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Evaluate the following integrals:

(a) $\mathbf{∫}\mathbf{(}{\mathbf{r}}^{\mathbf{2}}\mathbf{+}\mathbf{r}\mathbf{·}\mathbf{a}\mathbf{+}{\mathbf{a}}^{\mathbf{2}}\mathbf{)}{\mathbf{δ}}^{\mathbf{3}}\mathbf{(}\mathbf{r}\mathbf{-}\mathbf{a}\mathbf{)}\mathbf{»åÏ„}$, where a is a fixed vector, a is its magnitude.

(b) ${\mathbf{∫}}_{\mathbf{v}}{\mathbf{|}\mathbf{r}\mathbf{-}\mathbf{d}\mathbf{|}}^{\mathbf{2}}{\mathbf{δ}}^{\mathbf{3}}\mathbf{\left(}\mathbf{5}\mathbf{r}\mathbf{\right)}\mathbf{»åÏ„}$, where V is a cube of side 2, centered at origin and $\mathbf{b}\mathbf{=}\mathbf{4}\hat{\mathbf{y}}\mathbf{+}\mathbf{3}\hat{\mathbf{z}}$.

(c) ${\mathbf{∫}}_{\mathbf{v}}{\mathbf{r}}^{\mathbf{4}}\mathbf{+}{\mathbf{r}}^{\mathbf{2}}\mathbf{(}\mathbf{r}\mathbf{·}\mathbf{c}\mathbf{)}\mathbf{+}{\mathbf{c}}^{\mathbf{4}}{\mathbf{δ}}^{\mathbf{3}}\mathbf{(}\mathbf{r}\mathbf{-}\mathbf{c}\mathbf{)}\mathbf{»åÏ„}$, where is a cube of side 6, about the origin, $\mathbf{c}\mathbf{=}\mathbf{5}\hat{\mathbf{x}}\mathbf{+}\mathbf{3}\hat{\mathbf{y}}\mathbf{+}\mathbf{2}\hat{\mathbf{z}}$and c is its magnitude.

(d) ${\mathbf{∫}}_{\mathbf{v}}\mathbf{r}\mathbf{·}\mathbf{(}\mathbf{d}\mathbf{-}\mathbf{r}\mathbf{)}{\mathbf{δ}}^{\mathbf{3}}\mathbf{(}\mathbf{e}\mathbf{-}\mathbf{r}\mathbf{)}\mathbf{»åÏ„}$, where $\mathbf{d}\mathbf{=}\mathbf{(}\mathbf{1}\mathbf{,}\mathbf{2}\mathbf{,}\mathbf{3}\mathbf{)}\mathbf{,}\mathbf{}\mathbf{e}\mathbf{=}\mathbf{(}\mathbf{3}\mathbf{,}\mathbf{2}\mathbf{,}\mathbf{1}\mathbf{)}$, and where v is a sphere of radius 1.5 centered at $(2,2,2)$.

Evaluate the integral

$\mathit{J}\mathbf{=}{\mathbf{∫}}_{\mathbf{v}}{\mathit{e}}^{\mathbf{-}\mathbf{r}}\mathbf{(}\mathbf{∇}\mathbf{·}\frac{\hat{\mathbf{r}}}{{\mathbf{r}}^{\mathbf{2}}}\mathbf{)}\mathit{d}\mathit{τ}$,

where V is a sphere of radius R centered at origin by two different methods as in Ex. 1.16..

(a) Let${\mathit{F}}_{\mathbf{1}}\mathbf{=}{\mathit{x}}^{\mathbf{2}}\mathit{i}$and${\mathit{F}}_{\mathbf{2}}\mathbf{=}\mathit{x}\mathit{i}\mathbf{+}\mathit{y}\mathit{j}\mathbf{+}\mathit{z}\mathit{k}$Calculate the divergence and curl of${\mathit{F}}_{\mathbf{1}}$and${\mathit{F}}_{\mathbf{2}}$which one can be written as the gradient of a scalar? Find a scalar potential that does the job. Which one can be written as the curl of a vector? Find a suitable vector potential.

(b) Show thatlocalid="1654510098914" ${\mathit{F}}_{\mathbf{3}}\mathbf{=}\mathit{y}\mathit{z}\mathbf{}\mathit{i}\mathbf{+}\mathit{z}\mathit{x}\mathbf{}\mathit{j}\mathbf{+}\mathit{x}\mathit{y}\mathbf{}\mathit{k}$can be written both as the gradient of a scalar and as the curl of a vector. Find scalar and vector potentials for this function.

For Theorem 1, show that$\mathbf{\left(}\mathbf{d}\mathbf{\right)}\mathbf{⇒}\mathbf{\left(}\mathbf{a}\mathbf{\right)}\mathbf{,}\mathbf{}\mathbf{\left(}\mathbf{a}\mathbf{\right)}\mathbf{⇒}\mathbf{\left(}\mathbf{c}\mathbf{\right)}\mathbf{,}\mathbf{}\mathbf{\left(}\mathbf{c}\mathbf{\right)}\mathbf{⇒}\mathbf{\left(}\mathbf{b}\mathbf{\right)}\mathbf{,}\mathbf{}\mathbf{\left(}\mathbf{b}\mathbf{\right)}\mathbf{⇒}\mathbf{\left(}\mathbf{c}\mathbf{\right)}$ and $\left(c\right)\mathbf{⇒}\left(a\right)$

For Theorem 2, show that$\left(d\right)⇒\left(a\right),\left(a\right)⇒\left(c\right),\left(c\right)⇒\left(b\right),\left(b\right)⇒\left(c\right)\mathrm{and}\left(c\right)⇒\left(a\right)$

For Theorem 2, show that $\left(d\right)⇒\left(a\right)$, $\left(a\right)⇒\left(c\right)$, $\left(c\right)⇒\left(b\right)$, $\left(b\right)⇒\left(c\right)$and$\left(c\right)⇒\left(a\right)$

(a) Which of the vectors in Problem 1.15 can be expressed as the gradient of a scalar? Find a scalar function that does the job.

(b) Which can be expressed as the curl of a vector? Find such a vector.

Check the divergence theorem for the function

$\mathit{v}\mathbf{=}\left(r^{2}cos\mathrm{θ}\right)\mathbf{\text{ }}\stackrel{\mathbf{âˆ\S }}{\mathbf{r}}\mathbf{+}\left(r^{2}cos\mathrm{Ï•}\right)\mathbf{\text{ }}\stackrel{\mathbf{âˆ\S }}{\mathbf{θ}}\mathbf{+}{\mathit{r}}^{\mathbf{2}}\mathbf{\text{ }}\mathit{c}\mathit{o}\mathit{s}\mathit{θ}\mathit{s}\mathit{i}\mathit{n}\mathit{Ï•}\stackrel{\mathbf{âˆ\S }}{\mathbf{Ï•}}$

using as your volume one octant of the sphere of radius R(Fig. 1.48). Make sure you include the entiresurface. [Answer:$\mathit{Ï€}{\mathit{R}}^{\mathbf{4}}\mathbf{/}\mathbf{4}$]

Check Stokes' theorem using the function $\mathit{v}\mathbf{=}\mathit{a}\mathit{y}\mathit{i}\mathbf{ }\mathbf{+}\mathit{b}\mathit{x}\mathbf{ }\mathit{j}$(aand bare constants) and the circular path of radius R,centered at the origin in the xyplane. [Answer:$\mathit{π}{\mathit{R}}^{\mathbf{2}}\left(b-a\right)$ ],

Compute the line integral of

$v=6i\text{ }+y^2\text{ }j+\left(3y+z\right)k$

along the triangular path shown in Fig. 1.49. Check your answer using Stokes' theorem. [Answer:8/3]