Chapter 15.3, Problem 34ES

(a) In Example 3 of Section 15.1 we showed that $\varphi \left(x,y\right)=-\frac{c}{{\left(x^2+y^2\right)}^{1/2}}$ is a potential function for the two-dimensional inverse-square filed $\text{F}\left(x,y\right)=\frac{c}{{\left(x^2+y^2\right)}^{3/2}}\left(x\text{i}+y\text{j}\right)$ but we did not explain how the potential function $\varphi \left(x,y\right)$ was obtained. Use Theorem 15.3.3 to show that the two-dimensional inverse-square filed is conservative everywhere except at the origin, and then use the method of Example 4 to derive the formula for $\varphi \left(x,y\right)$ .

(b) Use an appropriate generalization of the method of Example 4 to derive the potential function $\varphi \left(x,y,z\right)=-\frac{c}{{\left(x^2+y^2+z^2\right)}^{1/2}}$ for the three-dimensional inverse-square filed given by Formula (5) of Section 15.1.