91Ó°ÊÓ

$\mathbf{âˆ\neg }\mathbf{V}\mathbf{×}\mathbf{²Ô»åσ}$over the closed surface of the tin can bounded by if ${\mathbf{x}}^{\mathbf{2}}\mathbf{+}{\mathbf{y}}^{\mathbf{2}}\mathbf{=}\mathbf{9}\mathbf{,}\mathbf{z}\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{z}\mathbf{=}\mathbf{5}$if$\mathbf{V}\mathbf{=}\mathbf{2}\mathit{x}\mathit{y}\mathit{i}\mathbf{−}{\mathbf{y}}^{\mathbf{2}}\mathbf{j}\mathbf{+}\mathbf{(}\mathbf{z}\mathbf{+}\mathit{x}\mathit{y}\mathbf{)}\mathbf{k}$.

Evaluate each of the integrals in Problems $\mathbf{3}$to $\mathbf{8}$as either a volume integral or a surface integral, whichever is easier.

$\mathbf{∭}\mathbf{∇}\mathbf{×}\mathbf{³Õ»åÏ„}$ over the unit cube in the first octant, where $\mathbf{V}\mathbf{=}\left(x^3−x^2\right)\mathbf{yi}\mathbf{+}\left(y^3−2y^2+y\right)\mathbf{xj}\mathbf{+}\left(z^2−1\right)\mathbf{k}$

Let F = 2i - 3j + k act at the point (5, 1, 3)

(a) Find the torque of F about the point (4, 1, 0)

(b) Find the torque of F about the line r = 4i + j + (2i + j - 2k)t.

Evaluate each of the integrals in Problems $\mathbf{3}$to $\mathbf{8}$as either a volume integral or a surface integral, whichever is easier.

$\mathbf{âˆ\neg }\mathbf{r}\mathbf{×}\mathbf{²Ô»åσ}$over the entire surface of the cone with base ${\mathbf{x}}^{\mathbf{2}}\mathbf{+}{\mathbf{y}}^{\mathbf{2}}\mathbf{≤}\mathbf{16}\mathbf{,}\mathbf{z}\mathbf{=}\mathbf{0}$ and vertex at$\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{0}\mathbf{,}\mathbf{3}\mathbf{)}$ where $\mathbf{r}\mathbf{=}\mathbf{ix}\mathbf{+}\mathbf{jy}\mathbf{+}\mathbf{kz}$

Use Problem 6 to show that the area inside the ellipse $\mathit{x}\mathbf{=}\mathit{a}\mathit{c}\mathit{o}\mathit{s}\mathit{θ}\mathbf{,}\mathit{y}\mathbf{=}\mathit{b}\mathit{s}\mathit{i}\mathit{n}\mathit{θ}\mathbf{,}\mathbf{0}\mathbf{≤}\mathit{θ}\mathbf{≤}\mathbf{2}\mathit{π}\mathbf{,}\mathbf{}\mathit{i}\mathit{s}\mathit{A}\mathbf{=}\mathit{π}\mathit{a}\mathit{b}\mathbf{.}$

${\mathbf{∭}}^{\mathbf{′}}\mathbf{curlV}\mathbf{)}\mathbf{×}\mathbf{\text{nd}}\mathbf{σ}$over any surface whose bounding curve is in the $\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{}\mathbf{y}\mathbf{)}$plane, where $\mathbf{V}\mathbf{=}\left(x−x^{2}z\right)\mathbf{i}\mathbf{+}\left({\mathrm{yz}}^3−y^2\right)\mathbf{j}\mathbf{+}\left(x^{2}y−\mathrm{xz}\right)\mathbf{k}$.

Find the direction of the line normal to the surface ${\mathit{x}}^{\mathbf{2}}\mathit{y}\mathbf{+}{\mathit{y}}^{\mathbf{2}}\mathit{z}\mathbf{+}{\mathit{z}}^{\mathbf{2}}\mathit{x}\mathbf{+}\mathbf{1}\mathbf{=}\mathbf{0}$at the point$\mathbf{(}\mathbf{1}\mathbf{,}\mathbf{2}\mathbf{,}\mathbf{}\mathbf{-}\mathbf{}\mathbf{1}\mathbf{)}$. Write the equations of the tangent plane and normal line at this point.

The force F = i - 2jacts at the point (0, 1, 2) Find the torque of F about the line $\mathbf{=}\mathbf{(}\mathbf{2}\mathbf{i}\mathbf{-}\mathbf{j}\mathbf{)}\mathbf{t}$.

(a) Find the directional derivative of $\mathit{Ï•}\mathbf{=}{\mathit{x}}^{\mathbf{2}}\mathbf{+}\mathit{s}\mathit{i}\mathit{n}\mathit{y}\mathbf{-}\mathit{x}\mathit{z}$ in the direction $\mathit{i}\mathbf{}\mathbf{+}\mathbf{}\mathbf{2}\mathit{j}\mathbf{}\mathbf{-}\mathbf{}\mathbf{2}\mathit{k}$ at the point $\mathbf{(}\mathbf{1}\mathbf{,}\frac{\mathbf{Ï€}}{\mathbf{2}}\mathbf{,}\mathbf{-}\mathbf{3}\mathbf{)}$.

(b)Find the equation of the tangent plane and the equations of the normal line to $\mathit{Ï•}\mathbf{=}\mathbf{5}$ at the point $\mathbf{(}\mathbf{1}\mathbf{,}\frac{\mathbf{Ï€}}{\mathbf{2}}\mathbf{,}\mathbf{-}\mathbf{3}\mathbf{)}$.

Evaluate each of the integrals in Problems$\mathbf{3}\mathbf{}$to $\mathbf{8}$as either a volume integral or a surface integral, whichever is easier.

$\mathbf{∭}\mathbf{∇}\mathbf{×}\mathbf{³Õ»åÏ„}$over the volumerole="math" localid="1657334446941" ${\mathbf{x}}^{\mathbf{2}}\mathbf{+}{\mathbf{y}}^{\mathbf{2}}\mathbf{≤}\mathbf{4}\mathbf{,}\mathbf{0}\mathbf{≤}\mathbf{z}\mathbf{≤}\mathbf{5}\mathbf{,}\mathbf{V}\mathbf{=}\left(\sqrt{x^2+y^2}\right)\mathbf{(}\mathbf{ix}\mathbf{+}\mathbf{jy}\mathbf{)}$