Chapter 6: Vector Analysis

As in Problem 17, find the following gradients in two ways and show that your answers are equivalent $\mathbf{\nabla }{\mathit{r}}^{\mathbf{2}}$

Find vector fields $\mathbf{A}\mathbf{}$such that $\mathbf{V}\mathbf{}\mathbf{=}\mathbf{}\mathbf{curl}\mathbf{}\mathbf{A}$for each given$\mathbf{V}\mathbf{=}\mathbf{i}\left({\mathrm{ze}}^{\mathrm{zy}}+\mathrm{xsin}\mathrm{zx}\right)\mathbf{+}\mathbf{jxcos}\mathbf{}\mathbf{xz}\mathbf{-}\mathbf{kzsin}\mathbf{}\mathbf{zx}$

$\mathbf{\iint }\left(\nabla \times V\right)\mathbf{.}\mathit{n}\mathit{d}\mathit{\sigma }$over the surface consisting of the four slanting faces of a pyramid whose base is the square in the (x,y) plane with corners at $\left(0,0\right)\mathbf{,}\left(0,2\right)\mathbf{,}\left(2,0\right)\mathbf{,}\left(2,2\right)$, and whose top vertex is at (1,1,2) where$\mathit{V}\mathbf{=}\left(x^{2}z-2\right)\mathit{i}\mathbf{+}\left(x+y+z\right)\mathit{j}\mathbf{-}\mathit{x}\mathit{y}\mathit{z}\mathit{k}$.

Verify equation (6.8); that is, find∇fin spherical coordinates as we did for cylindrical coordinates.

Consider a uniform distribution of total mass m’ over a spherical shell of radius r’. The potential energy φ of a mass m in the gravitational field of the spherical shell is

$$\begin{gathered} \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{const}\mathbf{.}\mathbf{,}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{if}\mathbf{}\mathbf{}\mathbf{m}\mathbf{}\mathbf{is}\mathbf{}\mathbf{inside}\mathbf{}\mathbf{the}\mathbf{}\mathbf{spherical}\mathbf{}\mathbf{shell} \\ \mathbf{\varphi }\mathbf{=}\left\{-\frac{\mathrm{Cm}\text{'}}{r},\mathrm{if}m\mathrm{is}\mathrm{outside}\mathrm{the}\mathrm{spherical}\mathrm{shell},\mathrm{where}r\mathrm{is}\mathrm{the}\mathrm{distance}\right\} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{from}\mathbf{}\mathbf{the}\mathbf{}\mathbf{centre}\mathbf{}\mathbf{of}\mathbf{}\mathbf{the}\mathbf{}\mathbf{sphere}\mathbf{}\mathbf{to}\mathbf{}\mathbf{m}\mathbf{,}\mathbf{and}\mathbf{}\mathbf{C}\mathbf{}\mathbf{is}\mathbf{}\mathbf{constant} \end{gathered}$$

Assuming that the earth is a spherical ball of radius R and constant density, find the potential and the force on a mass m outside and inside the earth. Evaluate the constants in terms of the acceleration of gravity g, to get

role="math" localid="1664278476490" $\mathit{F}\mathbf{=}\mathbf{-}\frac{\mathbf{m}\mathbf{g}{\mathbf{R}}^{\mathbf{2}}}{{\mathbf{r}}^{\mathbf{2}}}{\mathit{e}}_{\mathbf{r}}$ androle="math" localid="1664278464442" $\mathit{\varphi }\mathbf{=}\mathbf{-}\frac{\mathbf{m}\mathbf{g}{\mathbf{R}}^{\mathbf{2}}}{{\mathbf{r}}^{\mathbf{2}}}$ m outside the earth

role="math" localid="1664278454050" $\mathit{F}\mathbf{=}\mathbf{-}\frac{\mathbf{m}\mathbf{g}\mathbf{r}}{\mathbf{R}}{\mathit{e}}_{\mathbf{r}}$ and $\mathit{\varphi }\mathbf{=}\mathbf{-}\frac{\mathbf{m}\mathbf{g}}{\mathbf{2}\mathbf{R}}\left(r^2-3R^2\right)$ m outside the earth.

Find vector fields A such that $\mathbf{V}\mathbf{=}\mathbf{curlA}$for each given V.

$\mathbf{V}\mathbf{=}\mathbf{-}\mathbf{K}$

$\mathbf{\iint }\mathbf{V}\mathbf{\times }\mathbf{nd\sigma }$over the entire surface of the sphere, iflocalid="1657353129148" $\mathbf{(}\mathbf{x}\mathbf{-}\mathbf{2}{\mathbf{)}}^{\mathbf{2}}\mathbf{+}\mathbf{(}\mathbf{y}\mathbf{+}\mathbf{3}{\mathbf{)}}^{\mathbf{2}}\mathbf{+}\mathbf{(}\mathbf{z}{\mathbf{)}}^{\mathbf{2}}\mathbf{=}\mathbf{9}\mathbf{\text{, if}}\mathbf{V}\mathbf{=}\mathbf{(}\mathbf{3}\mathbf{x}\mathbf{-}\mathit{y}\mathit{z}\mathbf{)}\mathbf{i}\mathbf{+}\mathbf{(}{\mathbf{z}}^{\mathbf{2}}\mathbf{-}{\mathbf{y}}^{\mathbf{2}}\mathbf{)}\mathbf{j}\mathbf{+}\mathbf{(}\mathbf{2}\mathbf{y}\mathbf{z}\mathbf{+}{\mathbf{x}}^{\mathbf{2}}\mathbf{)}\mathbf{k}$

Find vector fields$\mathbf{A}\mathbf{}$ such that $\mathbf{V}\mathbf{}\mathbf{=}\mathbf{}\mathbf{curl}\mathbf{}\mathbf{A}$for each given V.$\mathbf{V}=(\mathbf{y}+\mathbf{z})\mathbf{i}+(\mathbf{x}-\mathbf{z})\mathbf{j}+\left({\mathbf{x}}^2+{\mathbf{y}}^2\right)\mathbf{k}$

$\mathbf{\iint }\mathbf{F}\mathbf{\times }\mathbf{nd\sigma }$where $\mathbf{F}\mathbf{=}\left(y^2-x^2\right)\mathbf{i}\mathbf{+}\mathbf{(}\mathbf{2}\mathbf{xy}\mathbf{-}\mathbf{y}\mathbf{)}\mathbf{j}\mathbf{+}\mathbf{3}\mathbf{zk}$and $\mathrm{\sigma }$is the entire surface of the tin can bounded by the cylinder

role="math" localid="1657353627256" ${\mathbf{x}}^{\mathbf{2}}\mathbf{+}{\mathbf{y}}^{\mathbf{2}}\mathbf{=}\mathbf{16}$

role="math" localid="1657353639412" $\mathbf{z}\mathbf{=}\mathbf{3}$

$\mathbf{\iint }\mathbf{r}\mathbf{\times }\mathbf{nd\sigma }$over the entire surface of the hemisphere${\mathrm{x}}^2+{\mathrm{y}}^2+{\mathrm{z}}^2=9$,$\mathrm{z}\ge 0$

where $\mathrm{r}=\mathrm{xi}+\mathrm{yj}+\mathrm{zk}$.