Chapter 6: Vector Analysis

Expand the triple product for $\mathbf{a}\mathbf{=}\mathbf{\omega }\mathbf{\times }\left(\omega \times r\right)$ given in the discussion of Figure 3.8. If is perpendicular to $\mathbf{w}$(Problem 16), show that $\mathbf{a}\mathbf{=}\mathbf{‐}{\mathbf{\omega }}^{\mathbf{2}}\mathbf{r}$, and so find the elementary result that the acceleration is toward the center of the circle and of magnitude $\raisebox{1ex}{${\mathbf{v}}^{\mathbf{2}}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{r}$}\right.$.

Which, if either, of the two force fields

$$\begin{gathered} {\mathbf{F}}_{\mathbf{1}}\mathbf{=}\mathbf{-}\mathit{y}\mathbf{i}\mathbf{+}\mathit{x}\mathbf{j}\mathbf{+}\mathit{z}\mathbf{k}\mathbf{,}\mathbf{}\mathbf{}\mathbf{} \\ {\mathbf{F}}_{\mathbf{2}}\mathbf{=}\mathit{y}\mathbf{i}\mathbf{+}\mathit{x}\mathbf{j}\mathbf{+}\mathit{z}\mathbf{k} \end{gathered}$$

is conservative? Calculate for each field the work done in moving a particle around the circle$\mathbf{x}\mathbf{=}\mathbf{cos}\mathbf{t}\mathbf{,}\mathbf{y}\mathbf{=}\mathbf{sin}\mathbf{t}$in the (x, y) plane.

$\mathit{H}\mathit{i}\mathit{n}\mathit{t}\mathbf{:}\mathit{I}\mathit{n}\mathit{t}\mathit{e}\mathit{g}\mathit{r}\mathit{a}\mathit{t}\mathit{e}\mathbf{\left(}\mathit{g}\mathbf{\right)}$Derive the following vector integral theorems

(a) ${\mathbf{\int }}_{\mathbf{v}\mathbf{o}\mathbf{l}\mathbf{u}\mathbf{m}\mathbf{e}\mathbf{}\mathbf{\tau }}\mathbf{\nabla }\mathit{\varphi }\mathit{d}\mathit{\tau }\mathbf{=}{\mathbf{\oint }}_{\underset{\mathbf{i}\mathbf{n}\mathbf{c}\mathbf{l}\mathbf{o}\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{g}\mathbf{}\mathbf{\tau }}{\mathbf{s}\mathbf{u}\mathbf{r}\mathbf{f}\mathbf{a}\mathbf{c}\mathbf{e}}}\mathit{\varphi }\mathit{n}\mathit{d}\mathit{\sigma }$

Hint: In the divergence theorem (10.17), substitute $\mathbf{V}\mathbf{=}\mathbf{\varphi C}$where is an arbitrary constant vector, to obtain $\mathit{C}\mathbf{\cdot }\mathbf{\int }\mathbf{\nabla }\mathbf{}\mathit{\varphi }\mathit{d}\mathit{\tau }\mathbf{=}\mathit{C}\mathbf{\cdot }\mathbf{\oint }\mathit{\varphi }\mathbf{}\mathit{n}\mathit{d}\mathit{\sigma }$Since C is arbitrary, let C=i to show that the x components of the two integrals are equal; similarly, let C=j and C=k to show that the y components are equal and the z components are equal.

(b) ${\mathbf{\int }}_{\mathbf{volume}\mathbf{}\mathbf{\tau }}\mathbf{\nabla }\mathbf{\times }\mathit{V}\mathit{d}\mathit{\tau }\mathbf{=}{\mathbf{\oint }}_{\underset{\mathbf{inclosing}\mathbf{}\mathbf{\tau }}{\mathbf{surface}}}\mathit{n}\mathbf{\times }\mathit{V}\mathit{d}\mathit{\sigma }$

Hint: Replace in the divergence theorem by where is an arbitrary constant vector. Follow the last part of the hint in (a).

(c) localid="1659323284980" ${\mathbf{\int }}_{\underset{\mathbf{b}\mathbf{o}\mathbf{u}\mathbf{n}\mathbf{d}\mathbf{i}\mathbf{n}\mathbf{g}\mathbf{\sigma }}{\mathbf{c}\mathbf{u}\mathbf{r}\mathbf{v}\mathbf{e}}}\mathit{\varphi }\mathit{d}\mathit{r}\mathbf{=}{\mathbf{\oint }}_{\mathbf{s}\mathbf{u}\mathbf{r}\mathbf{f}\mathbf{a}\mathbf{c}\mathbf{e}}\mathit{\sigma }\mathbf{(}\mathit{n}\mathbf{\times }\mathbf{\nabla }\mathit{\varphi }\mathbf{)}\mathit{d}\mathit{\sigma }\mathbf{.}$

(d) ${\mathbf{\int }}_{\underset{\mathbf{bounding\sigma }}{\mathbf{curve}}}\mathit{\varphi }\mathit{d}\mathit{r}\mathbf{\times }\mathit{V}\mathbf{=}{\mathbf{\oint }}_{\mathbf{surface}}\mathbf{(}\mathit{n}\mathbf{\times }\mathbf{\nabla }\mathbf{)}\mathbf{\times }\mathit{V}\mathit{d}\mathit{\sigma }$

Hints for (c) and (d): Use the substitutions suggested in (a) and (b) but in Stokes' theorem (11.9) instead of the divergence theorem.

(e) ${\mathbf{\int }}_{\mathbf{volume}\mathbf{}\mathbf{\tau }}\mathbf{\nabla }\mathit{\varphi }\mathit{d}\mathit{\tau }\mathbf{=}{\mathbf{\oint }}_{\underset{\mathbf{inclosing}\mathbf{}\mathbf{\tau }}{\mathbf{surface}}}\mathit{\varphi }\mathit{V}\mathbf{·}\mathit{n}\mathit{d}\mathit{\sigma }\mathbf{-}{\mathbf{\oint }}_{\underset{\mathbf{inclosing}\mathbf{}\mathbf{\tau }}{\mathbf{surface}}}\mathit{\varphi }\mathit{V}\mathbf{·}\mathbf{\nabla }\mathit{\varphi }\mathit{n}\mathit{d}\mathit{\tau }\mathbf{.}$

Hint: Integrate (7.6) over volume and use the divergence theorem.

(f) localid="1659324199695" ${\mathbf{\int }}_{\mathbf{volume}\mathbf{}\mathbf{\tau }}\mathit{V}\mathbf{·}\mathbf{(}\mathbf{\nabla }\mathbf{\times }\mathbf{\cup }\mathbf{)}\mathit{d}\mathit{\tau }\mathbf{=}{\mathbf{\int }}_{\mathbf{volume}\mathbf{}\mathbf{\tau }}\mathit{V}\mathbf{·}\mathbf{(}\mathbf{\nabla }\mathbf{\times }\mathbf{\cup }\mathbf{)}\mathit{d}\mathit{\tau }\mathbf{+}{\mathbf{\oint }}_{\underset{\mathbf{inclosing}\mathbf{}\mathbf{\tau }}{\mathbf{surface}}}\mathbf{(}\mathbf{\cup }\mathbf{\times }\mathbf{V}\mathbf{)}\mathbf{·}\mathit{n}\mathit{d}\mathit{\sigma }$

Hint: Integrate (h) in the Table of Vector Identities (page 339) and use the divergence theorem.

(g) ${\mathbf{\int }}_{\mathbf{s}\mathbf{u}\mathbf{r}\mathbf{f}\mathbf{a}\mathbf{c}\mathbf{e}\mathbf{}\mathbf{o}\mathbf{f}\mathbf{}\mathbf{\sigma }}\mathbf{}\mathit{\varphi }\mathbf{(}\mathbf{\nabla }\mathbf{\times }\mathit{V}\mathbf{)}\mathbf{\cdot }\mathit{n}\mathit{d}\mathit{\sigma }\mathbf{=}{\mathbf{\int }}_{\mathbf{s}\mathbf{u}\mathbf{r}\mathbf{f}\mathbf{a}\mathbf{c}\mathbf{e}\mathbf{}\mathbf{o}\mathbf{f}\mathbf{}\mathbf{\sigma }}\mathbf{(}\mathbf{\nabla }\mathbf{\times }\mathbf{\nabla }\mathit{\varphi }\mathbf{)}\mathbf{\cdot }\mathit{n}\mathit{d}\mathit{\sigma }\mathbf{+}{\mathbf{\oint }}_{\underset{\mathbf{b}\mathbf{o}\mathbf{u}\mathbf{n}\mathbf{d}\mathbf{i}\mathbf{n}\mathbf{g}}{\mathbf{c}\mathbf{u}\mathbf{r}\mathbf{v}\mathbf{e}}}\mathit{\varphi }\mathit{V}\mathbf{\cdot }\mathit{d}\mathit{r}$

$\mathit{H}\mathit{i}\mathit{n}\mathit{t}\mathbf{:}\mathit{I}\mathit{n}\mathit{t}\mathit{e}\mathit{g}\mathit{r}\mathit{a}\mathit{t}\mathit{e}\mathbf{\left(}\mathit{g}\mathbf{\right)}$in the Table of Vector Identities (page 339) and use Stokes' Theorem.

Find, $\mathbf{\nabla }\mathbf{r}$, where$\mathbf{r}\mathbf{=}\sqrt{{\mathbf{x}}^{\mathbf{2}}\mathbf{+}{\mathbf{y}}^{\mathbf{2}}}$ , using (6.7) and also using (6.3). Show that your results are the same by using (4.11) and (4.12).

Is F = yi+xzj+zk conservative? Evaluate $\mathbf{\int }\mathit{F}\mathbf{.}\mathit{d}\mathit{r}$from along the paths

(a) broken line (0,0,0)to (1,1,1) to (1,1,0) to (1,1,1)

(b) Straight line connecting the points.

Find vector fields $\mathbf{A}$such that role="math" localid="1657346627450" $\mathbf{V}\mathbf{}\mathbf{=}\mathbf{}\mathbf{curl}\mathbf{}\mathbf{A}\mathbf{}$for each givenrole="math" localid="1657346639484" $\mathbf{V}\mathbf{.}$

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

For the force field $\mathit{F}\mathbf{=}\mathbf{-}\mathit{y}\mathit{i}\mathbf{+}\mathit{x}\mathit{j}\mathbf{+}\mathit{z}\mathit{k}\mathbf{,}$ calculate the work done in moving a particle from (1,0,0) torole="math" localid="1664273455603" $\left(-1,0,\pi \right)$

(a) along the helix$\mathit{x}\mathbf{=}\mathit{c}\mathit{o}\mathit{s}\mathit{t}\mathbf{,}\mathit{y}\mathbf{=}\mathit{s}\mathit{i}\mathit{n}\mathit{t}\mathbf{,}\mathit{z}\mathbf{=}\mathit{t}\mathbf{;}$

(b) along the straight line joining the points.

Given

$$\begin{gathered} {\mathbf{F}}_{\mathbf{1}}\mathbf{=}\mathbf{-}\mathbf{2}\mathbf{yi}\mathbf{+}\left(z-2x\right)\mathbf{j}\mathbf{+}\left(y+z\right)\mathbf{k} \\ {\mathbf{F}}_{\mathbf{2}}\mathbf{=}\mathbf{yi}\mathbf{+}\mathbf{2}\mathbf{xj} \end{gathered}$$:

(a) Is ${\mathit{F}}_{\mathbf{1}}$conservative? Is ${\mathbf{\text{F}}}_{\mathbf{2}}$conservative?

(b) Find the work done by 2 on a particle that moves around the ellipse $\mathit{x}\mathbf{=}\mathit{c}\mathit{o}\mathit{s}\mathit{\theta }$, $\mathbf{y}\mathbf{=}\mathbf{2}\mathbf{sin\theta }$from $\mathit{\theta }\mathbf{=}\mathbf{0}\mathbf{}\mathit{t}\mathit{o}\mathbf{}\mathit{\theta }\mathbf{=}\mathbf{2}\mathit{\pi }$

(c) For any conservative force in this problem find a potential function Vsuch

that $\mathbf{F}\mathbf{=}\mathbf{-}\mathbf{\nabla }\mathbf{V}$ (d) Find the work done by on a particle that moves along the straight line from $\left(0,1,0\right)\mathbf{}\mathbf{to}\mathbf{}\left(0,2,5\right)$

(e) Use Green’s theorem and the result of Problem 9.7 to do Part (b) above.

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