Vector Analysis

In Exercises $25-40$, evaluate the integral

${\iint }_S\mathbf{F}(x,y,z)·\mathbf{n}dS$

for the specified function $\mathbf{F}(x,y,z)$ and the given surface $S$. In each integral, $\mathbf{n}$ is the outwards-pointing normal vector.

$\mathbf{F}(x,y,z)=x{y}^2\mathbf{i}+y(z-3x)\mathbf{j}+4xyz\mathbf{k}$, and $S$ is the surface of the region $W$ bounded by the planes $y=0, y=z, z=3, x=0$, and $x=4$.

In Exercises $25-40$, evaluate the integral

${\iint }_S\mathbf{F}(x,y,z)·\mathbf{n}dS$

for the specified function $\mathbf{F}(x,y,z)$ and the given surface $S$. In each integral, $\mathbf{n}$ is the outwards-pointing normal vector.

$\mathbf{F}(x,y,z)=x{y}^2\mathbf{i}+y(z-3x)\mathbf{j}+4xyz\mathbf{k}$, and $S$ is the surface of the region $W$ bounded by the planes $y=0, y=z, z=3, x=0$, and $x=4$.

Work as an integral of force and distance: Find the work done in moving an object along the x-axis from the origin to $x=\frac{\mathrm{\pi }}{2}$ if the force acting on the object at a given value of x is $\mathrm{F}\left(x\right)=x\sin x$.

What are the inputs of a vector field in the Cartesian plane?

Calculus of vector-valued functions: Calculate each of the following.

$\frac{d}{dt}\left(r\right(t\left)\right), \mathrm{where} r\left(t\right)=3{\cos }^{2}t\mathbf{i}+5t\mathbf{j}+\frac{t}{t^2+1}\mathbf{k}$

What are the inputs of a vector field in ${\mathrm{ℝ}}^3$?

Calculus of vector-valued functions: Calculate each of the following.

$\int r\left(t\right)\mathrm{d}t, \mathrm{where} r\left(t\right)=e^t\mathbf{i}+t^3\mathbf{j}-4\mathbf{k}$

What are the outputs of a vector field in the Cartesian plane?

What are the outputs of a vector field in ${\mathrm{ℝ}}^3$?

What does it mean to say that a vector field is conservative?

Do the vectors in the range of $\mathbf{F}(x,y)=x\mathbf{i}+y\mathbf{j}$ point towards or away from the origin?

What is the difference between the graphs of

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

What is the difference between the graphs of

$\mathrm{G}(x,y,z)=-\mathbf{i}-\mathbf{j}-\mathbf{k} and \mathrm{F}(x,y,z)=\mathbf{i}+\mathbf{j}+\mathbf{k}$

What is the difference between the graphs of

$\mathrm{G}(x,y)=x\mathbf{i}+\mathrm{y}\mathbf{j} and \mathrm{F}(x,y)=-x\mathbf{i}+(-y)\mathbf{j}$

What is the difference between the graphs of

$\mathrm{G}(x,y,z)=2\mathbf{i}-3\mathbf{j}+z\mathbf{k} and \mathrm{F}(x,y,z)=-2\mathbf{i}+3\mathbf{j}-z\mathbf{k}$

Consider the vector field $\mathrm{F}(x,y,z)=(yz,xz,xy)$. Find a vector field $\mathrm{G}(x,y,z)$ with the property that, for all points in ${\mathrm{ℝ}}^3,\mathrm{G}(x,y,z)=2\mathrm{F}(x,y,z)$.

How would you show that a given vector field in ${\mathrm{ℝ}}^2$ is not conservative?

How would you show that a given vector field in ${\mathrm{ℝ}}^3$ is not conservative?

In Exercises 17–24, find a potential function for the given vector field.

$\mathbf{F}(x, y)=(3x^2\cos y,-x^3\sin y)$

In Exercises 17–24, find a potential function for the given vector field.

$\mathbf{F}(x, y)=(e^y{\sec }^{2}x, e^y\tan x)$

In Exercises 17–24, find a potential function for the given vector field.

$\mathbf{G}(\mathrm{x},\mathrm{y})=(5{\mathrm{x}}^4+\mathrm{y},\mathrm{x}-12{\mathrm{y}}^3)$

In Exercises 17–24, find a potential function for the given vector field.

$\mathbf{G}(x,y)=\mathbf{i}-\mathbf{j}$

In Exercises 17–24, find a potential function for the given vector field.

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

In Exercises 17–24, find a potential function for the given vector field.

$\mathbf{F}(x,y,z)=e^{y^2}\mathbf{i}+(2xy{e}^{y^2}+\sin z)\mathbf{j}+y\cos z\mathbf{k}$

Write ${\int }_{\mathrm{C}}\mathbf{F}(x,y,z)·\mathrm{d}r$ explicitly as an integral of t, where $\mathbf{F}(x,y,z)=(zy-x, xz-y, xy-z)$ and $\mathbf{r}\left(t\right)=(t, \cos t, \sin t)$ for $a \le t \le b$.

In Exercises 17–24, find a potential function for the given vector field.

$\mathbf{G}(x,y,z)=\cos y\mathbf{i}+(\sin z-x\sin y)\mathrm{j}+y\cos z\mathbf{k}$

In Exercises 17–24, find a potential function for the given vector field.

$\mathbf{G}(x,y,z)=(y{e}^{xy+z}, x{e}^{xy+z}, e^{xy+z})$

Sketch the vector fields in Exercises 25–32.

$\mathbf{F}(x,y)=2\mathbf{i}+0\mathbf{j}$

Sketch the vector fields in Exercises 25–32.

$\mathbf{F}(x,y)=0\mathbf{i}+2\mathbf{j}$

Sketch the vector fields in Exercises 25–32.

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

Sketch the vector fields in Exercises 25–32.

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

Sketch the vector fields in Exercises 25–32.

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

Sketch the vector fields in Exercises 25–32.

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

Sketch the vector fields in Exercises 25–32.

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

Sketch the vector fields in Exercises 25–32.

$\mathbf{F}(x,y)=-2x\mathbf{i}-3\mathrm{y}\mathbf{j}$

Show that the vector fields in Exercises 33–40 are not conservative.

$\mathbf{F}(x,y)=(xy,-y)$

Show that the vector fields in Exercises 33–40 are not conservative.

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

Show that the vector fields in Exercises 33–40 are not conservative.

$\mathbf{G}(x,y)=\frac{1}{x^2+y}\mathbf{i}+\frac{y}{x}\mathbf{j}$

Show that the vector fields in Exercises 33–40 are not conservative.

$\mathbf{G}(x,y)=y\mathbf{i}-x\mathbf{j}$

Show that the vector fields in Exercises 33–40 are not conservative.

$\mathbf{F}(x,y,z)=2\mathbf{i}-z\mathbf{j}+{\mathrm{e}}^{\mathrm{yz}}\mathbf{k}$

Show that the vector fields in Exercises 33–40 are not conservative.

$\mathbf{F}(x,y,z)=\tan \left(yz\right)\mathbf{i}+\left(xzse{c}^2\right(yz)-2)\mathbf{j}+4z^3\mathbf{k}$

Show that the vector fields in Exercises 33–40 are not conservative.

$\mathbf{G}(x,y,z)=(3,yz,z+12)$

Show that the vector fields in Exercises 33–40 are not conservative.

$\mathbf{G}(x,y,z)=(e^y+z,x{e}^y+z, x+y)$

Determine whether or not each of the vector fields in Exercises 41–48 is conservative. If the vector field is conservative, find a potential function for the field.

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

Determine whether or not each of the vector fields in Exercises 41–48 is conservative. If the vector field is conservative, find a potential function for the field.

$\mathbf{F}(x,y)=\left({\tan }^{-1}y, \frac{x}{1+y^2}\right)$

Determine whether or not each of the vector fields in Exercises 41–48 is conservative. If the vector field is conservative, find a potential function for the field.

$\mathbf{G}(x,y)=(2x+y\cos (xy), x\cos (xy)-1)$

Determine whether or not each of the vector fields in Exercises 41–48 is conservative. If the vector field is conservative, find a potential function for the field.

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

Determine whether or not each of the vector fields in Exercises 41–48 is conservative. If the vector field is conservative, find a potential function for the field.

$\mathbf{F}(x,y,z)=(y{e}^{2z}+1,x{e}^{2z},2xy{e}^{2z})$

Determine whether or not each of the vector fields in Exercises 41–48 is conservative. If the vector field is conservative, find a potential function for the field.

$\mathbf{F}(x,y,z)=\mathbf{i}+2\mathbf{j}-3\mathbf{k}$

Determine whether or not each of the vector fields in Exercises 41–48 is conservative. If the vector field is conservative, find a potential function for the field.

$\mathbf{G}(x,y,z)=(z-y)\mathbf{i}-xy\mathbf{j}+(xz+y)\mathbf{k}$