$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

# 3.6: Line Integrals

$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

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

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

##### 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

## 3.7.1 Properties of line integrals

1. Independent of parametrization.

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

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

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

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

## 3.7.2 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

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

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

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

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 $$F$$.

## Path 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 $$(x, y)$$.