Vector Calculus

15.3 Assess Your Understanding
Printed Page 1000

Concepts and Vocabulary

True or False If $\nabla \! \ f=\mathbf{F}$ for some function $f$ that has continuous first-order partial derivatives, then $\mathbf{F}$ is a potential function for $f$ .

False

Multiple Choice If $\int_{C_{1}}(P\,dx+Q\,dy)=\int_{C_{2}}(P\,dx+Q\,dy)$ for any two piecewise-smooth curves $C_{1}$ and $C_{2}$ with the same end points lying entirely in a region $R,$ then $\int_{C}(P\,dx+Q\,dy)$ is said to be [(a) one-to-one, (b) singular, (c) independent of $P$ and $Q$ , (d) independent of the path.]

(d)

True or False The line integral $\int_{C} \mathbf{F} \, {\cdot}\, d\mathbf{r}$ , where $C$ is a piecewise smooth curve and $\mathbf{F}$ is a conservative vector field $\mathbf{,}$ is independent of the path.

True

Multiple Choice Suppose $\mathbf{F}=\mathbf{F}(x,y)$ is a conservative vector field and $\nabla\! \ f=\mathbf{F}$ . The function $f$ is called a(n) [(a) gradient function, (b) independent function, (c) potential function, (d) gravity function] for $\mathbf{F}$ .

(c)

A piecewise-smooth curve $C$ is ____________ if its initial point $(x_{0},y_{0})$ and its terminal point $(x_{1},y_{1})$ are the same.

Closed

If $\mathbf{F}$ is a conservative vector field on some open region $R$ , and $C$ is a closed, piecewise-smooth curve that lies entirely in $R$ , then $\int_{C}\mathbf{F}\,{\cdot}\, d\mathbf{r}=$ ____________.

True or False An open region $R$ is connected if there are two points $(x_{1},y_{1})$ and $(x_{2},y_{2})$ in $R$ that can be joined by a piecewise-smooth curve $C$ that lies entirely in $R$ .

False

True or False The converse of the Fundamental Theorem of Line Integrals is true if the open region $R$ is connected.

True

True or False Let $\mathbf{r}=\mathbf{r}(t)$ , $a\leq t\leq b$ , trace out a piecewise-smooth curve $C$ . Then $C$ is closed if there is a least one point where the curve intersects itself.

False

True or False A parabola is a simple curve.

True

True or False A region $R$ is called simply connected if it is closed.

False

Let $\mathbf{F}=\mathbf{F}(x,y)=P(x,y)\mathbf{i}+Q(x,y)\mathbf{j }$ be a vector field, where the functions $P$ and $Q$ are continuous on some simply connected,

open region $R$ . Suppose $\dfrac{\partial P}{\partial y}$ and $\dfrac{\partial Q}{\partial x}$ are also continuous on $R$ .

Then $\mathbf{F}(x,y)=P(x,y)\mathbf{i}+Q(x,y)\mathbf{j}$ is a conservative vector field on $R$ if and only if ____________.

$\displaystyle{{\partial P} \over {\partial y}} = \displaystyle{{\partial Q} \over {\partial x}}$

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Skill Building

Find the line integral $\int_{C}(e^{x}\cos y\,dx-e^{x}\sin y\,dy)$ for each curve $C$ :
1. $C$ consists of the line segments from $\left( 0,\dfrac{\pi }{3} \right)$ to $\left( 1,\dfrac{\pi }{3}\right)$ and from $\left( 1,\dfrac{ \pi }{3}\right)$ to $(1,\pi )$ .
2. $C$ consists of the line segment from $\left( 0,\dfrac{\pi }{3}\right)$ to $(1,\pi )$ .
3. $C$ consists of the line segments from $\left( 0,\dfrac{\pi }{3} \right)$ to $\left( 0,\dfrac{\pi }{2}\right)$ and from $\left( 0,\dfrac{\pi }{2}\right)$ to $(1,\pi )$ .

$- \displaystyle{1 \over 2} - e$
$- \displaystyle{1 \over 2} - e$
$- \displaystyle{1 \over 2} - e$

Find the line integral $\int_{C}(e^{y}\cos x\,dx+e^{y}\sin x\,dy)$ for each curve $C$ :
1. $C$ consists of the line segment from $(0,0)$ to $(\pi ,1)$ .
2. $C$ consists of the line segments from $(0,0)$ to $\left( \dfrac{\pi }{2},\dfrac{1}{2}\right)$ and from $\left( \dfrac{\pi }{2}, \dfrac{1}{2}\right)$ to $(\pi ,1)$ .
3. $C$ consists of the line segments from $(0,0)$ to $( 0,1)$ and from $( 0,1)$ to $(\pi ,1)$ .

In Problems 15–20, $\mathbf{F=F}(x,y) =P(x,y) \,\mathbf{i}+Q(x,y) \,\mathbf{j}$ is a conservative vector field since $\mathbf{F}$ is the gradient of $f(x,y)$ . Use this fact to find $\int_{C}[ P(x,y) dx+Q(x,y)\, dy]$ , where $C$ is any piecewise smooth curve joining the two given points.

$\int_{C} [ ( 2x-y)\, dx+(2y-x)\, dy]$ from $( 3,-1)$ to $( 5,4);\quad f(x,y) =x^{2}-xy+y^{2}$

$\int_{C} [( y^{2}+2x)\, dx+2x y\,dy]$ from $( 2,2)$ to $( -3,8);\quad (f(x,y) =xy^{2}+x^{2}$

$\int_{C} ( 3x^{2}y\,dx+x^{3}dy)$ from $(0,-1)$ to $( 3,4);\quad f(x,y) =x^{3}y$

108

$\int_{C} \big( 2xe^{x^{2}+y^{2}}dx+2y e^{x^{2}+y^{2}}dy\big)$ from $(0,0)$ to $(1,1); f(x,y) =e^{x^{2}+y^{2}}$

$\int_{C} \left( \dfrac{x}{x^{2}+y^{2}}dx+\dfrac{y}{x^{2}+y^{2}}dy\right)$ from $( 3,4)$ to $( 5,12)$ ; $f(x,y) =\ln \sqrt{x^{2}+y^{2}}$

$\ln 13 - \ln 5$

$\int_{C} \left( \dfrac{x}{\sqrt{x^{2}+y^{2}}}dx+\dfrac{y}{\sqrt{x^{2}+y^{2}}}dy\right)$ from $( 0,-4)$ to $( 3,4)$ ; $f(x,y) =\sqrt{x^{2}+y^{2}}$

In Problems 21–28, $\mathbf{F=F}(x,y) =P(x,y) \,\mathbf{i}+Q(x,y) \,\mathbf{j}$ is a conservative vector field. Use this fact to find $\int_{C}[ P(x,y)\, dx+Q(x,y)\, dy] ,$ where $C$ is any piecewise smooth curve joining the two given points.

$\int_{C}( x^{3}dx+y^{3}dy)$ from $(1,1)$ to $(2,4)$

$\displaystyle{{135} \over 2}$

$\int_{c}( 2x^{2}dx-4y\,dy)$ from $(-1,2)$ to $( 2,3)$

$\int_{C}[ ( x^{2}+y^{2})\, dx+ ( 2xy+\sin y)\, dy]$ from $(0,0)$ to $( 2,0)$

$\displaystyle{8 \over 3}$

$\int_{C}[ ( x-\cos y)\, dx+x\sin y\,dy]$ from $\left( 1,\dfrac{\pi }{2}\right)$ to $( 2,\pi )$

$\int_{C}( y^{2}e^{x}\,dx+2ye^{x}dy)$ from $(0,0)$ to $(\ln 2,2)$

$\int_{C}\left[ xe^{y}dx+\dfrac{1}{2}( x^{2}-4) e^{y}dy\right]$ from $(0,0)$ to $(2,1)$

$\int_{C}( ye^{x}\,dx+ e^{x}\,dy)$ on a closed curve $C$

$\int_{C}( y\cos x\,dx+\sin x\,dy)$ on a closed curve $C$

In Problems 29–32, $\mathbf{F=F}(x,y) =P(x,y) \,\mathbf{i}+Q(x,y) \,\mathbf{j}$ is a conservative vector field. Find a potential function $f$ for $\mathbf{F}$ .

$\mathbf{F}(x,y) =(5x-2y+1) \,\mathbf{i}+( 4-2x) \,\mathbf{j}$

$f(x,y) = \displaystyle{5 \over 2}x^2 - 2xy + x + 4y + K$

$\mathbf{F}(x,y) =( 4xy) \,\mathbf{i}+(2x^{2}+y) \,\mathbf{j}$

$\mathbf{F}(x,y) =( \ln y+2x) \,\mathbf{i}+\!\left( \dfrac{x}{y}\right)\, \mathbf{j}$

$f(x,y) = x\ln y + x^2 + K$

$\mathbf{F}(x,y) =(ye^{x}) \,\mathbf{i}+e^{x}\,\mathbf{j}$

In Problems 33–36, determine whether each vector field $\mathbf{F}$ is a conservative vector field.

$\mathbf{F}(x,y)=x^{2}\,\mathbf{i}+y^{2}\mathbf{j}$

$f$ is a conservative vector field.

$\mathbf{F}(x,y)=xy\mathbf{i}+xy\mathbf{j}$

$\mathbf{F}(x,y)=xe^{y}\,\mathbf{i}+\dfrac{1}{2} x^{2}e^{y}\,\mathbf{j}$

$f$ is a conservative vector field.

$\mathbf{F}(x,y)=(x^{2}+y^{2})\mathbf{i}+(2xy-\sin y)\mathbf{j}$

In Problems 37–42:

(a) Show that the line integral $\int_{C}(P\,dx+Q\,dy)$ is independent of the path.
(b) Find a function $f$ so that $\nabla\! \ f=P(x,y)\mathbf{i}+Q(x,y)\mathbf{j}$ .
(c) Find $\int_{C}(P\,dx+Q\,dy)$ for the given points.

$\int_{C}(x\,dx+y\,dy),$ where $C$ is a piecewise-smooth curve joining the points $(1,3)$ and $(2,5)$

$P, Q, \dfrac{\partial P}{\partial y}$ , and $\dfrac{\partial Q}{\partial x}$ are continuous everywhere in the xy-plane, and $\displaystyle{{\partial P} \over {\partial y}} = \displaystyle{{\partial Q} \over {\partial x}} = 0$ .
$f(x,y) = \displaystyle{1 \over 2}x^2 + \displaystyle{1 \over 2}y^2 + K$
$\displaystyle{{19} \over 2}$

$\int_{C}[2xy\,dx+(x^{2}+1)\,dy],$ where $C$ is a piecewise-smooth curve joining the points $(1,-4)$ and $(-2,3)$

$\int_{C}[(x^{2}+3y)\,dx+3x\,dy],$ where $C$ is a piecewise-smooth curve joining the points $(1,2)$ and $(-3,5)$

$P, Q, \dfrac{\partial P}{\partial y}$ , and $\dfrac{\partial Q}{\partial x}$ are continuous everywhere in the xy-plane, and $\displaystyle{{\partial P} \over {\partial y}} = \displaystyle{{\partial Q} \over {\partial x}} = 3$ .
$f(x,y) = \displaystyle{1 \over 3}x^3 + 3xy + K$
$- \displaystyle{{181} \over 3}$

$\int_{C}[(2x+y+1)\,dx+(x+3y+2)\,dy],$ where $C$ is a piecewise-smooth curve joining the points $(0,0)$ and $(1,2)$

$\int_{C}[(4x^{3}+20xy^{3}-3y^{4})\,dx+(30x^{2}y^{2}-12xy^{3}+5y^{4})\,dy]$ $,$ where $C$ is a piecewise-smooth curve joining the points $(0,0)$ and $(1,1)$

$P, Q, \dfrac{\partial P}{\partial y}$ , and $\dfrac{\partial Q}{\partial x}$ are continuous everywhere in the xy-plane, and $\displaystyle{{\partial P} \over {\partial y}} \,{=}\, \displaystyle{{\partial Q} \over {\partial x}} \,{=}\, 60xy^2 - 12y^3$ .
$f(x,y) = x^4 + 10x^2y^3 - 3xy^4 + y^5 + K$
(c) 9

$\int_{C}[(2yx^{-1})\,dx+(\ln x^{2})\,dy],$ where $C$ is a piecewise-smooth curve in the first quadrant joining the points $(1,1)$ and $(5,5)$

In Problems 43–54:

(a) Show that each line integral $\int_{C}\mathbf{F}\,{\cdot}\, d\mathbf{r}=\int_{C}(P\,dx+Q\,dy)$ is independent of the path in the entire $xy$ -plane.
(b) Find a potential function $f$ for $\mathbf{F}$ .

$\int_{C}(3x^{2}y^{2}\,dx+2x^{3}y\,dy)$

$P, Q, \dfrac{\partial P}{\partial y}$ , and $\dfrac{\partial Q}{\partial x}$ are continuous everywhere in the xy-plane, and $\displaystyle{{\partial P} \over {\partial y}} = \displaystyle{{\partial Q} \over {\partial x}} = 6x^2y$ .
$f(x,y) = x^3y^2 + K$

$\int_{C}[(2x+y)\,dx+(x-2y)\,dy]$

$\int_{C}[(x+3y)\,dx+3x\,dy]$

$P, Q, \dfrac{\partial P}{\partial y}$ , and $\dfrac{\partial Q}{\partial x}$ are continuous everywhere in the xy-plane, and $\displaystyle{{\partial P} \over {\partial y}} = \displaystyle{{\partial Q} \over {\partial x}} = 3$ .
$f(x,y) = \displaystyle{1 \over 2}x^2 + 3xy + K$

$\int_{C}[(2x+y)\,dx+(2y+x)\,dy]$

1002

$\int_{C}[(2xy-y^{2})\,dx+(x^{2}-2xy)\,dy]$

$P, Q, \dfrac{\partial P}{\partial y}$ , and $\dfrac{\partial Q}{\partial x}$ are continuous everywhere in the xy-plane, and $\displaystyle{{\partial P} \over {\partial y}} = \displaystyle{{\partial Q} \over {\partial x}} = 2x - 2y$ .
$f(x,y) = x^2y - xy^2 + K$

$\int_{C}[y^{2}\,dx+(2yx-e^{y})\,dy]$

$\int_{C}[(x^{2}-x+y^{2})\,dx-(ye^{y}-2xy)\,dy]$

$P, Q, \dfrac{\partial P}{\partial y}$ , and $\dfrac{\partial Q}{\partial x}$ are continuous everywhere in the xy-plane, and $\displaystyle{{\partial P} \over {\partial y}} = \displaystyle{{\partial Q} \over {\partial x}} = 2y$ .
$f(x,y) = \displaystyle{1 \over 3}x^3 - \displaystyle{1 \over 2}x^2 + xy^2 + e^y - ye^y + K$

$\int_{C}[(3x^{2}y+xy^{2}+e^{x})\,dx+(x^{3}+x^{2}y+\sin y)\,dy]$

$\int_{C}[(y\cos x-2\sin y)\,dx-(2x\cos y-\sin x)\,dy]$

$P, Q, \dfrac{\partial P}{\partial y}$ , and $\dfrac{\partial Q}{\partial x}$ are continuous everywhere in the xy-plane, and $\displaystyle{{\partial P} \over {\partial y}} = \displaystyle{{\partial Q} \over {\partial x}} = \cos x - 2\cos y$ .
$f(x,y) = y\sin x - 2x\sin y + K$

$\int_{C}[(e^{x}\sin y+2y\sin x)\,dx+(e^{x}\cos y-2\cos x)\,dy]$

$\int_{C}[(2x+y\cos x)\,dx+\sin x\,dy]$

$P, Q, \dfrac{\partial P}{\partial y}$ , and $\dfrac{\partial Q}{\partial x}$ are continuous everywhere in the xy-plane, and $\displaystyle{{\partial P} \over {\partial y}} = \displaystyle{{\partial Q} \over {\partial x}} = \cos x$ .
$f(x,y) = x^2 + y\sin x + K$

$\int_{C}[ (\cos y-\cos x)\,dx+(e^{y}-x\sin y)\,dy]$

Applications and Extensions

Show that $\int_{C}\left( -\dfrac{y}{x^{2}+y^{2}}\,dx+\dfrac{x}{x^{2}+y^{2}}\,dy\right)$ is independent of the path in the rectangle $R$ whose vertices are $\left( \dfrac{1}{a},-a\right)$ , $(a,-a)$ , $\left(\dfrac{1}{a},-\dfrac{1}{a}\right)$ , and $\left( a,-\dfrac{1}{a}\right),$ $a>1$ . Find a potential function $f$ for $\mathbf{F}$ .

$P, Q, \dfrac{\partial P}{\partial y}$ , and $\dfrac{\partial Q}{\partial x}$ are continuous everywhere in the rectangle, and $\displaystyle{{\partial P} \over {\partial y}} = \displaystyle{{\partial Q} \over {\partial x}} = \displaystyle{{y^2 - x^2} \over {(x^2 + y^2)^2}}$ ; $f(x,y) = \tan ^{ - 1}\left(\displaystyle{y \over x}\right) + K$ .

Given the constant vector $\mathbf{c}$ , show that $\mathbf \nabla (\mathbf{c}\,{\cdot}\, \mathbf{r})=\mathbf{c}$ , where $\mathbf{r}=x \mathbf{i}+y\mathbf{j}+z\mathbf{k}$ . Use the result to prove that $\int_{C} \mathbf{c}\,{\cdot}\, d\mathbf{r}=0$ , where $C$ is any closed piecewise smooth curve in space.

Suppose that $\mathbf{F}(x,y)$ is a force field directed toward the origin with magnitude inversely proportional to the square of the distance from the origin. (Such “inverse square law” forces are common in nature. See Examples 4 and 5 in Section 15.1, pp. 975-976.)
1. Show that $\mathbf{F}$ is a conservative vector field.
2. Find a potential function $f$ for $\mathbf{F}$ .

(a) See Student Solutions Manual.
$f(x,y) = \displaystyle{k \over {\sqrt {x^2 + y^2} }} + K$

Suppose $f$ and $g$ are differentiable functions of one variable. Show that $\int_{C}[f(x)\,dx+g(y)\,dy]=0$ , where $C$ is any circle in the $xy$ -plane.

Challenge Problem

Let $f$ and $g$ have continuous partial derivatives in a plane region $R$ , and let $C$ be a piecewise-smooth curve in $R$ going from $A$ to $B$ . Show that $\int_{C}f\,{\nabla }g\,{\cdot}\, d\mathbf{r}=f(B)g(B)-f(A)g(A)-\int_{C}g\, \nabla\! \ f\,{\cdot}\, d\mathbf{r}$

See Student Solutions Manual.

15.3 Assess Your UnderstandingPrinted Page 1000

15.3 Assess Your Understanding
Printed Page 1000