벡터 미적분 전체 문제

For the following exercises, consider a rigid body that is rotating about the x-axis counterclockwise with constant angular velocity $\omega =\langle a,b,c\rangle .$ If P is a point in the body located at $\text{r}=x\text{i}+y\text{j}+z\text{k},$ the velocity at P is given by vector field $\text{F}=\omega \,\times \,\text{r}.$

Find $\text{the div of}\,\text{F}.$

For the following exercises, consider a rigid body that is rotating about the x-axis counterclockwise with constant angular velocity $\omega =\langle a,b,c\rangle .$ If P is a point in the body located at $\text{r}=x\text{i}+y\text{j}+z\text{k},$ the velocity at P is given by vector field $\text{F}=\omega \,\times \,\text{r}.$

Find $\text{the curl of}\,\text{F}$

[T] Consider rotational velocity field $\text{v}=\langle 0,10z,-10y\rangle .$ If a paddlewheel is placed in plane $x+y+z=1$ with its axis normal to this plane, using a computer algebra system, calculate how fast the paddlewheel spins in revolutions per unit of time.

The portion of cylinder ${x}^{2}+{y}^{2}=9$ in the first octant, for $0\le z\le 3$

$\text{F}(x,y,z)={x}^{2}\text{i}+{y}^{2}\text{j}+{z}^{2}\text{k},$ and S is the portion of plane $z=y+1$ that lies inside cylinder ${x}^{2}+{y}^{2}=1.$

Evaluate ${\iint }_{S}({x}^{2}+y-z)dS,$ where $S$ is the surface defined parametrically by $\text{r}(u,v)=(2u+v)\text{i}+(u-2v)\text{j}+(u+3v)\text{k}$ for $0\le u\le 1,\,\text{and}\,0\le v\le 2.$

For the following exercises, approximate the mass of the lamina that has the shape of given surface S. Round to four decimal places.

[T] Evaluate ${\iint }_{S}(x-{y}^{2}+z)dS,$ where S is the surface defined by $\text{r}(u,v)={u}^{2}\text{i}+v\text{j}+u\text{k},\,0\le u\le 1,\,0\le v\le 1.$

[T] Evaluate ${\iint }_{S}{x}^{2}zdS,$ where S is the portion of cone ${z}^{2}={x}^{2}+{y}^{2}$ that lies between planes $z=1$ and $z=4.$

[T] Evaluate ${\iint }_{S}(xz/y)dS,$ where S is the portion of cylinder $x={y}^{2}$ that lies in the first octant between planes $z=0,z=5,y=1,$ and $y=4.$

[T] Evaluate ${\iint }_{S}(z+y)dS,$ where S is the part of the graph of $z=\sqrt{1-{x}^{2}}$ in the first octant between the xz-plane and plane $y=3.$

Compute ${\iint }_{S}\text{F}\cdot \text{N}dS,$ where $\text{F}(x,y,z)=x\text{i}-5y\text{j}+4z\text{k}$ and N is an outward normal vector of S, where S is the union of two squares ${S}_{1}:x=0,\,0\le y\le 1,\,0\le z\le 1$ and ${S}_{2}:z=1,\,0\le x\le 1,\,0\le y\le 1.$

${\iint }_{S}({x}^{2}-2y+z)dS;$ S is the portion of the graph of $4x+y=8$ bounded by the coordinate planes and plane $z=6.$

${\iint }_{S}({x}^{2}-2y+z)dS;$ S is the portion of the graph of $4x+y=8$ bounded by the coordinate planes and plane $z=6.$

A lamina has the shape of a portion of sphere ${x}^{2}+{y}^{2}+{z}^{2}={a}^{2}$ that lies within cone $z=\sqrt{{x}^{2}+{y}^{2}}.$ Let S be the spherical shell centered at the origin with radius a, and let C be the right circular cone with a vertex at the origin and an axis of symmetry that coincides with the z-axis. Determine the mass of the lamina if $\rho (x,y,z)={x}^{2}{y}^{2}z.$

A lamina has the shape of a portion of sphere ${x}^{2}+{y}^{2}+{z}^{2}={a}^{2}$ that lies within cone $z=\cot {\varphi }_{0}\sqrt{{x}^{2}+{y}^{2}}$ Let S be the spherical shell centered at the origin with radius a, and let C be the right circular cone with a vertex at the origin and an axis of symmetry that coincides with the z-axis. Suppose the angle between the sides of the cone and the z-axis is ${\phi }_{0},\,\text{with}\,0\le {\phi }_{0}<\frac{\pi }{2}.$ Determine the mass of that portion of the shape enclosed in the intersection of S and C. Assume $\rho (x,y,z)={x}^{2}{y}^{2}z.$

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 $p=0.$

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.

Use Stokes’ theorem to evaluate $\int_{C}\left(\frac{1}{2}{y}^{2}dx+zdy+xdz\right),$ where C is the curve of intersection of plane $x+z=1$ and ellipsoid ${x}^{2}+2{y}^{2}+{z}^{2}=1,$ oriented clockwise from the origin.

Let $\text{F}(x,y,z)=xy\text{i}+({e}^{{z}^{2}}+y)\text{j}+(x+y)\text{k}$ and let S be the graph of function $y=\frac{{x}^{2}}{9}+\frac{{z}^{2}}{9}-1$ with $y\le 0$ oriented so that the normal vector of S has a positive j component. Use Stokes’ theorem to compute integral ${\iint }_{S}\text{curl}\,\text{F}\cdot d\text{S}.$

Use the surface integral in Stokes’ theorem to calculate the circulation of field F, $\text{F}(x,y,z)={x}^{2}{y}^{3}\text{i}+\text{j}+z\text{k}$ around C, which is the intersection of cylinder ${x}^{2}+{y}^{2}=4$ and hemisphere ${x}^{2}+{y}^{2}+{z}^{2}=16,z\ge 0,$ oriented counterclockwise when viewed from above.