다변수 미분 전체 문제

(a) In polar coordinates $x=r\cos\theta$, $y=r\sin\theta$

$$\begin{aligned} \frac{2x^2 + x^2y - y^2x + 2y^2}{x^2 + y^2} &=\frac{2r^2\cos^2\theta + r^3\cos^2\theta\sin\theta - r^3\cos\theta\sin^2\theta + 2r^2\sin^2\theta}{r^2} \\ &=2+ r\big[\cos^2\theta\sin\theta - \sin^2\theta\cos\theta \big] \end{aligned}$$

As

$$r\big|\cos^2\theta\sin\theta - \sin^2\theta\cos\theta \big| \le 2r \rightarrow 0\text{ as }r\rightarrow 0$$

we have

$$\lim_{(x,y)\rightarrow(0,0)} \frac{2x^2 + x^2y - y^2x + 2y^2}{x^2 + y^2}=2$$

(b) Since

$$\begin{aligned} \frac{x^2y^2 -2 x^2y + x^2} {(x^2 + y^2-2y+1)^2} =\frac{x^2(y-1)^2} {\big[x^2 + (y-1)^2\big]^2} \end{aligned}$$

and, in polar coordinates centred on $(0,1)$, $x=r\cos\theta$, $y=1+r\sin\theta$,

$$\frac{x^2(y-1)^2} {\big[x^2 + (y-1)^2\big]^2} =\frac{r^4\cos^2\theta\sin^2\theta}{r^4} =\cos^2\theta\sin^2\theta$$

we have that the limit does not exist. For example, if we send $(x,y)$ to $(0,1)$ along the line $y=1$, so that $\theta=0$, we get the limit $0$, while if we send $(x,y)$ to $(0,1)$ along the line $y=x+1$, so that $\theta=\frac{\pi}{4}$, we get the limit $\frac{1}{4}$.

(a) Since, in polar coordinates,

$$\frac{xy}{x^2+y^2}=\frac{r^2\cos\theta\sin\theta}{r^2} =\cos\theta\sin\theta$$

we have that the limit does not exist. For example,

• if we send $(x,y)$ to $(0,0)$ along the positive $x$-axis, so that $\theta=0$, we get the limit $\sin\theta\cos\theta\big|_{\theta=0}=0$,
• while if we send $(x,y)$ to $(0,0)$ along the line $y=x$ in the first quadrant, so that $\theta=\frac{\pi}{4}$, we get the limit $\sin\theta\cos\theta\big|_{\theta=\pi/4}=\frac{1}{2}$.

(b) This limit does not exist, since if it were to exist the limit

$$\lim_{(x,y)\rightarrow(0,0)}\frac{xy}{x^2+y^2} =\lim_{(x,y)\rightarrow(0,0)}\frac{xy}{\sin(xy)}\ \frac{\sin(xy)}{x^2+y^2} =\lim_{(x,y)\rightarrow(0,0)}\frac{xy}{\sin(xy)}\ \lim_{(x,y)\rightarrow(0,0)}\frac{\sin(xy)}{x^2+y^2}$$

would also exist. (Recall that $\displaystyle \lim_{t\rightarrow 0}\tfrac{\sin t}{t} =1$.)

(c) Since

$$\begin{aligned} \lim_{(x,y)\rightarrow(-1,1)}\big[x^2+2xy^2+y^4\big] &=(-1)^2+2(-1)(1)^2+(1)^4=0 \\ \lim_{(x,y)\rightarrow(-1,1)}\big[1+y^4\big] &=1+(1)^4=2 \end{aligned}$$

and the second limit is nonzero,

$$\lim_{(x,y)\rightarrow(-1,1)}\frac{x^2+2xy^2+y^4}{1+y^4}=\frac{0}{2}=0$$

(d) Since the limit along the positive $x$-axis

$$\lim_{\genfrac{}{}{0pt}{}{t\rightarrow 0}{t>0}}|y|^x\Big|_{(x,y)=(t,0)} =\lim_{\genfrac{}{}{0pt}{}{t\rightarrow 0}{t>0}}0^t =\lim_{\genfrac{}{}{0pt}{}{t\rightarrow 0}{t>0}}0 =0$$

and the limit along the $y$-axis

$$\lim_{t\rightarrow 0}|y|^x\Big|_{(x,y)=(0,t)} =\lim_{t\rightarrow 0}|t|^0 =\lim_{t\rightarrow 0}1 =1$$

are different, the limit as $(x,y)\rightarrow 0$ does not exist.

As $z(x,y)=f(x^2+y^2)$

$$\begin{aligned} \frac{\partial z}{\partial x}(x,y)&=2xf'(x^2+y^2) \\ \frac{\partial z}{\partial y}(x,y)&=2yf'(x^2+y^2) \end{aligned}$$

by the (ordinary single variable) chain rule. So

$$y\frac{\partial z}{\partial x}-x\frac{\partial z}{\partial y} =y(2x)f'(x^2+y^2)-x(2y)f'(x^2+y^2)=0$$

and the differential equation is always satisfied, assuming that $f$ is differentiable, so that the chain rule applies.

By definition

$$\begin{aligned} \frac{\partial f}{\partial x}(0,0) &=\lim_{\Delta x\rightarrow 0}\frac{f(\Delta x,0)-f(0,0)}{\Delta x} \\ &=\lim_{\Delta x\rightarrow 0}\frac{\frac{(\Delta x+2\times 0)^2}{\Delta x+0}-0}{\Delta x} \\ &=\lim_{\Delta x\rightarrow 0}\frac{\Delta x}{\Delta x} \\ &=1 \end{aligned}$$

and

$$\begin{aligned} \frac{\partial f}{\partial y}(0,0) &=\lim_{\Delta y\rightarrow 0}\frac{f(0,\Delta y)-f(0,0)}{\Delta y} \\ &=\lim_{\Delta y\rightarrow 0}\frac{\frac{(0+2\Delta y)^2}{0+\Delta y}-0}{\Delta y} \\ &=\lim_{\Delta y\rightarrow 0}\frac{4\Delta y}{\Delta y} \\ &=4 \end{aligned}$$

(b) $f(x,y)$ is not continuous at $(0,0)$, even though both partial derivatives exist there. To see this, make a change of coordinates from $(x,y)$ to $(X,y)$ with $X=x+y$ (the denominator). Of course, $(x,y)\rightarrow (0,0)$ if and only if $(X,y)\rightarrow (0,0)$. Now watch what happens when $(X,y)\rightarrow(0,0)$ with $X$ a lot smaller than $y$. For example, $X=ay^2$. Then

$$\begin{aligned} \frac{(x+2y)^2}{x+y}=\frac{(X+y)^2}{X}=\frac{(ay^2+y)^2}{ay^2} =\frac{(1+ay)^2}{a}\rightarrow\frac{1}{a} \end{aligned}$$

This depends on $a$. So approaching $(0,0)$ along different paths gives different limits. (You can see the same effect without changing coordinates by sending $(x,y)\rightarrow (0,0)$ with $x=-y+ay^2$.) Even more dramatically, watch what happens when $(X,y)\rightarrow(0,0)$ with $X=y^3$. Then

$$\frac{(x+2y)^2}{x+y}=\frac{(X+y)^2}{X}=\frac{(y^3+y)^2}{y^3} =\frac{{(1+y^2)}^2}{y}\rightarrow\pm\infty$$

Solution 1

Let's start by finding an equation for this surface. Every level curve is a horizontal circle of radius one, so the equation should be of the form

$$(x-f_1)^2+(y-f_2)^2=1$$

where $f_1$ and $f_2$ are functions depending only on $z$. Since the centre of the circle at height $z$ is at position $x=0$, $y=z$, we see that the equation of our surface is

$$x^2+(y-z)^2=1$$

The height of the surface at the point $(x,y)$ is the $z(x,y)$ found by solving that equation. That is,

$$x^2+\big(y-z(x,y)\big)^2=1 \tag{$*$}$$

We differentiate this equation implicitly to find $z_x(x,y)$ and $z_y(x,y)$ at the desired point $(x,y)= (0,-1)$. First, differentiating $(*)$ with respect to $y$ gives

$$\begin{aligned} 0+2\big(y-z(x,y)\big)\big(1- z_y(x,y)\big)&=0 \\ 2(-1-0)\big(1-z_y(0,-1)\big)&=0& & \text{ at } (0,-1,0) \end{aligned}$$

so that the slope looking in the positive $y$ direction is $z_y(0,-1)=1$. Similarly, differentiating $(*)$ with respect to $x$ gives

$$\begin{aligned} 2x+2\big(y-z(x,y)\big)\cdot\big(0-z_x(x,y)\big)&=0 \\ 2x&=2\big(y-z(x,y)\big)\cdot z_x(x,y)\\ z_x(x,y)&=\frac{x}{y-z(x,y)}\\ z_x(0,-1)&=0 &\text{ at } (0,-1,0) \end{aligned}$$

The slope looking in the positive $x$ direction is $z_x(0,-1)=0$.

Solution 2

Standing at $(0,-1,0)$ and looking in the positive $y$ direction, the surface follows the straight line that

• passes through the point $(0,-1,0)$, and
• is parallel to the central line $z=y, x=0$ of the cylinder.

Shifting the central line one unit in the $y$-direction, we get the line $z=y+1$, $x=0$. (As a check, notice that $(0,-1,0)$ is indeed on $z=y+1$, $x=0$.) The slope of this line is 1.

Standing at $(0,-1,0)$ and looking in the positive $x$ direction, the surface follows the circle $x^2+y^2=1$, $z=0$, which is the intersection of the cylinder with the $xy$-plane. As we move along that circle our $z$ coordinate stays fixed at $0$. So the slope in that direction is 0.

As

$$\begin{aligned} u_t(x,y,z,t) &=-\frac{3}{2}\frac{1}{t^{5/2}} e^{-(x^2+y^2+z^2)/(4\alpha t)} +\frac{1}{4\alpha \,t^{7/2}}(x^2+y^2+z^2) e^{-(x^2+y^2+z^2)/(4\alpha t)} \\ u_x(x,y,z,t) &=-\frac{x}{2\alpha \,t^{5/2}} e^{-(x^2+y^2+z^2)/(4\alpha t)} \\ u_{xx}(x,y,z,t) &=-\frac{1}{2\alpha \,t^{5/2}} e^{-(x^2+y^2+z^2)/(4\alpha t)} +\frac{x^2}{4\alpha ^2\,t^{7/2}} e^{-(x^2+y^2+z^2)/(4\alpha t)} \\ u_{yy}(x,y,z,t) &=-\frac{1}{2\alpha \,t^{5/2}} e^{-(x^2+y^2+z^2)/(4\alpha t)} +\frac{y^2}{4\alpha ^2\,t^{7/2}} e^{-(x^2+y^2+z^2)/(4\alpha t)} \\ u_{zz}(x,y,z,t) &=-\frac{1}{2\alpha \,t^{5/2}} e^{-(x^2+y^2+z^2)/(4\alpha t)} +\frac{z^2}{4\alpha ^2\,t^{7/2}} e^{-(x^2+y^2+z^2)/(4\alpha t)} \end{aligned}$$