Study guide and
7 practice problems
on:
Conservative vector fields and potential functions
$\newcommand{\bfA}{\mathbf{A}}$ $\newcommand{\bfB}{\mathbf{B}}$ $\newcommand{\bfC}{\mathbf{C}}$ $\newcommand{\bfF}{\mathbf{F}}$ $\newcommand{\bfI}{\mathbf{I}}$ $\newcommand{\bfa}{\mathbf{a}}$ $\newcommand{\bfb}{\mathbf{b}}$ $\newcommand{\bfc}{\mathbf{c}}$ $\newcommand{\bfd}{\mathbf{d}}$ $\newcommand{\bfe}{\mathbf{e}}$ $\newcommand{\bfi}{\mathbf{i}}$ $\newcommand{\bfj}{\mathbf{j}}$ $\newcommand{\bfk}{\mathbf{k}}$ $\newcommand{\bfn}{\mathbf{n}}$ $\newcommand{\bfr}{\mathbf{r}}$ $\newcommand{\bfu}{\mathbf{u}}$ $\newcommand{\bfv}{\mathbf{v}}$ $\newcommand{\bfw}{\mathbf{w}}$ $\newcommand{\bfx}{\mathbf{x}}$ $\newcommand{\bfy}{\mathbf{y}}$ $\newcommand{\bfz}{\mathbf{z}}$
A vector field $\mathbf{F}(x,y)$ is defined to be conservative if there exists a $\phi(x,y)$ such that $\mathbf{F}(x,y) = \nabla \phi(x,y)$. $\phi$ is known as a potential function.
Study Guide
To check if $\bfF = \langle u, v \rangle$ is conservative, verify $\partial_y u(x,y) = \partial_x v(x,y).$
(5 problems)
If $\bfF$ is conservative, then its potential function $\phi$ can be found by integrating each component of $\bfF(x,y) = \nabla \phi(x,y)$ and combining into a single function $\phi$.
(4 problems)
Related topics
Vector fields
(10 problems)
Multivariable calculus
(147 problems)
Practice problems
Let $\mathbf{F}(x,y) = 2 x \log y \ \mathbf{i} + x^2 / y \ \mathbf{j}$. Is $\mathbf{F}(x,y)$ conservative?
Solution
$\mathbf{F}(x,y) = 2 \mathbf{i} + 3 \mathbf{j}$ is a conservative vector field. Find a potential function for it.
Solution
The vector field $\mathbf{F}(x,y) = -y \mathbf{i} + x \mathbf{j}$ is not conservative. Try to find the potential function for it by integrating each component. What goes wrong?
Solution
Is $\mathbf{F}(x,y) = \frac{1}{x+y} \mathbf{i} + \frac{1}{x+y} \mathbf{j}$ conservative?
If so, find a $\phi(x,y)$ such that $\mathbf{F}(x,y) = \nabla \phi(x,y)$?
Solution
Find $\int_C \bigl( \frac{1}{x+y} \mathbf{i} + \frac{1}{x+y} \mathbf{j} \bigr) \cdot d\mathbf{r}$, where $C$ is the segment of the unit circle going counterclockwise from $(1,0)$ to $(0,1)$.
Solution
Let $\mathbf{F}(x,y) = \langle 2, 3 \rangle$. Suppose $C$ is a curve connecting $(0,0)$ to $(1,1)$. Does the value of $\int_C \mathbf{F}\cdot d\mathbf{r}$ depend on the shape of the curve $C$? If not, find the value of the integral.
Solution
Let $C$ be a circle of radius $a$ centered at the origin, traversed counterclockwise. For what nonzero value of $a$ is $\oint_C \bigl(-y + \frac{1}{3} y^3 + x^2 y \bigr) dx = 0$?
Solution