Chapter 6

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

Determine whether the statement is true or false. Vector field \(\mathbf{F}=\left\langle 3 x^{2}, 1\right\rangle\) is a gradient field for both \(\phi_{1}(x, y)=x^{3}+y\) and \(\phi_{2}(x, y)=y+x^{3}+100\).

Determine whether the statement is true or false. Vector field \(\mathbf{F}=\frac{\langle y, x\rangle}{\sqrt{x^{2}+y^{2}}}\) is constant in direction and magnitude on a unit circle.

Determine whether the statement is true or false. Vector field \(\mathbf{F}=\frac{\langle y, x\rangle}{\sqrt{x^{2}+y^{2}}}\) is neither a radial field nor a rotation.

Describe each vector field by drawing some of its vectors. $$ \mathbf{F}(x, y)=x \mathbf{i}+y \mathbf{j} $$

Describe each vector field by drawing some of its vectors. $$ \mathbf{F}(x, y)=-y \mathbf{i}+x \mathbf{j} $$

Describe each vector field by drawing some of its vectors. $$ \mathbf{F}(x, y)=x \mathbf{i}-y \mathbf{j} $$

Describe each vector field by drawing some of its vectors. $$ \mathbf{F}(x, y)=\mathbf{i}+\mathbf{j} $$

Describe each vector field by drawing some of its vectors. $$ \mathbf{F}(x, y)=2 x \mathbf{i}+3 y \mathbf{j} $$

Describe each vector field by drawing some of its vectors. $$ \mathbf{F}(x, y)=3 \mathbf{i}+x \mathbf{j} $$

Describe each vector field by drawing some of its vectors. $$ \mathbf{F}(x, y)=y \mathbf{i}+\sin x \mathbf{j} $$

Describe each vector field by drawing some of its vectors. $$ \mathbf{F}(x, y, z)=x \mathbf{i}+y \mathbf{j}+z \mathbf{k} $$

Describe each vector field by drawing some of its vectors. $$ \mathbf{F}(x, y, z)=2 x \mathbf{i}-2 y \mathbf{j}-2 z \mathbf{k} $$

Find the gradient vector field of each function \(f\). $$ f(x, y)=x \sin y+\cos y $$

Find the gradient vector field of each function \(f\). $$ f(x, y, z)=z e^{-x y} $$

Find the gradient vector field of each function \(f\). $$ f(x, y, z)=x^{2} y+x y+y^{2} z $$

Find the gradient vector field of each function \(f\). $$ f(x, y)=x^{2} \sin (5 y) $$

Find the gradient vector field of each function \(f\). $$ f(x, y)=\ln \left(1+x^{2}+2 y^{2}\right) $$

Find the gradient vector field of each function \(f\). $$ f(x, y, z)=x \cos \left(\frac{y}{z}\right) $$

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

Write formulas for the vector fields with the given properties. All vectors are parallel to the \(x\) -axis and all vectors on a vertical line have the same magnitude.

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

Is vector field \(\mathbf{F}(x, y)=(P(x, y), Q(x, y))=(\sin x+y) \mathbf{i}+(\cos y+x) \mathbf{j}\) a gradient field?

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

Assume that an electric field in the \(x y\) -plane caused by an infinite line of charge along the \(x\) -axis is a gradient field with potential function \(V(x, y)=c \ln \left(\frac{r_{0}}{\sqrt{x^{2}+y^{2}}}\right), \quad\) where \(c>0\) is a constant and \(r_{0}\) is a reference distance at which the potential is assumed to be zero. Find the components of the electric field in the \(x\) - and \(y\) -directions, where \(\mathbf{E}(x, y)=-\nabla V(x, y)\).

Assume that an electric field in the \(x y\) -plane caused by an infinite line of charge along the \(x\) -axis is a gradient field with potential function \(V(x, y)=c \ln \left(\frac{r_{0}}{\sqrt{x^{2}+y^{2}}}\right), \quad\) 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 \(x y\) -plane is directed outward from the origin and has magnitude \(\mathbf{E} \mid=\frac{c}{r}, \quad\) where \(r=\sqrt{x^{2}=y^{2}}\)

Show that the given curve \(\mathbf{c}(t)\) is a flow line of the given velocity vector field \(\mathbf{F}(x, y, z)\). $$ \mathbf{c}(t)=\left(e^{2 t}, \ln |t|, \frac{1}{t}\right), t \neq 0 ; \mathbf{F}(x, y, z)=\left\langle 2 x, z,-z^{2}\right\rangle $$

Show that the given curve \(\mathbf{c}(t)\) is a flow line of the given velocity vector field \(\mathbf{F}(x, y, z)\). $$ \mathbf{c}(t)=\left(\sin t, \cos t, e^{t}\right) ; \mathbf{F}(x, y, z)=\langle y,-x, z\rangle $$

True or False? Line integral \(\int_{C} f(x, y) d s\) is equal to a definite integral if \(C\) is a smooth curve defined on \([a, b]\) and if function \(f\) is continuous on some region that contains curve \(C\).

True or False? Vector functions \(\mathbf{r}_{1}=t \mathbf{i}+t^{2} \mathbf{j}\), \(0 \leq t \leq 1,\) and \(\mathbf{r}_{2}=(1-t) \mathbf{i}+(1-t)^{2} \mathbf{j}, \quad 0 \leq t \leq 1\) define the same oriented curve.

True or False? A piecewise smooth curve \(C\) consists of a finite number of smooth curves that are joined together end to end.

For the following exercises, use a computer algebra system (CAS) to evaluate the line integrals over the indicated path. \([\mathrm{T}] \int_{C}(x+y) d s \quad C: x=t, y=(1-t), z=0\) from (0,1,0) to (1,0,0)

For the following exercises, use a computer algebra system (CAS) to evaluate the line integrals over the indicated path. [T] \(\int_{C}(x-y) d s \quad C: \mathbf{r}(t)=4 t \mathbf{i}+3 t \mathbf{j} \quad\) when \(0 \leq t \leq 2\)

For the following exercises, use a computer algebra system (CAS) to evaluate the line integrals over the indicated path. [T] \(\quad \int_{C}\left(x^{2}+y^{2}+z^{2}\right) d s\) \(C: \mathbf{r}(t)=\sin t \mathbf{i}+\cos t \mathbf{j}+8 t \mathbf{k}\) when \(0 \leq t \leq \frac{\pi}{2}\)

For the following exercises, use a computer algebra system (CAS) to evaluate the line integrals over the indicated path. \([\mathbf{T}]\) Evaluate \(\int_{C} x y^{4} d s, \quad\) where \(C\) is the right half of circle \(x^{2}+y^{2}=16\) and is traversed in the clockwise

For the following exercises, use a computer algebra system (CAS) to evaluate the line integrals over the indicated path. \([\mathrm{T}]\) Evaluate \(\int_{C} 4 x^{3} d s,\) where \(C\) is the line segment from (-2,-1) to (1,2)

For the following exercises, use a computer algebra system (CAS) to evaluate the line integrals over the indicated path. Find the work done by vector field \(\mathbf{F}(x, y, z)=x \mathbf{i}+3 x y \mathbf{j}-(x+z) \mathbf{k}\) on a particle moving along a line segment that goes from (1,4,2) to (0,5,1) .

Find the work done by a person weighing \(150 \mathrm{lb}\) walking exactly one revolution up a circular, spiral staircase of radius \(3 \mathrm{ft}\) if the person rises \(10 \mathrm{ft}\).

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

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

Evaluate \(\int_{C} \mathbf{F} \cdot d \mathbf{r}, \quad\) where \(\mathbf{F}(x, y)=-1 \mathbf{j}, \quad\) and \(C\) is the part of the graph of \(y=\frac{1}{2} x^{3}-x\) from (2,2) to (-2,-2).

Evaluate \(\int_{\gamma}\left(x^{2}+y^{2}+z^{2}\right)^{-1} d s,\) where \(\gamma\) is the helix \(x=\cos t, y=\sin t, z=t(0 \leq t \leq T)\).

Evaluate \(\int_{C} y z d x+x z d y+x y d z\) over the line segment from (1,1,1) to (3,2,0)

[T] Use a computer algebra system to evaluate the line integral \(\int_{C}\left(x+3 y^{2}\right) d y\) over the path \(C\) given by \(x=2 t, y=10 t,\) where \(0 \leq t \leq 1\).

[T] Use a CAS to evaluate line integral \(\int_{C} x y d x+y d y\) over path \(C\) given by \(x=2 t, y=10 t,\) where \(0 \leq t \leq 1\).

Evaluate line integral \(\int_{C}(2 x-y) d x+(x+3 y) d y\), where \(C\) lies along the \(x\) -axis from \(x=0\) to \(x=5\).

[T] Use a CAS to evaluate \(\int_{C} x y d s,\) where \(C\) is \(x=t^{2}, y=4 t, 0 \leq t \leq 1\)

In the following exercises, find the work done by force field \(\mathbf{F}\) on an object moving along the indicated path. \(\mathbf{F}(x, y)=-x \mathbf{i}-2 y \mathbf{j}\) \(C: y=x^{3}\) from (0,0) to (2,8)

In the following exercises, find the work done by force field \(\mathbf{F}\) on an object moving along the indicated path. \(\mathbf{F}(x, y)=2 x i+y \mathbf{j}\) C: counterclockwise around the triangle with vertices \((0,0),(1,0),\) and (1,1)

In the following exercises, find the work done by force field \(\mathbf{F}\) on an object moving along the indicated path. \(\mathbf{F}(x, y, z)=x \mathbf{i}+y \mathbf{j}-5 z \mathbf{k}\) \(C: \mathbf{r}(t)=2 \cos t \mathbf{i}+2 \sin t \mathbf{j}+t \mathbf{k}, 0 \leq t \leq 2 \pi\)