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Chapter Review Exercises

True or False? Justify your answer with a proof or a counterexample.

Vector field $\text{F}(x,y)={x}^{2}y\text{i}+{y}^{2}x\text{j}$ is conservative.

For vector field $\text{F}(x,y)=P(x,y)\text{i}+Q(x,y)\text{j},$ if ${P}_{y}(x,y)={Q}_{x}(x,y)$ in open region $D,$ then ${\int }_{C}Pdx+Qdy=0.$

The divergence of a vector field is a vector field.

Chapter Review Exercises

True or False? Justify your answer with a proof or a counterexample.

If $\text{curl}\,\text{F}=0,$ then $\text{F}$ is a conservative vector field.

Draw the following vector fields.

$\text{F}(x,y)=\frac{1}{2}\text{i}+2x\text{j}$

Draw the following vector fields.

$\text{F}(x,y)=\frac{<y,3x>}{\sqrt{{x}^{2}+{y}^{2}}}$

Are the following the vector fields conservative? If so, find the potential function $f$ such that $\text{F}=\nabla f.$

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

Are the following the vector fields conservative? If so, find the potential function $f$ such that $\text{F}=\nabla f.$

$\text{F}(x,y)=(6xy)\text{i}+(3{x}^{2}-y{e}^{y})\text{j}$

Are the following the vector fields conservative? If so, find the potential function $f$ such that $\text{F}=\nabla f.$

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

Are the following the vector fields conservative? If so, find the potential function $f$ such that $\text{F}=\nabla f.$

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

Evaluate the following integrals.

$\int_{C}{x}^{2}dy+(2x-3xy)dx,$ along $C:y=\frac{1}{2}x$ from (0, 0) to (4, 2)

Evaluate the following integrals.

$\int_{C}ydx+x{y}^{2}dy,$ where $C:x=\sqrt{t},y=t-1,0\le t\le 1$

Evaluate the following integrals.

${\iint }_{S}x{y}^{2}dS,$ where S is surface $z={x}^{2}-y,0\le x\le 1,0\le y\le 4$

Find the divergence and curl for the following vector fields.

$\text{F}(x,y,z)=3xyz\text{i}+xy{e}^{z}\text{j}-3xy\text{k}$

Find the divergence and curl for the following vector fields.

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

$\int_{C}3xydx+2x{y}^{2}dy,$ where C is a square with vertices (0, 0), (0, 2), (2, 2) and (2, 0) oriented counterclockwise.

${\int }_{C}3ydx+(x+{e}^{y})dy,$ where C is a circle centered at the origin with radius 3

$\text{F}(x,y,z)=y\text{i}-x\text{j}+z\text{k},$ where $S$ is the upper half of the unit sphere

$\text{F}(x,y,z)=y\text{i}+xyz\text{j}-2zx\text{k},$ where $S$ is the upward-facing paraboloid $z={x}^{2}+{y}^{2}$ lying in cylinder ${x}^{2}+{y}^{2}=1$

Use the divergence theorem to evaluate ${\iint }_{S}\text{F}\cdot dS.$

$\text{F}(x,y,z)=({x}^{3}y)\text{i}+(3y-{e}^{x})\text{j}+(z+x)\text{k},$ over cube $S$ defined by $-1\le x\le 1,$ $0\le y\le 2,$ $0\le z\le 2$