벡터 미적분 전체 문제

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 ${x}^{2}+{y}^{2}+{z}^{2}={a}^{2}.$

For the following exercises, consider the radial fields $\text{F}=\frac{\langle x,y,z\rangle }{{({x}^{2}+{y}^{2}+{z}^{2})}^{\frac{p}{2}}}=\frac{\text{r}}{{||\text{r}||}^{p}},$ where p is a real number. Let S consist of spheres A and B centered at the origin with radii $0<a<b.$ The total outward flux across S consists of the outward flux across the outer sphere B less the flux into S across inner sphere A.

Find the total flux across S with $p=0.$

Show that for $p=3$ the flux across S is independent of a and b.

For the following exercises, without using Stokes’ theorem, calculate directly both the flux of $\text{curl}\,\text{F}\cdot \text{N}$ over the given surface and the circulation integral around its boundary, assuming all boundaries have positive orientation.

$\text{F}(x,y,z)={y}^{2}\text{i}+{z}^{2}\text{j}+{x}^{2}\text{k};$ S is the first-octant portion of plane $x+y+z=1.$

For the following exercises, without using Stokes’ theorem, calculate directly both the flux of $\text{curl}\,\text{F}\cdot \text{N}$ over the given surface and the circulation integral around its boundary, assuming all boundaries have positive orientation.

$\text{F}(x,y,z)=z\text{i}+x\text{j}+y\text{k};$ S is hemisphere $z={({a}^{2}-{x}^{2}-{y}^{2})}^{1/2}.$

For the following exercises, without using Stokes’ theorem, calculate directly both the flux of $\text{curl}\,\text{F}\cdot \text{N}$ over the given surface and the circulation integral around its boundary, assuming all boundaries have positive orientation.

$\text{F}(x,y,z)={y}^{2}\text{i}+2x\text{j}+5\text{k};$ S is hemisphere $z={(4-{x}^{2}-{y}^{2})}^{1/2}.$

For the following exercises, without using Stokes’ theorem, calculate directly both the flux of $\text{curl}\,\text{F}\cdot \text{N}$ over the given surface and the circulation integral around its boundary, assuming all boundaries have positive orientation.

$\text{F}(x,y,z)=z\text{i}+2x\text{j}+3y\text{k};$ S is upper hemisphere $z=\sqrt{9-{x}^{2}-{y}^{2}}.$

For the following exercises, without using Stokes’ theorem, calculate directly both the flux of $\text{curl}\,\text{F}\cdot \text{N}$ over the given surface and the circulation integral around its boundary, assuming all boundaries have positive orientation.

$\text{F}(x,y,z)=(x+2z)\text{i}+(y-x)\text{j}+(z-y)\text{k};$ S is a triangular region with vertices (3, 0, 0), (0, 3/2, 0), and (0, 0, 3).

$\text{F}(x,y,z)=2y\text{i}-6z\text{j}+3x\text{k};$ S is a portion of paraboloid $z=4-{x}^{2}-{y}^{2}$ and is above the xy-plane.

$\text{F}(x,y,z)=xy\text{i}-z\text{j}$ and S is the surface of the cube $0\le x\le 1,0\le y\le 1,0\le z\le 1,$ except for the face where $z=0,$ and using the outward unit normal vector.

$\text{F}(x,y,z)=xy\text{i}+{x}^{2}\text{j}+{z}^{2}\text{k};$ and S is the part of paraboloid $z={x}^{2}+{y}^{2}$ below plane $z=y,$ and using the outward normal vector.

$\text{F}(x,y,z)=4y\text{i}+z\text{j}+2y\text{k}$ and S is the part of sphere ${x}^{2}+{y}^{2}+{z}^{2}=4$ above plane $z=0,$ and using the outward normal vector

Use Stokes’ theorem to evaluate $\int_{C}[2x{y}^{2}zdx+2{x}^{2}yzdy+({x}^{2}{y}^{2}-2z)dz],$ where C is the curve given by $x=\cos \,t,y=\sin \,t,z=\sin \,t,0\le t\le 2\pi ,$ traversed in the direction of increasing t.

[T] Use a computer algebraic system (CAS) and Stokes’ theorem to approximate line integral $\int_{C}(ydx+zdy+xdz),$ where C is the intersection of plane $x+y=2$ and surface ${x}^{2}+{y}^{2}+{z}^{2}=2(x+y),$ traversed counterclockwise viewed from the origin.

[T] Use a CAS and Stokes’ theorem to approximate line integral $\int_{C}(3ydx+2zdy-5xdz),$ where C is the intersection of the xy-plane and hemisphere $z=\sqrt{1-{x}^{2}-{y}^{2}},$ traversed counterclockwise viewed from the top—that is, from the positive z-axis toward the xy-plane.

[T] Use a CAS and Stokes’ theorem to approximate line integral $\int_{C}[(1+y)zdx+(1+z)xdy+(1+x)ydz],$ where C is a triangle with vertices $(1,0,0),$ $(0,1,0),$ and $(0,0,1)$ oriented counterclockwise.

Use Stokes’ theorem to evaluate ${\iint }_{S}\text{curl}\,\text{F}\cdot d\text{S},$ where $\text{F}(x,y,z)={e}^{xy}\cos \,z\text{i}+{x}^{2}z\text{j}+xy\text{k},$ and S is half of sphere $x=\sqrt{1-{y}^{2}-{z}^{2}},$ oriented out toward the positive x-axis.

[T] Use a CAS and Stokes’ theorem to evaluate ${\iint }_{S}(\text{curl}\,\text{F}\cdot \text{N})dS,$ where $\text{F}(x,y,z)={x}^{2}y\text{i}+x{y}^{2}\text{j}+{z}^{3}\text{k}$ and S is the curve of the part of plane $3x+2y+z=6$ above cylinder ${x}^{2}+{y}^{2}=4,$ oriented clockwise when viewed from above.