Chapter 6: Vector CalculusContent check — Calculus Vol. 3, Chapter 6: Vector Calculus

Let $\text{G}(x,y)=$ $x^{2}y\text{i}-$ $(x+y)\text{j}$ be a vector field in ${ℝ}^2.$ What vector is associated with the point $\left(\mathrm{-2},3\right)?$

Draw the radial field $\text{F}(x,y)=$ $-$ $\frac{x}{3}\text{i}-$ $\frac{y}{3}\text{j}.$

Sketch vector field $\text{F}(x,y)=$ $\langle \mathrm{-2}y,$ $2x\rangle .$ Is the vector field radial, rotational, or neither?

Vector field $v(x,y)=$ $\langle 4\left|x\right|,$ $1\rangle$ models the velocity of water on the surface of a river. What is the speed of the water at point $\left(2,3\right)?$ Use meters per second as the units.

Is vector field $\text{F}(x,y)=$ $\langle \text{−}y,$ $x\rangle$ a unit vector field?

Sketch vector field $\text{G}(x,y,z)=$ $\langle 2,$ $\frac{z}{2},$ $1\rangle .$

The mass of asteroid 1 is 750,000 kg and the mass of asteroid 2 is 130,000 kg. Assume asteroid 1 is located at the origin, and asteroid 2 is located at $\left(15,\mathrm{-5},10\right),$ measured in units of 10 to the eighth power kilometers. Given that the universal gravitational constant is $G=$ $6.67384\times {10}^{\mathrm{-11}}{\text{m}}^3{\text{kg}}^{\mathrm{-1}}{\text{s}}^{\mathrm{-2}},$ find the gravitational force vector that asteroid 1 exerts on asteroid 2.

Use technology to plot the gradient vector field of $f(x,y)=$ $\text{sin}x\text{cos}y.$

Is $f(x,y,z)=$ $x^2\text{cos}(yz)+$ $y^2{z}^2$ a potential function for $\text{F}(x,y,z)=$ $\langle 2x\text{cos}(yz),$ $\text{−}{x}^{2}z\text{sin}(yz)+$ $2y{z}^2,$ $y^2\rangle ?$

Verify that $f(x,y)=$ $x^3{y}^2+$ $1$ is a potential function for velocity field $\text{v}(x,y)=$ $⟨3x^2{y}^2,$ $2x^{3}y⟩.$

Show that the vector field $\text{F}(x,y)=$ $y\text{i}-$ $x^{2}xy\text{j}$ is not conservative.

Is vector field $G(x,y,z)=$ $\langle y,$ $x,$ $xyz\rangle$ conservative?

The domain of vector field $\text{F}=\text{F}\left(x,y\right)$ is a set of points $\left(x,y\right)$ in a plane, and the range of F is a set of what in the plane?

For the following exercises, determine whether the statement is true or false.

Vector field $\text{F}=$ $\frac{\langle y,x\rangle }{\sqrt{x^2+y^2}}$ is constant in direction and magnitude on a unit circle.

For the following exercises, describe each vector field by drawing some of its vectors.

[T] $\text{F}(x,y)=$ $x\text{i}+$ $y\text{j}$

For the following exercises, describe each vector field by drawing some of its vectors.

[T] $\text{F}(x,y)=$ $x\text{i}-$ $y\text{j}$

For the following exercises, describe each vector field by drawing some of its vectors.

[T] $\text{F}(x,y)=$ $2x\text{i}+$ $3y\text{j}$

For the following exercises, describe each vector field by drawing some of its vectors.

[T] $\text{F}(x,y)=$ $y\text{i}+$ $\text{sin}x\text{j}$

For the following exercises, describe each vector field by drawing some of its vectors.

[T] $\text{F}(x,y,z)=$ $2x\text{i}-$ $2y\text{j}-$ $2z\text{k}$

For the following exercises, find the gradient vector field of each function $f.$

$f(x,y)=$ $x\text{sin}y+$ $\text{cos}y$

For the following exercises, find the gradient vector field of each function $f.$

$f(x,y,z)=$ $x^{2}y+$ $xy+$ $y^{2}z$

For the following exercises, find the gradient vector field of each function $f.$

$f(x,y)=$ $\text{ln}(1+x^2+2y^2)$

For the following exercises, find the gradient vector field of each function $f.$

What is vector field $\text{F}\left(x,y\right)$ with a value at $\left(x,y\right)$ that is of unit length and points toward $\left(1,0\right)?$

For the following exercises, write formulas for the vector fields with the given properties.

All vectors point toward the origin and have constant length.

For the following exercises, write formulas for the vector fields with the given properties.

Give a formula $\text{F}(x,y)=$ $M(x,y)\text{i}+$ $N(x,y)\text{j}$ for the vector field in a plane that has the properties that $\text{F}=0$ at $\left(0,0\right)$ and that at any other point $\left(a,b\right),$ F is tangent to circle $x^2+y^2=a^2+b^2$ and points in the clockwise direction with magnitude $F=\sqrt{a^2+b^2}.$

For the following exercises, write formulas for the vector fields with the given properties.

Find a formula for vector field $\text{F}(x,y)=$ $M(x,y)\text{i}+$ $N(x,y)\text{j}$ given the fact that for all points $(x,y),$ F points toward the origin and $F=\frac{10}{x^2+y^2}.$

For the following exercises, assume that an electric field in the xy-plane caused by an infinite line of charge along the x-axis is a gradient field with potential function $V(x,y)=$ $c\text{ln}\left(\frac{r_0}{\sqrt{x^2+y^2}}\right),$ where $c>0$ is a constant and $r_0$ is a reference distance at which the potential is assumed to be zero.

Show that the electric field at a point in the xy-plane is directed outward from the origin and has magnitude $E=\frac{c}{r}$, where $r=\sqrt{x^2+y^2}$.

A flow line (or streamline) of a vector field $\text{F}$ is a curve $\text{r}\left(t\right)$ such that $d\text{r}\text{/}dt=$ $\text{F}\left(\text{r}\right(t\left)\right).$ If $\text{F}$ represents the velocity field of a moving particle, then the flow lines are paths taken by the particle. Therefore, flow lines are tangent to the vector field. For the following exercises, show that the given curve $\text{c}\left(t\right)$ is a flow line of the given velocity vector field $\text{F}\left(x,y,z\right).$

$\text{c}\left(t\right)=$ $\text{sin}t,\text{cos}t,e^t;$ $\text{F}(x,y,z)=$ $⟨y,$ $\text{−}x,$ $z⟩$

For the following exercises, let $\text{F}=x\text{i}+y\text{j},$ $\text{G}=\text{−}y\text{i}+x\text{j},$ and $\text{H}=x\text{i}-y\text{j}.$ Match F, G, and H with their graphs.

For the following exercises, let $\text{F}=x\text{i}+y\text{j},$ $\text{G}=\text{−}y\text{i}+x\text{j},$ and $\text{H}=x\text{i}–y\text{j}.$ Match the vector fields in a through d with their graphs.

For the following exercises, let $\text{F}=x\text{i}+y\text{j},$ $\text{G}=\text{−}y\text{i}+x\text{j},$ and $\text{H}=x\text{i}–y\text{j}.$ Match the vector fields in a through d with their graphs.

Find the value of ${\displaystyle {\int }_C\left(x+y\right)ds,}$ where $C$ is the curve parameterized by $x=t,$ $y=t,$ $0\le t\le 1.$

Evaluate ${\displaystyle {\int }_C\left(x^2+y^2+z\right)ds,}$ where C is the curve with parameterization $\text{r}\left(t\right)=$ $⟨\text{sin}\left(3t\right),$ $\text{cos}\left(3t\right)\text{,t}⟩,$ $0\le t\le 2\mathrm{\pi }.$

Evaluate line integral ${\displaystyle {\int }_C\left(x^2+yz\right)ds},$ where $C$ is the line with parameterization $\text{r}(t)=$ $\langle 2t,$ $5t,$ $\text{−}t\rangle ,$ $0\le t\le 10.$ Reparameterize C with parameterization $\text{s}(t)=$ $\langle 4t,$ $10t,$ $\mathrm{-2}t\rangle ,$ $0\le t\le 5,$ recalculate line integral ${\displaystyle {\int }_C\left(x^2+yz\right)ds},$ and notice that the change of parameterization had no effect on the value of the integral.

Find the length of a wire with parameterization $\text{r}\left(t\right)=$ $\langle 3t+$ $1,$ $4-$ $2t,$ $5+$ $2t\rangle ,$ $0\le t\le 4.$

Let $\text{F}=x\text{i}+y\text{j}$ be a vector field and let C be the curve with parameterization $\langle t,t^2\rangle$ for $0\le t\le 2.$ Which is greater: ${\displaystyle {\int }_C\text{F}·\text{T}ds}$ or ${\displaystyle {\int }_{\text{−}C}\text{F}·\text{T}ds}?$

Find the value of ${\displaystyle {\int }_{C}4xdx+zdy+4y^{2}dz,}$ where $C$ is the curve parameterized by $\text{r}\left(t\right)=$ $\langle 4\text{cos}\left(2t\right),$ $2\text{sin}\left(2t\right),$ $3\rangle ,$ $0\le t\le \frac{\pi }{4}.$

Calculate line integral ${\displaystyle {\int }_C\text{F}}·d\text{r},$ where F is vector field $\langle y^2,2xy+1\rangle$ and C is a triangle with vertices $\left(0,0\right),$ $\left(4,0\right),$ and $\left(0,5\right),$ oriented counterclockwise.

Calculate the mass of a spring in the shape of a helix parameterized by $\text{r}\left(t\right)=$ $\langle \text{cos}t,$ $\text{sin}t,$ $t\rangle ,$ $0\le t\le 6\pi ,$ with a density function given by $\rho (x,y,z)=$ $x+$ $y+$ $z$ kg/m.

Calculate the flux of $\text{F}=\langle x+y,2y\rangle$ across the line segment from $(0,0)$ to $(2,3),$ where the curve is oriented from left to right.

Calculate the circulation of $\text{F}(x,y)=$ $\langle -$ $\frac{y}{x^2+y^2},$ $\frac{x}{x^2+y^2}\rangle$ along a unit circle oriented counterclockwise.

Calculate the work done by field $\text{F}(x,y)=$ $\langle 2x,$ $3y\rangle$ on a particle that traverses the unit circle. Assume the particle begins its movement at $(\mathrm{-1},0).$

True or False? Line integral ${\displaystyle {\int }_C^{}f(x,y)ds}$ is equal to a definite integral if C is a smooth curve defined on $\left[a,b\right]$ and if function $f$ is continuous on some region that contains curve C.

True or False? ${\displaystyle {\int }_{\text{−}C}^{}(Pdx+Qdy)={\displaystyle {\int }_C^{}(Pdx-Qdy)}}$

True or False? If C is given by $x(t)=$ $t\text{,}y(t)=$ $t\text{, 0}\le \text{t}\le 1,$ then ${\displaystyle {\int }_C^{}xyds={\displaystyle {\int }_0^1{t}^{2}dt}}.$

For the following exercises, use a computer algebra system (CAS) to evaluate the line integrals over the indicated path.

[T] ${\displaystyle {\int }_C^{}(x-y)ds}$

$C\text{:}\text{r}(t)=$ $4t\text{i}+$ $3t\text{j}$ when $0\le t\le 2$

For the following exercises, use a computer algebra system (CAS) to evaluate the line integrals over the indicated path.

[T] Evaluate ${\displaystyle {\int }_C^{}x{y}^{4}ds},$ where C is the right half of circle $x^2+y^2=16$ and is traversed in the clockwise direction.

For the following exercises, find the work done.

Find the work done by vector field $\text{F}(x,y,z)=$ $x\text{i}+$ $3xy\text{j}-$ $(x+z)\text{k}$ on a particle moving along a line segment that goes from $(1,4,2)$ to $(0,5,1).$

For the following exercises, find the work done.

Find the work done by force field $\text{F}(x,y,z)=$ $-$ $\frac{1}{2}x\text{i}-$ $\frac{1}{2}y\text{j}+$ $\frac{1}{4}\text{k}$ on a particle as it moves along the helix $\text{r}(t)=$ $\text{cos}t\text{i}+$ $\text{sin}t\text{j}+$ $t\text{k}$ from point $(1,0,0)$ to point $\left(\mathrm{-1},0,3\pi \right).$

For the following exercises, find the work done.

Find the work done by force $\text{F}(x,y)=$ $2y\text{i}+$ $3x\text{j}+$ $(x+y)\text{k}$ in moving an object along curve $\text{r}(t)=$ $\text{cos}(t)\text{i}+$ $\text{sin}(t)\text{j}+$ $\frac{1}{6}\text{k},$ where $0\le t\le 2\pi .$

For the following exercises, evaluate the line integrals.

Evaluate ${\displaystyle {\int }_C\text{F}·d\text{r}},$ where $\text{F}(x,y)=\mathrm{-1}\text{j},$ and C is the part of the graph of $y=\frac{1}{2}{x}^3-x$ from $(2,2)$ to $(\mathrm{-2},\mathrm{-2}).$

For the following exercises, evaluate the line integrals.

Evaluate ${\displaystyle {\int }_C^{}yzdx+xzdy+xydz}$ over the line segment from $(1,1,1)$ to $(3,2,0).$

For the following exercises, evaluate the line integrals.

[T] Use a computer algebra system to evaluate the line integral ${\displaystyle {\int }_C{y}^{2}dx+xdy},$ where C is the arc of the parabola $x=4-y^2$ from (−5, −3) to (0, 2).

For the following exercises, evaluate the line integrals.

[T] Use a CAS to evaluate line integral ${\displaystyle {\int }_C^{}xydx+ydy}$ over path C given by $x=2t\text{,}y=10t\text{,}$ where $0\le t\le 1.$

For the following exercises, evaluate the line integrals.

[T] Use a CAS to evaluate ${\displaystyle {\int }_C^{}\frac{y}{2x^2-y^2}ds},$ where C is $x=$ $t\text{,}y=$ $t\text{,}1\le t\le 5.$

In the following exercises, find the work done by force field F on an object moving along the indicated path.

$\text{F}(x,y)=$ $\text{−}x\text{i}-$ $2y\text{j}$

$C\text{:}y=$ $x^3\text{from (0, 0) to (2, 8)}$

In the following exercises, find the work done by force field F on an object moving along the indicated path.

$\text{F}(x\text{,}y\text{,}z)=$ $x\text{i}+$ $y\text{j}-$ $5z\text{k}$

$\text{C}\text{:}\text{r}(t)=$ $2\text{cos}t\text{i}+$ $2\text{sin}t\text{j}+$ $t\text{k}\text{,}0\le t\le 2\pi$

In the following exercises, find the work done by force field F on an object moving along the indicated path.

Compute the work done by force $\text{F}(x,y,z)=$ $2x\text{i}+$ $3y\text{j}-$ $z\text{k}$ along path $\text{r}(t)=$ $t\text{i}+$ $t^2\text{j}+$ $t^3\text{k},$ where $0\le t\le 1.$

In the following exercises, find the work done by force field F on an object moving along the indicated path.

Force $\text{F}(x,y,z)=$ $zy\text{i}+$ $x\text{j}+$ $z^{2}x\text{k}$ acts on a particle that travels from the origin to point (1, 2, 3). Calculate the work done if the particle travels:

a. along the path $(0,0,0)\to$ $(1,0,0)\to$ $(1,2,0)\to$ $(1,2,3)$ along straight-line segments joining each pair of endpoints;

b. along the straight line joining the initial and final points.

c. Is the work the same along the two paths?

In the following exercises, find the work done by force field F on an object moving along the indicated path.

How much work is required to move an object in vector field $\text{F}(x,y)=$ $y\text{i}+$ $3x\text{j}$ along the upper part of ellipse $\frac{x^2}{4}+y^2=1$ from (2, 0) to $(\mathrm{-2},0)?$

In the following exercises, find the work done by force field F on an object moving along the indicated path.

Evaluate the line integral of scalar function $xy$ along parabolic path $y=x^2$ connecting the origin to point (1, 1).

In the following exercises, find the work done by force field F on an object moving along the indicated path.

Find ${\displaystyle {\int }_C^{}{y}^{2}dx+\left(xy-x^2\right)dy}$ along C: $y^2=9x$ from (0, 0) to (1, 3).

For the following exercises, use a CAS to evaluate the given line integrals.

[T] Evaluate line integral ${\displaystyle {\int }_{\gamma }^{}x{e}^{y}ds}$ where, $\gamma$ is the arc of curve $x=e^y$ from $(1,0)$ to $(e,1).$

For the following exercises, use a CAS to evaluate the given line integrals.

[T] Evaluate line integral ${\displaystyle {\int }_{\gamma }^{}\left(y^2-xy\right)ds},$ where $\gamma$ is curve $y=\text{ln}x$ from (1, 0) toward $(e\text{,}1).$

For the following exercises, use a CAS to evaluate the given line integrals.

[T] Evaluate ${\displaystyle {\int }_C^{}\text{F}·d\text{r}},$ where $\text{F}(x,y,z)=$ $x^{2}y\text{i}+$ $(x-z)\text{j}+$ $xyz\text{k}$ and

C: $\text{r}(t)=$ $t\text{i}+$ $t^2\text{j}+$ $2\text{k}\text{,}0\le t\le 1.$

For the following exercises, use a CAS to evaluate the given line integrals.

Find the line integral of $\text{F}(x,y,z)=$ $12x^2\text{i}-$ $5xy\text{j}+$ $xz\text{k}$ over path C defined by $y=x^2,$ $z=x^3$ from point (0, 0, 0) to point (2, 4, 8).

For the following exercises, find the flux.

Compute the flux of $\text{F}=x^2\text{i}+y\text{j}$ across a line segment from (0, 0) to (1, 2).

For the following exercises, find the flux.

Let $\text{F}=5\text{j}$ and let C be curve $y=0,0\le x\le 4.$ Find the flux across C.

For the following exercises, find the flux.

Let $\text{F}=$ $(x^2+y^3)\text{i}+$ $(2xy)\text{j}.$ Calculate flux F orientated counterclockwise across curve C: $x^2+y^2=9.$

For the following exercises, find the flux.

A spring is made of a thin wire twisted into the shape of a circular helix $x=$ $2\text{cos}t\text{,}y=$ $2\text{sin}t\text{,}z=$ $t.$ Find the mass of two turns of the spring if the wire has constant mass density.

For the following exercises, find the flux.

An object moves in force field $\text{F}(x,y,z)=$ $y^2\text{i}+$ $2(x+1)y\text{j}$ counterclockwise from point (2, 0) along elliptical path $x^2+4y^2=4$ to $(\mathrm{-2},0),$ and back to point (2, 0) along the x-axis. How much work is done by the force field on the object?

For the following exercises, find the flux.

If an inverse force field F is given by $\text{F}(x,y,z)=$ $\frac{k}{{\Vert \text{r}\Vert }^3}\text{r},$ where k is a constant, find the work done by F as its point of application moves along the x-axis from $A(1,0,0)\text{to}B(2,0,0).$

Is the curve given by parameterization $\text{r}\left(t\right)=$ $\langle 2\text{cos}t,$ $3\text{sin}t\rangle ,$ $0\le t\le 6\pi ,$ a simple closed curve?

Is the region in the below image connected? Is the region simply connected?

Given that $f(x,y)=$ ${\left(x-1\right)}^{2}y+$ ${\left(y+1\right)}^{2}x$ is a potential function for $\text{F}=$ $\langle 2xy-$ $2y+$ ${\left(y+1\right)}^2,$ ${\left(x-1\right)}^2+$ $2yx+$ $2x\rangle ,$ calculate integral ${\displaystyle {\int }_C\text{F}·d\text{r}},$ where C is the lower half of the unit circle oriented counterclockwise.

Show that $\text{F}(x,y)=$ $\langle xy,$ $x^2{y}^2\rangle$ is not path independent by considering the line segment from $(0,0)$ to $(2,2)$ and the piece of the graph of $y=\frac{x^2}{2}$ that goes from $(0,0)$ to $(2,2).$

Find a potential function for $\text{F}(x,y)=$ $\langle e^x{y}^3+$ $y,$ $3e^x{y}^2+$ $x\rangle .$

Find a potential function for $\text{F}(x,y,z)=$ $\langle 12x^2,$ $\text{cos}y\text{cos}z,$ $1-$ $\text{sin}y\text{sin}z\rangle .$

Find a potential function $f$ for the three-dimensional gravitational force $\text{F}(x,y,z)=$ $\langle \frac{\text{−}Gx}{{\left(x^2+y^2+z^2\right)}^{3\text{/}2}},$ $\frac{\text{−}Gy}{{\left(x^2+y^2+z^2\right)}^{3\text{/}2}},$ $\frac{\text{−}Gz}{{\left(x^2+y^2+z^2\right)}^{3\text{/}2}}\rangle .$

Determine whether $\text{F}(x,y)=$ $\langle \text{sin}x\text{cos}y,$ $\text{cos}x\text{sin}y\rangle$ is conservative.

Calculate integral ${\displaystyle {\int }_C\text{F}·d\text{r}},$ where $\text{F}(x,y)=$ $\langle \text{sin}x\text{sin}y,$ $5-$ $\text{cos}x\text{cos}y\rangle$ and C is a semicircle with starting point $(0,\pi )$ and endpoint $(0,\text{−}\pi ).$

Let $\text{F}(x,y)=$ $\langle 4x^3{y}^4,$ $4x^4{y}^3\rangle ,$ and suppose that a particle moves from point $(4,4)$ to $(1,1)$ along any smooth curve. Is the work done by F on the particle positive, negative, or zero?

True or False? If vector field F is conservative on the open and connected region D, then line integrals of F are path independent on D, regardless of the shape of D.

True or False? Vector field $\text{F}(x,y,z)=$ $\left(y\text{sin}z\right)\text{i}+$ $\left(x\text{sin}z\right)\text{j}+$ $\left(xy\text{cos}z\right)\text{k}$ is conservative.

Verify the Fundamental Theorem of Line Integrals for ${\displaystyle {\int }_C\text{F}·d\text{r}}$ by computing the integral using a parameterization and, separately, by finding a potential function for the case when $\text{F}(x,y)=$ $(2x+2y)\text{i}+$ $(2x+2y)\text{j}$ and C is a portion of circle $x^2+y^2=25$ oriented counterclockwise from (5, 0) to (3, 4).

[T] Evaluate line integral ${\displaystyle {\int }_C^{}\text{F}·d\text{r}},$ where $\text{F}(x,y)=$ $(e^x\text{sin}y-y)\text{i}+$ $(e^x\text{cos}y-x-2)\text{j},$ and C is the path given by $r(t)=$ $\left[t^3\text{sin}\frac{\pi t}{2}\right]\text{i}-$ $\left[\frac{\pi }{2}\text{cos}\right(\frac{\pi t}{2}+\frac{\pi }{2}\left)\right]\text{j}$ for $0\le t\le 1.$

For the following exercises, determine whether the vector field is conservative and, if it is, find the potential function.

$\text{F}(x,y)=$ $(\text{−}y+e^x\text{sin}y)\text{i}+$ $\left[\right(x+2\left)e^x\text{cos}y\right]\text{j}$

For the following exercises, determine whether the vector field is conservative and, if it is, find the potential function.

$\text{F}(x,y)=$ $(6x+5y)\text{i}+$ $(5x+4y)\text{j}$

For the following exercises, determine whether the vector field is conservative and, if it is, find the potential function.

$\text{F}(x,y)=$ $[y{e}^x+\text{sin}(y)]\text{i}+$ $[e^x+x\text{cos}(y)]\text{j}$

For the following exercises, evaluate the line integrals using the Fundamental Theorem of Line Integrals.

${\displaystyle {\int }_C(2ydx+2xdy)},$ where C is the line segment from (0, 0) to (4, 4)

For the following exercises, evaluate the line integrals using the Fundamental Theorem of Line Integrals.

Find the conservative vector field for the potential function

$$f(x,y)=5x^2+3xy+{10y}^2.$$

For the following exercises, determine whether the vector field is conservative and, if so, find a potential function.

$\text{F}(x,y)=$ $\left(e^x\text{cos}y\right)\text{i}+$ $6\left(e^x\text{sin}y\right)\text{j}$

For the following exercises, determine whether the vector field is conservative and, if so, find a potential function.

$\text{F}(x,y\text{,}z)=$ $\left(y{e}^z\right)\text{i}+$ $\left(x{e}^z\right)\text{j}+$ $\left(xy{e}^z\right)\text{k}$

For the following exercises, determine whether the vector field is conservative and, if so, find a potential function.

$\text{F}(x,y\text{,}z)=$ $(–\frac{1}{y})\text{i}+$ $\left(\frac{x}{y^2}\right)\text{j}+$ $(2z-1)\text{k}$

For the following exercises, determine whether the vector field is conservative and, if so, find a potential function.

$\text{F}(x,y\text{,}z)=$ $\left(2xy\right)\text{i}+$ $(x^2+2yz)\text{j}+$ $y^2\text{k}$

For the following exercises, determine whether the vector field is conservative and, if so, find a potential function.

$\text{F}(x,y)=$ $\left(y{e}^{x^2}\right)\text{i}+$ $\left(x^2{e}^{y^2}\right)\text{j}$

For the following exercises, evaluate the integral using the Fundamental Theorem of Line Integrals.

[T] Evaluate ${\displaystyle {\int }_C\nabla f·d\text{r}},$ where $f(x,y)=$ $xy+$ $e^x$ and C is a straight line from $\left(0,0\right)$ to $\left(2,1\right).$

For the following exercises, evaluate the integral using the Fundamental Theorem of Line Integrals.

Evaluate ${\displaystyle {\int }_C\nabla f·d\text{r}},$ where $f(x,y\text{,}z)=$ $xy{z}^2-$ $yz$ and C has initial point (1, 2, 3) and terminal point (3, 5, 1).

For the following exercises, let $\text{F}(x,y)=$ $2x{y}^2\text{i}+$ $(2y{x}^2+2y)\text{j}$ and $G(x,y)=$ $(y+x)\text{i}+$ $(y-x)\text{j},$ and let C₁ be the curve consisting of the circle of radius 2, centered at the origin and oriented counterclockwise, and C₂ be the curve consisting of a line segment from (0, 0) to (1, 1) followed by a line segment from (1, 1) to (3, 1).

Calculate the line integral of G over C₁.

For the following exercises, let $\text{F}(x,y)=$ $2x{y}^2\text{i}+$ $(2y{x}^2+2y)\text{j}$ and $G(x,y)=$ $(y+x)\text{i}+$ $(y-x)\text{j},$ and let C₁ be the curve consisting of the circle of radius 2, centered at the origin and oriented counterclockwise, and C₂ be the curve consisting of a line segment from (0, 0) to (1, 1) followed by a line segment from (1, 1) to (3, 1).

Calculate the line integral of G over C₂.

For the following exercises, let $\text{F}(x,y)=$ $2x{y}^2\text{i}+$ $(2y{x}^2+2y)\text{j}$ and $G(x,y)=$ $(y+x)\text{i}+$ $(y-x)\text{j},$ and let C₁ be the curve consisting of the circle of radius 2, centered at the origin and oriented counterclockwise, and C₂ be the curve consisting of a line segment from (0, 0) to (1, 1) followed by a line segment from (1, 1) to (3, 1).

[T] Find line integral ${\displaystyle {\int }_C\text{F}·dr}$ of vector field $\text{F}(x,y,z)=$ $3x^{2}z\text{i}+$ $z^2\text{j}+$ $(x^3+2yz)\text{k}$ along curve C parameterized by $r(t)=$ $\left(\frac{\text{ln}t}{\text{ln}2}\right)\text{i}+$ $t^{3\text{/}2}\text{j}+$ $t\text{cos}\left(\pi t\right)k\text{,}1\le t\le 4.$

For the following exercises, show that the following vector fields are conservative by using a computer. Calculate ${\displaystyle {\int }_C\text{F}·d\text{r}}$ for the given curve.

$\text{F}=$ $\frac{2x}{y^2+1}\text{i}-$ $\frac{2y\left(x^2+1\right)}{{\left(y^2+1\right)}^2}\text{j};$ C is parameterized by $x=$ $t^3-$ $1,$ $y=$ $t^6-$ $t,$ $0\le t\le 1.$