벡터 미적분 전체 문제

Find the area of the region bounded by hypocycloid $\text{r}(t)={\cos }^{3}(t)\text{i}+{\sin }^{3}(t)\text{j}.$ The curve is parameterized by $t\in [0,2\pi ].$

Find the area of a pentagon with vertices $(0,4),(4,1),(3,0),(-1,-1),$ and $(-2,2).$

Use Green’s theorem to find the area of one loop of a four-leaf rose $r=3\,\sin \,2\theta .$ (Hint: $xdy-ydx={r}^{2}d\theta ).$

If the coordinate functions of $\text{F}:{\mathbb{R}}^{3}\to {\mathbb{R}}^{3}$ have continuous second partial derivatives, then $\text{curl}\,(\text{div}(\text{F}))$ equals zero.

In the following exercises, suppose a solid object in ${\mathbb{R}}^{3}$ has a temperature distribution given by $T(x,y,z)=100{e}^{-{x}^{2}+{y}^{2}+{z}^{2}}$. The heat flow vector field in the object is $\text{F}=-k\nabla T,$ where $k>0$ is a property of the material. The heat flow vector points in the direction opposite to that of the gradient, which is the direction of greatest temperature decrease. The divergence of the heat flow vector is $\nabla \cdot \text{F}=-k\nabla \cdot \nabla T=-k{\nabla }^{2}T.$

Compute the heat flow vector field.

In the following exercises, suppose a solid object in ${\mathbb{R}}^{3}$ has a temperature distribution given by $T(x,y,z)=100{e}^{-{x}^{2}+{y}^{2}+{z}^{2}}$. The heat flow vector field in the object is $\text{F}=-k\nabla T,$ where $k>0$ is a property of the material. The heat flow vector points in the direction opposite to that of the gradient, which is the direction of greatest temperature decrease. The divergence of the heat flow vector is $\nabla \cdot \text{F}=-k\nabla \cdot \nabla T=-k{\nabla }^{2}T.$

Compute the divergence.

For the following exercises, use a computer algebra system to approximate the area of the following surfaces using a parametric description of the surface.

[T] Half cylinder $\{(r,\theta ,z):r=4,0\le \theta \le \pi ,0\le z\le 7\}$

For the following exercises, use a computer algebra system to approximate the area of the following surfaces using a parametric description of the surface.

[T] Plane $z=10-x-y$ above square $|x|\le 2,|y|\le 2$

A paper cup has the shape of an inverted right circular cone of height 6 in. and radius of top 3 in. If the cup is full of water weighing $62.5\,\text{lb}/{\text{ft}}^{3},$ find the total force exerted by the water on the inside surface of the cup.

For the following exercises, the heat flow vector field for conducting objects is $\text{F}=-k\nabla T,\,\text{where}\,T(x,y,z)$ is the temperature in the object and $k>0$ is a constant that depends on the material. Find the outward flux of F across the following surfaces S for the given temperature distributions and assume $k=1.$

$T(x,y,z)=100{e}^{-x-y};$ S consists of the faces of cube $|x|\le 1,|y|\le 1,|z|\le 1.$

For the following exercises, the heat flow vector field for conducting objects is $\text{F}=-k\nabla T,\,\text{where}\,T(x,y,z)$ is the temperature in the object and $k>0$ is a constant that depends on the material. Find the outward flux of F across the following surfaces S for the given temperature distributions and assume $k=1.$

$T(x,y,z)=-\ln ({x}^{2}+{y}^{2}+{z}^{2});$ S is sphere ${x}^{2}+{y}^{2}+{z}^{2}={a}^{2}.$

Velocity field $\text{v}=\langle 0,1-{x}^{2},0\rangle ,$ for $|x|\le 1\,\text{and}\,|z|\le 1,$ represents a horizontal flow in the y-direction. Compute the curl of v in a clockwise rotation.

Compute the flux of water through parabolic cylinder $S:y={x}^{2},$ from $0\le x\le 2,0\le z\le 3,$ if the velocity vector is $\text{F}(x,y,z)=3{z}^{2}\text{i}+6\text{j}+6xz\text{k}.$

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 $k=1.$

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

Find the amount of work performed by a 50-kg woman ascending a helical staircase with radius 2 m and height 100 m. The woman completes five revolutions during the climb. Use 9.8 m/s² as the acceleration due to gravity.

In Exercises 13–16, find the work performed by the force field $\vec F$ moving a particle along the path $C$.

$\vec F = \langle y,x^2\rangle$ N; $C$ is the segment of the line $y=x$ from $(0,0)$ to $(1,1)$, where distances are measured in meters.

In Exercises 13–16, find the work performed by the force field $\vec F$ moving a particle along the path $C$.

$\vec F = \langle y,x^2\rangle$ N; $C$ is the portion of $y=\sqrt x$ from $(0,0)$ to $(1,1)$, where distances are measured in meters.

In Exercises 13–16, find the work performed by the force field $\vec F$ moving a particle along the path $C$.

$\vec F = \langle 2xy,x^2,1\rangle$ lbs; $C$ is the path from $(0,0,0)$ to $(2,4,8)$ via $\vec r(t) = \langle t, t^2, t^3\rangle$ on $0\leq t\leq 2$, where distance are measured in feet.

In Exercises 13–16, find the work performed by the force field $\vec F$ moving a particle along the path $C$.

$\vec F = \langle 2xy,x^2,1\rangle$ lbs; $C$ is the path from $(0,0,0)$ to $(2,4,8)$ via $\vec r(t) = \langle t,2t, 4t\rangle$ on $0\leq t\leq 2$, where distance are measured in feet.