벡터 미적분 전체 문제

Use the divergence theorem to evaluate ${\iint }_{S}\text{F}\cdot d\text{S},$ where $\text{F}(x,y,z)=xy\text{i}-\frac{1}{2}{y}^{2}\text{j}+z\text{k}$ and S is the surface consisting of three pieces: $z=4-3{x}^{2}-3{y}^{2},1\le z\le 4$ on the top; ${x}^{2}+{y}^{2}=1,0\le z\le 1$ on the sides; and $z=0$ on the bottom.

[T] Use a CAS and the divergence theorem to evaluate ${\iint }_{S}\text{F}\cdot d\text{S},$ where $\text{F}(x,y,z)=(2x+y\,\cos \,z)\text{i}+({x}^{2}-y)\text{j}+{y}^{2}z\text{k}$ and S is sphere ${x}^{2}+{y}^{2}+{z}^{2}=4$ orientated outward.

Use the divergence theorem to evaluate ${\iint }_{S}\text{F}\cdot d\text{S},$ where $\text{F}(x,y,z)=x\text{i}+y\text{j}+z\text{k}$ and S is the boundary of the solid enclosed by paraboloid $y={x}^{2}+{z}^{2}-2,$ cylinder ${x}^{2}+{z}^{2}=1,$ and plane $x+y=2,$ and S is oriented outward.

For the following exercises, Fourier’s law of heat transfer states that the heat flow vector F at a point is proportional to the negative gradient of the temperature; that is, $\text{F}=-k\nabla T,$ which means that heat energy flows hot regions to cold regions. The constant $k>0$ is called the conductivity, which has metric units of joules per meter per second-kelvin or watts per meter-kelvin. A temperature function for region D is given. Use the divergence theorem to find net outward heat flux ${\iint }_{S}\text{F}\cdot \text{N}dS=-k{\iint }_{S}\nabla T\cdot \text{N}dS$ across the boundary S of D, where $k=1.$

$T(x,y,z)=100+x+2y+z;$ $D=\{(x,y,z):0\le x\le 1,0\le y\le 1,0\le z\le 1\}$

For the following exercises, Fourier’s law of heat transfer states that the heat flow vector F at a point is proportional to the negative gradient of the temperature; that is, $\text{F}=-k\nabla T,$ which means that heat energy flows hot regions to cold regions. The constant $k>0$ is called the conductivity, which has metric units of joules per meter per second-kelvin or watts per meter-kelvin. A temperature function for region D is given. Use the divergence theorem to find net outward heat flux ${\iint }_{S}\text{F}\cdot \text{N}dS=-k{\iint }_{S}\nabla T\cdot \text{N}dS$ across the boundary S of D, where $k=1.$

$T(x,y,z)=100+{e}^{-z};$ $D=\{(x,y,z):0\le x\le 1,0\le y\le 1,0\le z\le 1\}$

$T(x,y,z)=100{e}^{-{x}^{2}-{y}^{2}-{z}^{2}};$ D is the sphere of radius a centered at the origin.

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.

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