Conservative vector field

A vector field $\mathbf F : \R^k \to \R^k$ is considered conservative if it is the gradient of some other function $f$. One of the important properties of a conservative vector field is path independence. Any line integral from point $\mathbf A$ to $\mathbf B$ across a conservative vector field is equal, regardless of the path taken.

Properties

If a vector field is conservative $\mathbf F = (P, Q) = \nabla f$. Then $P=f_x$ and $Q=f_y$. For a sufficiently nice function $f$, this means that $P_y = Q_x$. If this does not hold, then either $\mathbf F$ is not conservative or it is a gradient of a not-nice function. However, if $P_y = Q_x$ does hold, that’s not enough to conclude that $\mathbf F$ is conservative.

Checking if a field is conservative

We can check if a vector field $\mathbf F = (P, Q)$ is conservative using the following steps. The same approach works for fields in arbitrary dimensions, but it’s simplest to explain in 2-space.

1. We’re looking for $f$ such that $\nabla f = \mathbf F$, so set $f_x = P$ and $f_y = Q$.

2. Integrate $f = \int Pdx + C_1(y)$ and $f = \int Qdy + C_2(x)$ to find two potential expressions for $f$. Note the integration constants.

3. Since when integrating $\int Pdx$ we’re treating $y$ as a constant, the integration constant $C_1(y)$ might be a function of $y$. Likewise, the integration constant for $\int Qdy$, $C_2(x)$ might be a function of $x$.

4. Compare the results for $f$ to see if there are some $C_1(y)$ and $C_2(x)$ that make the expressions equal.