면적분과 플럭스

For the following exercises, determine whether the statements are true or false.

If surface S is given by $\{(x,y,z):0\le x\le 1,0\le y\le 1,z=10\},$ then ${\iint }_{S}f(x,y,z)dS={\int }_{0}^{1}{\int }_{0}^{1}f(x,y,10)dxdy.$

For the following exercises, determine whether the statements are true or false.

If surface S is given by $\{(x,y,z):0\le x\le 1,0\le y\le 1,z=x\},$ then ${\iint }_{S}f(x,y,z)dS={\int }_{0}^{1}{\int }_{0}^{1}f(x,y,x)dxdy.$

For the following exercises, determine whether the statements are true or false.

Surface $\text{r}=\langle v\,\cos \,u,v\,\sin \,u,{v}^{2}\rangle ,\,\text{for}\,0\le u\le \pi ,0\le v\le 2,$ is the same as surface $\text{r}=\langle \sqrt{v}\,\cos \,2u,\sqrt{v}\,\sin \,2u,v\rangle ,$ for $0\le u\le \frac{\pi }{2},0\le v\le 4.$

In the plane, flux is a measurement of how much of the vector field passes across a _____; in space, flux is a measurement of how much of the vector field passes across a _____.

When computing flux, what does it mean when the result is a negative number?

When $\mathcal{S}$ is a closed surface, we choose the normal vector so that it points to the _____ of the surface.

If $\mathcal{S}$ is a plane, and $\vec F$ is always parallel to $\mathcal{S}$, then the flux of $\vec F$ across $\mathcal{S}$ will be _____.