Conservative Field

A vector field $\boldsymbol{F}$ in 2-space or 3-space is conservative on a region $D$ iff it is the gradient field of some scalar function $\phi$ defined on $D$,

$\phi$ is a potential function, or scalar potential, of $\boldsymbol{F}$.

Curl Condition

If $\boldsymbol{F}$ is conservative on $D$ and $\phi$ has continuous second-order partial derivatives, then

The converse needs a condition on $D$. If $D$ is simply connected and $\text{curl}\,\boldsymbol{F} = \boldsymbol{0}$ on $D$, then $\boldsymbol{F}$ is conservative on $D$.

Conservative Field Test

Let $P(x,y)$ and $Q(x,y)$ have continuous first-order partial derivatives on an open, simply connected region $D$ in the plane. The vector field

is conservative on $D$ iff, for each point in $D$,

The region must be simply connected. On a region with a hole, $\partial P/\partial y = \partial Q/\partial x$ can hold everywhere without $\boldsymbol{F}$ being conservative.

Integral over a Closed Curve

Let $\boldsymbol{F}$ be conservative on $D$ and $C$ a piecewise smooth closed curve in $D$. Then

Equivalence of Conservative Fields

Let $\boldsymbol{F}(x,y,z) = f\hat{i} + g\hat{j} + h\hat{k}$ be continuous on an open connected region $D \subseteq \mathbb{R}^3$. The following are equivalent:

• $\boldsymbol{F}$ is conservative on $D$
• $\displaystyle\int_C \boldsymbol{F} \cdot \text{d}\boldsymbol{r} = 0$ for every piecewise smooth closed curve $C$ in $D$
• For any 2 points $P$, $Q$ in $D$, $\displaystyle\int_C \boldsymbol{F} \cdot \text{d}\boldsymbol{r}$ is independent of the piecewise smooth curve $C$ in $D$ from $P$ to $Q$

Finding the Potential Function

Let $\boldsymbol{F} = f\hat{i} + g\hat{j} + h\hat{k}$ be conservative on $D$. A potential function $\phi$ satisfies

Steps:

1. Integrate $f$ with respect to $x$ to get $\phi = \int f\,\text{d}x + a(y,z)$.
2. Differentiate $\phi$ with respect to $y$ and equate to $g$. Solve for $a_y$, then integrate to get $a(y,z) = \int a_y\,\text{d}y + b(z)$.
3. Differentiate $\phi$ with respect to $z$ and equate to $h$. Solve for $b'(z)$, then integrate to get $b(z)$.
4. Substitute back. $\phi$ is unique up to an additive constant.

In the plane, drop $z$ and stop after step 2.

Worked Example

Find a potential function of

Check the curl.

• $\partial h/\partial y = 0 = \partial g/\partial z$
• $\partial f/\partial z = 1 = \partial h/\partial x$
• $\partial g/\partial x = 2x = \partial f/\partial y$

So $\text{curl}\,\boldsymbol{F} = \boldsymbol{0}$ on $\mathbb{R}^3$, which is simply connected, and $\boldsymbol{F}$ is conservative.

Step 1.

Step 2.

Step 3.

Result.