3.6: Line Integrals

The ingredients for line (also called path or contour) integrals are the following:

• A vector field $F = (M, N)$
• A curve $\gamma (t) = (x(t), y(t))$ defined for $a \le t \le b$

Then the line integral of $F$ along $\gamma$ is defined by

$$\int_{\gamma} F \cdot dr = \int_a^b F(\gamma (t)) \cdot y'(t)dt = \int_{\gamma} M\ dx + N\ dy. \nonumber$$

Example $\PageIndex{1}$

Let $F = (-y/r^2, x/r^2)$ and let $\gamma$ be the unit circle. Compute line integral of $F$ along $\gamma$.

Solution

You should be able to supply the answer to this example

Properties of line integrals

1. Independent of parametrization.

2. Reverse direction on curve $\Rightarrow$ change sign. That is,

$$\int_{-C} F \cdot dr = -\int_{C} F \cdot dr. \nonumber$$

(Here, $-C$ means the same curve traversed in the opposite direction.)

3. If $C$ is closed then we sometimes indicate this with the notation $\oint_{C} F \cdot dr = \oint_{C} M\ dx + N\ dy$.

Fundamental theorem for gradient fields

Theorem $\PageIndex{1}$ Fundamental theorem for gradient fields

If $F = nabla f$ then $\int_{\gamma} F \cdot d r = f(P) - f(Q)$, where $Q, P$ are the beginning and endpoints respectively of $\gamma$.

Proof

By the chain rule we have

$$\dfrac{df(\gamma (t))}{dt} = \nabla f(\gamma (t)) \cdot \gamma '(t) = F(\gamma (t)) \cdot y'(t). \nonumber$$

The last equality follows from our assumption that $F = \nabla f$. Now we can this when we compute the line integral:

$$\begin{array} {rcl} {\int_{\gamma} F \cdot dr} & = & {\int_a^b F (\gamma (t)) \cdot y' (t)\ dt} \\ {} & = & {\int_a^b \dfrac{df(\gamma (t))}{dt} dt} \\ {} & = & {f(\gamma (b)) - f(\gamma (a))} \\ {} & = & {f(P) - f(Q)} \end{array} \nonumber$$

Notice that the third equality follows from the fundamental theorem of calculus.

Definition: Potential Function

If a vector field $F$ is a gradient field, with $F = \nabla f$, then we call $f$ a potential function for $F$.

Note: the usual physics terminology would be to call $f$ the potential function for $F$.

independence and conservative functions

Definition: Path Independence

For a vector field $F$, the line integral $\int F \cdot dr$ is called path independent if, for any two points $P$ and $Q$, the line integral has the same value for $every$ path between $P$ and $Q$.

Theorem $\PageIndex{2}$

$\int_C F \cdot dr$ is path independent is equivalent to $\oint_{C} F \cdot dr = 0$ for any closed path.

Sketch of Proof

Draw two paths from $Q$ to $P$. Following one from $Q$ to $P$ and the reverse of the other back to $P$ is a closed path. The equivalence follows easily. We refer you to the more detailed review of line integrals and Green’s theorem for more details.

Definition: Conservative Vector Field

A vector field with path independent line integrals, equivalently a field whose line integrals around any closed loop is 0 is called a conservative vector field.

Theorem $\PageIndex{3}$

We have the following equivalence: On a connected region, a gradient field is conservative and a conservative field is a gradient field.

Proof

Again we refer you to the more detailed review for details. Essentially, if $F$ is conservative then we can define a potential function $f(x, y)$ as the line integral of $F$ from some base point to $(x, y)$.