벡터 미적분 전체 문제

Evaluate ${\iint }_{S}({x}^{2}+{y}^{2})dS,$ where S is the surface of hemisphere $z=\sqrt{1-{x}^{2}-{y}^{2}},$ and above the plane $z=0.$

Evaluate ${\iint }_{S}({x}^{2}+{y}^{2}+{z}^{2})dS,$ where S is the portion of plane $z=x+1$ that lies inside cylinder ${x}^{2}+{y}^{2}=1.$

Evaluate ${\iint }_{S}xyzdS$ if S is the part of plane $z=x+y$ that lies over the triangular region in the xy-plane with vertices (0, 0, 0), (1, 0, 0), and (0, 2, 0).

Find the mass of a lamina of density $\rho (x,y,z)=z$ in the shape of hemisphere $z={({a}^{2}-{x}^{2}-{y}^{2})}^{1/2}.$

Compute ${\iint }_{S}\text{F}\cdot \text{N}dS,$ where $\text{F}(x,y,z)=xy\text{i}+z\text{j}+(x+y)\text{k}$ and N is an upward pointing normal vector $S$, where S is the triangular region of the plane $x+y+z=1$ in the first octant.

Compute ${\iint }_{S}\text{F}\cdot \text{N}dS,$ where $\text{F}(x,y,z)=2yz\text{i}+({\tan }^{-1}(xz))\text{j}+{e}^{xy}\text{k}$ and N is an outward normal vector of S, where S is the surface of sphere ${x}^{2}+{y}^{2}+{z}^{2}=1.$

Compute ${\iint }_{S}\text{F}\cdot \text{N}dS,$ where $\text{F}(x,y,z)=xyz\text{i}+xyz\text{j}+xyz\text{k}$ and N is an outward normal vector S, where S is the surface of the five faces of the unit cube $0\le x\le 1,\,0\le y\le 1,\,0\le z\le 1$ missing $z=0.$

For the following exercises, express the surface integral as an iterated double integral by using a projection on S on the yz-plane.

${\iint }_{S}x{y}^{2}{z}^{3}dS;$ S is the first-octant portion of plane $2x+3y+4z=12.$

For the following exercises, express the surface integral as an iterated double integral by using a projection on S on the xz-plane

${\iint }_{S}x{y}^{2}{z}^{3}dS;$ S is the first-octant portion of plane $2x+3y+4z=12.$

Evaluate surface integral ${\iint }_{S}yzdS,$ where S is the first-octant part of plane $x+y+z=\lambda ,$ where $\lambda$ is a positive constant.

For the following exercises, express the surface integral as an iterated double integral by using a projection on S on the xz-plane

Evaluate surface integral ${\iint }_{S}({x}^{2}z+{y}^{2}z)dS,$ where S is hemisphere ${x}^{2}+{y}^{2}+{z}^{2}={a}^{2},z\ge 0.$

For the following exercises, express the surface integral as an iterated double integral by using a projection on S on the xz-plane

Evaluate surface integral ${\iint }_{S}zdS,$ where S is surface $z=\sqrt{{x}^{2}+{y}^{2}},0\le z\le 2.$

Evaluate surface integral ${\iint }_{S}{x}^{2}yzdS,$ where S is the part of plane $z=1+2x+3y$ that lies above rectangle $0\le x\le 3\,\text{and}\,0\le y\le 2.$

Evaluate surface integral ${\iint }_{S}yzdS,$ where S is plane $x+y+z=1$ that lies in the first octant.

Evaluate surface integral ${\iint }_{S}yzdS,$ where S is the part of plane $z=y+3$ that lies inside cylinder ${x}^{2}+{y}^{2}=1.$

For the following exercises, use geometric reasoning to evaluate the given surface integrals.

${\iint }_{S}\sqrt{{x}^{2}+{y}^{2}+{z}^{2}}dS,$ where S is surface ${x}^{2}+{y}^{2}+{z}^{2}=4,z\ge 0$

${\iint }_{S}(x\text{i}+y\text{j})\cdot dS,$ where S is surface ${x}^{2}+{y}^{2}=4,1\le z\le 3,$ oriented with unit normal vectors pointing outward

${\iint }_{S}(z\text{k})\cdot dS,$ where S is disc ${x}^{2}+{y}^{2}\le 9$ on plane $z=4,$ oriented with unit normal vectors pointing upward

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 $z={({a}^{2}-{x}^{2}-{y}^{2})}^{1/2}.$