Chapter 15 Conceptual Project:

Showing Your Potential

Recall from Section 15.7 that if F is a vector field in ${ℝ}^3$ so that $\nabla \times \mathbf{F}=\mathbf{0}$ (such vector fields are called curl-free) on an open, simply connected domain in space, then F is conservative, that is, there is a scalar potential f so that $\nabla f=\mathbf{F}$. On the other hand, it can be shown that if F is divergence-free, that is, if $\nabla \cdot \mathbf{F}=0$, then there is a vector field P such that $\nabla \times \mathbf{P}=\mathbf{F}$ (such a vector field is called a vector potential for F). In this project you will discover a way of finding a vector potential for a given divergence-free vector field F.

1. Suppose

   $\mathbf{F}\left(x,y,z\right)=⟨F_1\left(x,y,z\right),F_2\left(x,y,z\right),F_3\left(x,y,z\right)⟩$

   and

   $\mathbf{P}\left(x,y,z\right)=⟨P_1\left(x,y,z\right),P_2\left(x,y,z\right),P_3\left(x,y,z\right)⟩$

   are vector fields so that $\nabla \times \mathbf{P}=\mathbf{F}$; that is, P is a vector potential for F. Show that for any differentiable scalar field f, $\nabla \times \left(\mathbf{P}+\nabla f\right)=\mathbf{F}$; that is, $\mathbf{P}+\nabla f$ is another vector potential for F. (Hint: See Exercise 41 of Section 15.4.)

2. If f is any scalar field such that ${\displaystyle \frac{\mathit{\partial }f}{\mathit{\partial }x}}=-P_1$, show that if we define $\hat{P}=\mathbf{P}+\nabla f$, then $\widehat{P_1}=0$.

3. Use Questions 1 and 2 to argue that if the vector field F has a vector potential P, then it has one whose first component is zero. In other words, we may assume throughout our discussion that $\mathbf{P}=⟨0,P_2,P_3⟩$.

In Questions 4–6, you will be guided to show that given a divergence-free vector field F, it is possible and fairly straightforward to find a vector potential of the form described in Question 3.

4. Assume that

   $\mathbf{F}\left(x,y,z\right)=⟨F_1\left(x,y,z\right),F_2\left(x,y,z\right),F_3\left(x,y,z\right)⟩$

   is a vector field such that $\nabla \cdot \mathbf{F}=0$, and P is any vector field of the form $\mathbf{P}=⟨0,P_2,P_3⟩$. Show that P is a vector potential for F if the following equalities hold.

   ${\displaystyle \frac{\mathit{\partial }{P}_3}{\mathit{\partial }y}}-{\displaystyle \frac{\mathit{\partial }{P}_2}{\mathit{\partial }z}}=F_1$    $-{\displaystyle \frac{\mathit{\partial }{P}_3}{\mathit{\partial }x}}=F_2$     $-{\displaystyle \frac{\mathit{\partial }{P}_2}{\mathit{\partial }x}}=F_3$

5. For the vector field F in Question 4, define $P_2\left(x,y,z\right)={\displaystyle {\int }_{x_0}^x{F}_3\left(t,y,z\right)\mathit{d}t+C_2\left(y,z\right)}$ and $P_3\left(x,y,z\right)=-{\displaystyle {\int }_{x_0}^x{F}_2\left(t,y,z\right)\mathit{d}t+C_3\left(y,z\right)}$, where $x_0$ is an arbitrary starting value and $C_2$ and $C_3$ are arbitrary functions of the variables y and z. Show that $\mathbf{P}\left(x,y,z\right)=⟨0,P_2\left(x,y,z\right),P_3\left(x,y,z\right)⟩$ satisfies the last two equations in Question 4.

6. Show that in Question 5, it is always possible to choose $C_2\left(y,z\right)$ and $C_3\left(y,z\right)$ to satisfy ${\displaystyle \frac{\mathit{\partial }{P}_3}{\mathit{\partial }y}}-{\displaystyle \frac{\mathit{\partial }{P}_2}{\mathit{\partial }z}}=F_1$, and conclude that $\mathbf{P}\left(x,y,z\right)=⟨0,P_2\left(x,y,z\right),P_3\left(x,y,z\right)⟩$ will then be a vector potential for F. (Hint: Use the fact that $\nabla \cdot \mathbf{F}=0$.)

7. Show that the vector field

   $\mathbf{F}\left(x,y,z\right)=⟨2x^{2}yz,-2x{y}^{2}z,x^{2}y⟩$

   is divergence‑free, and follow the steps outlined in Questions 5 and 6 to find a vector potential for F.(Answers may vary.)