접평면과 선형근사

A normal vector to the surface $x^2 + 9y^2 + 4z^2 = 17$ at the point $(x,y,z)$ is $\left< 2x\,,\, 18y\,,\,8z\right>$. A normal vector to the plane $x - 8z = 0$ is $\left< 1\,,\,0\,,\,-8\right>$. So we want $\left< 2x\,,\, 18y\,,\,8z\right>$ to be parallel to $\left< 1\,,\,0\,,\,-8\right>$, i.e. to be a nonzero constant times $\left< 1\,,\,0\,,\,-8\right>$. This is the case whenever $y=0$ and $z=-2x$ with $x\ne 0$. In addition, we want $(x,y,z)$ to lie on the surface $x^2 + 9y^2 + 4z^2 = 17$. So we want $y=0$, $z=-2x$ and

$$\begin{aligned} 17= x^2 + 9y^2 + 4z^2 =x^2 +4(-2x)^2=17x^2 \implies x=\pm 1 \end{aligned}$$

So the allowed points are $\pm(1,0,-2)$.

Let $(x_0,y_0,z_0)$ be a point on the hyperboloid $z^2=4x^2+y^2-1$ where the tangent plane is parallel to the plane $2x-y+z=0$. A normal vector to the plane $2x-y+z=0$ is $\left< 2,-1,1\right>$. Because the hyperboloid is $G(x,y,z)=4x^2+y^2-z^2-1$ and ${ \boldsymbol{\nabla} } G(x,y,z) = \left< 8x,2y,-2z\right>$, a normal vector to the hyperboloid at $(x_0,y_0,z_0)$ is ${ \boldsymbol{\nabla} } G(x_0,y_0,z_0)=\left< 8x_0,2y_0,-2z_0\right>$. So $(x_0,y_0,z_0)$ satisfies the required conditions if and only if there is a nonzero $t$ obeying

$$\begin{aligned} &\left< 8x_0,2y_0,-2z_0\right> =t\left< 2,-1,1\right> \text{ and } z_0^2=4x_0^2+y_0^2-1\\ &\iff x_0=\frac{t}{4},\ y_0=z_0=-\frac{t}{2}\text{ and } z_0^2=4x_0^2+y_0^2-1\\ &\iff \frac{t^2}{4}= \frac{t^2}{4}+ \frac{t^2}{4}-1\text{ and } x_0=\frac{t}{4},\ y_0=z_0=-\frac{t}{2}\\ & \iff t=\pm 2\qquad (x_0,y_0,z_0)=\pm \big(\tfrac{1}{2},-1,-1\big) \end{aligned}$$

One vector normal to the surface $F(x,y,z)=4x^2+9y^2-z^2=0$ at $(2,1,-5)$ is

$$\begin{aligned} { \boldsymbol{\nabla} } F(2,1,-5) = \left< 8x,18y,-2z\right>\Big|_{(2,1,-5)}=\left< 16,18,10\right> \end{aligned}$$

One vector normal to the surface $G(x,y,z)=6x+3y+2z=5$ at $(2,1,-5)$ is

$$\begin{aligned} { \boldsymbol{\nabla} } G(2,1,-5) = \left< 6,3,2\right> \end{aligned}$$

Now

• The curve lies in the surface $z^2=4x^2+9y^2$. So the tangent vector to the curve at $(2,1,-5)$ is perpendicular to the normal vector $\frac{1}{2}\left< 16,18,10\right>=\left< 8,9,5\right>$.
• The curve also lies in the surface $6x+3y+2z=5$. So the tangent vector to the curve at $(2,1,-5)$ is also perpendicular to the normal vector $\left< 6,3,2\right>$.
• So the tangent vector to the curve at $(2,1,-5)$ is parallel to

$$\begin{aligned} \left< 8,9,5\right>\times \left< 6,3,2\right> =\det\left[\begin{matrix} \hat{\boldsymbol{\imath}} & \hat{\boldsymbol{\jmath}} & \hat{\mathbf{k}} \\ 8 & 9 & 5 \\ 6 & 3 & 2 \end{matrix}\right] =\left< 3,14,-30\right> \end{aligned}$$

The desired vectors are

$$\pm\sqrt{3}\frac{\left< 3,14,-30\right>}{|\left< 3,14,-30\right>|} =\pm\sqrt{\frac{3}{1105}}\left< 3,14,-30\right>$$

Let $(x_0,y_0,z_0)$ be any point on the surface. A vector normal to the surface at $(x_0,y_0,z_0)$ is

$$\begin{aligned} &{ \boldsymbol{\nabla} }\Big(xy e^{-(x^2+y^2)/2}-z\Big)\bigg|_{(x_0,y_0,z_0)} \\& =\left< y_0 e^{-(x_0^2+y_0^2)/2}-x_0^2y_0 e^{-(x_0^2+y_0^2)/2}, x_0 e^{-(x_0^2+y_0^2)/2}-x_0y_0^2 e^{-(x_0^2+y_0^2)/2},-1\right> \end{aligned}$$

The tangent plane to the surface at $(x_0,y_0,z_0)$ is horizontal if and only if this vector is vertical, which is the case if and only if its $x$- and $y$-components are zero, which in turn is the case if and only if

$$\begin{aligned} &y_0(1-x_0^2)=0\text{ and }x_0(1-y_0^2)=0\\ &\iff\big\{y_0=0\text{ or }x_0=1\text{ or }x_0=-1\big\} \text{ and }\big\{x_0=0\text{ or }y_0=1\text{ or }y_0=-1\big\}\\ &\iff (x_0,y_0)=(0,0)\text{ or }(1,1)\text{ or }(1,-1) \text{ or }(-1,1)\text{ or }(-1,-1) \end{aligned}$$

The values of $z_0$ at these points are $0$, $e^{-1}$, $-e^{-1}$, $-e^{-1}$ and $e^{-1}$, respectively. So the horizontal tangent planes are $z=0$, $z=e^{-1}$ and $z=-e^{-1}$. At the highest and lowest points of the surface, the tangent plane is horizontal. So the largest and smallest values of $z$ are $e^{-1}$ and $-e^{-1}$, respectively.

(a) A normal vector to the surface at $(0,2,1)$ is

$$\begin{aligned} { \boldsymbol{\nabla} }\big(xy-2x+yz+x^2+y^2+z^3-7\big)\big|_{(0,2,1)} &=\left< y-2+2x\,,\,x+z+2y\,,\,y+3z^2\right>\big|_{(0,2,1)}\\ &=\left< 0, 5, 5\right> \end{aligned}$$

So the tangent plane is

$$\begin{aligned} 0(x-0)+5(y-2)+5(z-1)=0\text{ or }y+z=3 \end{aligned}$$

The vector parametric equations for the normal line are

$$\mathbf{r}(t)=\left< 0,2,1\right>+t\left< 0,5,5\right>$$

(b) Differentiating

$$xy-2x+y\,z(x,y)+x^2+y^2+z(x,y)^3=7$$

gives

$$\begin{aligned} y-2+y\,z_x(x,y)+2x+3z(x,y)^2z_x(x,y)&=0 & &\implies & z_x(x,y)&=\frac{2-2x-y}{y+3z(x,y)^2}\\ x+z(x,y)+y\,z_y(x,y)+2y+3z(x,y)^2z_y(x,y)&=0 & &\implies & z_y(x,y)&=-\frac{x+2y+z(x,y)}{y+3z(x,y)^2} \end{aligned}$$

In particular, at $(0,2,1)$, $z_y(0,2)=-\frac{4+1}{2+3}=-1$.

(c) Differentiating $z_x$ with respect to $y$ gives

$$\begin{aligned} z_{xy}(x,y) &=-\frac{1}{y+3z(x,y)^2} -\frac{2-2x-y}{{[y+3z(x,y)^2]}^2}\big(1+6z(x,y)z_y(x,y)\big)\\ &=-\frac{1}{y+3z(x,y)^2} -\frac{2-2x-y}{{[y+3z(x,y)^2]}^2} \left(1-6z(x,y)\frac{x+2y+z(x,y)}{y+3z(x,y)^2}\right) \end{aligned}$$

As an alternate solution, we could also differentiate $z_y$ with respect to $x$. This gives

$$\begin{aligned} z_{yx}(x,y) &=-\frac{1+z_x(x,y)}{y+3z(x,y)^2} +\frac{x+2y+z(x,y)}{{[y+3z(x,y)^2]}^2}6z(x,y)z_x(x,y)\\ &=-\frac{1}{y+3z(x,y)^2}\left(1+\frac{2-2x-y}{y+3z(x,y)^2}\right) +\frac{x+2y+z(x,y)}{{[y+3z(x,y)^2]}^2}6z(x,y)\frac{2-2x-y}{y+3z(x,y)^2} \end{aligned}$$

(a) A vector perpendicular to $x^2+z^2=10$ at $(1,1,3)$ is

$${ \boldsymbol{\nabla} }(x^2+z^2)\big|_{(1,1,3)} =(2x\hat{\boldsymbol{\imath}}+2z\hat{\mathbf{k}})\big|_{(1,1,3)} =2\hat{\boldsymbol{\imath}}+6\hat{\mathbf{k}}\text{ or } \frac{1}{2} \left< 2,0,6\right>=\left< 1,0,3\right>$$

(b) A vector perpendicular to $y^2+z^2=10$ at $(1,1,3)$ is

$${ \boldsymbol{\nabla} }(y^2+z^2)\big|_{(1,1,3)} =(2y\hat{\boldsymbol{\jmath}}+2z\hat{\mathbf{k}})\big|_{(1,1,3)} =2\hat{\boldsymbol{\jmath}}+6\hat{\mathbf{k}}\text{ or }\frac{1}{2} \left< 0,2,6\right>=\left< 0,1,3\right>$$

A vector is tangent to the specified curve at the specified point if and only if it perpendicular to both $(1,0,3)$ and $(0,1,3)$. One such vector is

$$\left< 0,1,3\right>\times\left<1,0,3\right> =\det\left[\begin{matrix} \hat{\boldsymbol{\imath}} & \hat{\boldsymbol{\jmath}} & \hat{\mathbf{k}} \\ 0 & 1 & 3 \\ 1 & 0 & 3 \end{matrix}\right] =\left< 3,3,-1\right>$$

(c) The specified tangent line passes through $(1,1,3)$ and has direction vector $\left< 3,3,-1\right>$ and so has vector parametric equation

$$\mathbf{r}(t)=\left< 1,1,3\right>+t\left< 3,3,-1\right>$$

$\mathbf{r}(t)=\left< x(t)\,,\,y(t)\,,\,z(t)\right>$ intersects $z^3 + xyz -2 = 0$ when

$$z(t)^3+x(t)\,y(t)\,z(t)-2=0\iff \big(t^2\big)^3 + \big(t^3)(t)\big(t^2\big)-2=0 \iff 2t^6=2\iff t=1$$

since $t$ is required to be positive. The direction vector for the curve at $t=1$ is

$$\mathbf{r}'(1)=3\,\hat{\boldsymbol{\imath}}+\hat{\boldsymbol{\jmath}}+2\,\hat{\mathbf{k}}$$

A normal vector for the surface at $\mathbf{r}(1)=\left< 1,1,1\right>$ is

$${ \boldsymbol{\nabla} }(z^3+xyz)\big|_{(1,1,1)}=[yz\hat{\boldsymbol{\imath}}+xz\hat{\boldsymbol{\jmath}}+(3z^2+xy)\hat{\mathbf{k}}]_{(1,1,1)} =\hat{\boldsymbol{\imath}}+\hat{\boldsymbol{\jmath}}+4\hat{\mathbf{k}}$$

The angle $\theta$ between the curve and the normal vector to the surface is determined by

$$\begin{aligned} \big|\left< 3,1,2\right>\big|\,\big|\left< 1,1,4\right>\big|\cos\theta =\left< 3,1,2\right> \cdot\left< 1,1,4\right> &\iff \sqrt{14}\sqrt{18}\cos\theta=12 \\ &\iff \sqrt{7\times 36}\cos\theta=12 \\ &\iff \cos\theta=\frac{2}{\sqrt{7}} \\ &\iff \theta=40.89^\circ \end{aligned}$$