I was solving some problems in Chapter 10 "Line Integrals" Section 10.18 from Apostol's Calculus. These problems involve finding potential functions for given functions $f$ which may or may not be gradients.
I was able to solve the following problem through essentially inspection but I am interested in understanding more systematic methods of finding the potential function.
- Let $S$ be the set of all $\pmb{x}\neq\pmb{0}$ in $\mathbb{R}^n$. Let $\pmb{r}=\lVert\pmb{x}\rVert$, and let $\pmb{f}$ be the vector field defined on $S$ by the equation
$$\pmb{f}(\pmb{x})=r^p\pmb{x}\tag{1}$$
where $p$ is a real constant. Find a potential function for $\pmb{f}$ on $S$. The case $p=-2$ should be treated separately.
I found a potential function for every case $p$ by simply inspecting the function and figuring out the antiderivative.
My question is how to find the potential using a more systematic method?
Here is how I did it by inspection. By writing out $\pmb{f}$
$$\pmb{f}(\pmb{x})=\Vert\pmb{x}\rVert^p\langle x_1,...,x_n\rangle\tag{2}$$
$$=\left (\sum\limits_{i=1}^n x_i^2 \right )^{p/2}\langle x_1, ..., x_n \rangle\tag{3}$$
$$=\left\langle \left (\sum\limits_{i=1}^n x_i^2 \right )^{p/2}\frac{2x_1}{2},\left (\sum\limits_{i=1}^n x_i^2 \right )^{p/2}\frac{2x_2}{2},...,\left (\sum\limits_{i=1}^n x_i^2 \right )^{p/2}\frac{2x_n}{n}\right\rangle\tag{4}$$
we can figure out by inspection that for $p\neq -2$ this is the gradient
$$=\nabla\left (\left (\sum\limits_{i=1}^n x_i^2 \right )^{\frac{p+2}{2}}\frac{2}{p+2}+C\right )\tag{5}$$
and for $p=-2$ we have
$$\pmb{f}(\pmb{x})=\left (\sum\limits_{i=1}^n x_i^2 \right )^{-1}\langle 2x_1,...,2x_n \rangle\frac{1}{2}\tag{6}$$
$$\nabla\left ( \ln{\sum\limits_{i=1}^n x_i^2} + C \right )\tag{7}$$
Systematic Way 1: Indefinite Integrals
If $\pmb{f}$ is a gradient of some unknown potential $\varphi$ then
$$f_1 = D_1\varphi, ..., f_n = D_n\varphi\tag{8}$$
where $f_k$ denotes the k-th component of $\pmb{f}$ and $D_k\varphi$ denotes the k-th partial derivative of $\varphi$.
Thus, we can integrate each of the $n$ equations in (8). Each time the result involves an unknown function of the other variables (the ones did not integrate over), and finding $\varphi$ means finding these functions such that all $n$ equations resulting from the $n$ integrals become the same expression.
In the current problem, these $n$ equations are
$$\int\left (\sum\limits_{i=1}^n x_i^2 \right )^{p/2}\frac{2x_1}{2}dx_1+A_1(x_2,...,x_n)=\left (\sum\limits_{i=1}^n x_i^2 \right )^{\frac{p+2}{2}}\frac{2}{p+2}+A_1(x_2,...,x_n)=\varphi(\pmb{x})$$
$$\int\left (\sum\limits_{i=1}^n x_i^2 \right )^{p/2}\frac{2x_2}{2}dx_2+A_2(x_1,x_3,...,x_n)=\left (\sum\limits_{i=1}^n x_i^2 \right )^{\frac{p+2}{2}}\frac{2}{p+2}+A_2(x_1,x_3,...,x_n)=\varphi(\pmb{x})$$
$$(...)$$
$$\int\left (\sum\limits_{i=1}^n x_i^2 \right )^{p/2}\frac{2x_n}{2}dx_n+A_n(x_1,...,x_{n-1})=\left (\sum\limits_{i=1}^n x_i^2 \right )^{\frac{p+2}{2}}\frac{2}{p+2}+A_n(x_1,...,x_{n-1})=\varphi(\pmb{x})$$
And we can see that for $p\neq -2$, by setting $A_1=A_2=...=A_n=C$ we have the result in all equations that
$$\varphi(\pmb{x})=\left (\sum\limits_{i=1}^n x_i^2 \right )^{\frac{p+2}{2}}\frac{2}{p+2} +C\tag{9}$$
For $p=-2$ we obtain
$$\varphi(\pmb{x})=\ln{\sum\limits_{i=1}^n x_i^2} + C$$
Systematic Way 2: Line Integral
I think there is another way which involves computing
$$\int_C \pmb{f}\cdot d\pmb{r}$$
where $C$ is some piecewise smooth path in the domain of $f$ between points $\pmb{a}$ and $\pmb{x}$. If we can compute this line integral we can equate it to $\varphi(\pmb{x})-\varphi(\pmb{a})$, and the justification I think would be the topic of this other question I asked.
Is this latter approach correct in this case?