Chapter 14.2, Problem 65E

a.

To determine

To find: The work done in moving an object in the radial field $\text{F}=\frac{\langle x,y,z\rangle }{{\left|r\right|}^p}$ along the curve $r\left(t\right)=\langle t,t,t\rangle$ for $1\le t\le a$ with $p=2$.

b.

To determine

To check: The work required to move an object in the radial field $\text{F}=\frac{\langle x,y,z\rangle }{{\left|r\right|}^2}$ along the curve $r\left(t\right)=\langle t,t,t\rangle$ for $1\le t\le a$ is finite or infinite as $a\to \infty$.

c.

To determine

To find: The work done in moving an object in the radial field $\text{F}=\frac{\langle x,y,z\rangle }{{\left|r\right|}^p}$ along the curve $r\left(t\right)=\langle t,t,t\rangle$ for $1\le t\le a$ with $p=4$.

d.

To determine

To check: The work required to move an object in the radial field $\text{F}=\frac{\langle x,y,z\rangle }{{\left|r\right|}^4}$ along the curve $r\left(t\right)=\langle t,t,t\rangle$ for $1\le t\le a$ is finite or infinite as $a\to \infty$.

e.

To determine

To find: The work done in moving an object in the radial field $\text{F}=\frac{\langle x,y,z\rangle }{{\left|r\right|}^p}$ along the curve $r\left(t\right)=\langle t,t,t\rangle$ for $1\le t\le a$ with $p>1$.

f.

To determine

To find: The value of $p$ for the work done in moving an object in the radial field $\text{F}=\frac{\langle x,y,z\rangle }{{\left|r\right|}^p}$ along the curve $r\left(t\right)=\langle t,t,t\rangle$ for $1\le t\le a$ with $p>1$ is finite as $a\to \infty$.