다변수 연쇄법칙

A closed box is in the shape of a rectangular solid with dimensions $x,y,\,\text{and}\,z.$ (Dimensions are in inches.) Suppose each dimension is changing at the rate of $0.5$ in./min. Find the rate of change of the total surface area of the box when $x=2\,\text{in}.,\,y=3\,\text{in.,}\,\text{and}\,z=1\,\text{in}.$

The total resistance in a circuit that has three individual resistances represented by $x,y,$ and $z$ is given by the formula $R(x,y,z)=\frac{xyz}{yz+xz+xy}.$ Suppose at a given time the $x$ resistance is $100\Omega ,$ the y resistance is $200\Omega ,$ and the $z$ resistance is $300\Omega .$ Also, suppose the $x$ resistance is changing at a rate of $2\Omega /\min ,$ the $y$ resistance is changing at the rate of $1\Omega /\min ,$ and the $z$ resistance has no change. Find the rate of change of the total resistance in this circuit at this time.

The temperature $T$ at a point $(x,y)$ is $T(x,y)$ and is measured using the Celsius scale. A fly crawls so that its position after $t$ seconds is given by $x=\sqrt{1+t}$ and $y=2+\frac{1}{3}t,$ where $x\,\text{and}\,y$ are measured in centimeters. The temperature function satisfies ${T}_{x}(2,3)=4$ and ${T}_{y}(2,3)=3.$ How fast is the temperature increasing on the fly’s path after $3$ sec?

The $x\,\text{and}\,y$ components of a fluid moving in two dimensions are given by the following functions: $u(x,y)=2y$ and $v(x,y)=-2x;$ $x\ge 0;y\ge 0.$ The speed of the fluid at the point $(x,y)$ is $s(x,y)=\sqrt{u{(x,y)}^{2}+v{(x,y)}^{2}}.$ Find $\frac{\partial s}{\partial x}$ and $\frac{\partial s}{\partial y}$ using the chain rule.

Let $u=u(x,y,z),$ where $x=x(w,t),y=y(w,t),z=z(w,t),w=w(r,s),\text{and}\,t=t(r,s).$ Use a tree diagram and the chain rule to find an expression for $\frac{\partial u}{\partial r}.$

What is the difference between a directional derivative and a partial derivative?

For what $\vec u$ is $D_{\vec u\,} f = f_x$?

For what $\vec u$ is $D_{\vec u\,} f = f_y$?

The gradient is _____ to level curves.

The gradient points in the direction of _____ increase.

It is generally more informative to view the directional derivative not as the result of a limit, but rather as the result of a _____ product.

In Exercises 7–12, a function $z=f(x,y)$ is given. Find $\nabla f$.

$f(x,y) = -x^2y+xy^2+xy$

In Exercises 7–12, a function $z=f(x,y)$ is given. Find $\nabla f$.

$\displaystyle f(x,y) = \sin x\cos y$

In Exercises 7–12, a function $z=f(x,y)$ is given. Find $\nabla f$.

$\displaystyle f(x,y) = \frac{1}{x^2+y^2+1}$

In Exercises 7–12, a function $z=f(x,y)$ is given. Find $\nabla f$.

$\displaystyle f(x,y) = -4x+3y$

In Exercises 7–12, a function $z=f(x,y)$ is given. Find $\nabla f$.

$\displaystyle f(x,y) = x^2+2y^2-xy-7x$

In Exercises 7–12, a function $z=f(x,y)$ is given. Find $\nabla f$.

$\displaystyle f(x,y) = x^2y^3-2x$

In Exercises 25–28, a function $w=F(x,y,z)$, a vector $\vec v$ and a point $P$ are given.

(a) Find $\nabla F(x,y,z)$. (b) Find $D_{\vec u\,}F$ at $P$, where $\vec u$ is the unit vector in the direction of $\vec v$.

$\displaystyle F(x,y,z) = 3x^2z^3+4xy-3z^2$, $\vec v = \left\langle 1,1,1\right\rangle$, $P = (3,2,1)$

In Exercises 25–28, a function $w=F(x,y,z)$, a vector $\vec v$ and a point $P$ are given.

(a) Find $\nabla F(x,y,z)$. (b) Find $D_{\vec u\,}F$ at $P$, where $\vec u$ is the unit vector in the direction of $\vec v$.

$\displaystyle F(x,y,z) = \sin(x)\cos(y)e^z$, $\vec v = \left\langle 2,2,1\right\rangle$, $P = (0,0,0)$

In Exercises 25–28, a function $w=F(x,y,z)$, a vector $\vec v$ and a point $P$ are given.

(a) Find $\nabla F(x,y,z)$. (b) Find $D_{\vec u\,}F$ at $P$, where $\vec u$ is the unit vector in the direction of $\vec v$.

$\displaystyle F(x,y,z) = x^2y^2-y^2z^2$, $\vec v = \left\langle -1,7,3\right\rangle$, $P = (1,0,-1)$