $z$-axis. We are only asked for the $z$-coordinate anyway. The $z$-coordinate of the centre of mass is the weighted average of $z$ over the cone. Since a density has not been specified, we assume that it is a constant. We may take the density to be $1$, so the $z$-coordinate of the centre of mass is $\iint _S z\,\mathrm{d}S/ \iint _S \mathrm{d}S$.

Since

$$\begin{aligned} \tfrac{\partial\mathbf{r}}{\partial\theta} &=\big(-\tfrac{2}{3}(3-z)\sin\theta\,,\, \tfrac{2}{3}(3-z)\cos\theta\,,\, 0 \big) \\ \tfrac{\partial\mathbf{r}}{\partial z} &=\big(-\tfrac{2}{3}\cos\theta\,,\, -\tfrac{2}{3}\sin\theta\,,\, 1 \big) \\ \tfrac{\partial\mathbf{r}}{\partial\theta} \times \tfrac{\partial\mathbf{r}}{\partial z} &=\big(\tfrac{2}{3}(3-z)\cos\theta\,,\, \tfrac{2}{3}(3-z)\sin\theta\,,\, \tfrac{4}{9}(3-z)\big) \end{aligned}$$

the element of surface area for this parametrization is

$$\begin{aligned} \mathrm{d}S &= \big|\tfrac{\partial\mathbf{r}}{\partial\theta}\times \tfrac{\partial\mathbf{r}}{\partial z} \big|\mathrm{d}\theta\mathrm{d}z =\tfrac{2}{3}(3-z)\big|(\cos\theta\,,\,\sin\theta\,,\,\tfrac{2}{3}\big)\big| \mathrm{d}\theta\mathrm{d}z \\ &=\tfrac{2\sqrt{13}}{9}(3-z) \mathrm{d}\theta\mathrm{d}z \end{aligned}$$

So the surface area, $\iint _S \mathrm{d}S$, of the cone is

$$\begin{aligned} \int_0^3\mathrm{d}z\int_0^{2\pi}\mathrm{d}\theta\ \tfrac{2\sqrt{13}}{9}(3-z) &=\tfrac{4\sqrt{13}}{9}\pi\int_0^3\mathrm{d}z\ (3-z) =-\tfrac{2\sqrt{13}}{9}\pi(3-z)^2\Big|_0^3 \\ &=2\sqrt{13}\pi \end{aligned}$$

and the $z$-coordinate of the centre of mass is

$$\begin{aligned} \bar z &=\frac{1}{2\sqrt{13}\pi} \int_0^3\mathrm{d}z\int_0^{2\pi}\mathrm{d}\theta\ \tfrac{2\sqrt{13}}{9}(3-z) z =\frac{2}{9}\int_0^3\mathrm{d}z\ (3z-z^2) =\frac{2}{9}{\Big[\frac{3z^2}{2}-\frac{z^3}{3}\Big]}_0^3 \\ &=\frac{2}{9}\frac{27}{6} =1 \end{aligned}$$

This is a little less than half way up the cone, which is reasonable since the cone is “bottom heavy”.

Write

$$V=\big\{ \ (x,y,z)\ \big|\ 1\le x^2+y^2+z^2\le 4\ \big\}$$

The boundary of $V$ consists of two parts — the sphere, $S_2$, of radius $2$, centred on the origin, with (outward) normal $\hat{\mathbf{n}}=\frac{\mathbf{r}}{|\mathbf{r}|}=\frac{\mathbf{r}}{2}$, and the sphere $S_1$ of radius $1$, centred on the origin, with (inward) normal $\hat{\mathbf{n}} = -\mathbf{r}$, So,

$$\begin{aligned} \iint _{\partial V} \mathbf{F}\cdot\hat{\mathbf{n}}\,\mathrm{d}S &=\iint _{S_2} \frac{\mathbf{r}}{|\mathbf{r}|}\cdot\frac{\mathbf{r}}{2}\,\mathrm{d}S -\iint _{S_1} \frac{\mathbf{r}}{|\mathbf{r}|}\cdot\mathbf{r}\,\mathrm{d}S \\ &=\iint _{S_2} \mathrm{d}S -\iint _{S_1} \mathrm{d}S \\ &=4\pi(2)^2 -4\pi(1)^2 \\ &=12\pi \end{aligned}$$