line integral

A line integral or path integral is an integral where the function to be integrated is evaluated along a curve (defined as one-manifold with boundaries). Line integral of complex-valued functions on the complex plane is often called contour integral.

Many formulae in physics, such as work (), have a line integral analogues, in this case being , which computes the work done on the object moving through a force field along the path .

Line Integral with respect to Arc Length

The line integral of a function is denoted by:

where is the differential of arc length. We used here because we move along the curve .

To evaluate this integral, we typically parameterized the curve path:

For example, in 2D where ,

Line Integral of a Vector Field

The line integral of a vector field along a path is defined as:

Geometric Reasoning

Line integrals are intrinsically geometric, so we can sometimes use geometric reasoning to avoid tedious calculations.

Note that The velocity vector will be tangent to the trajectory. Thus,

And we can have our integral as

Let be a circle of radius at origin counterclockwise and . Find

() at all point of the circle. So the integral is zero.

Example: Let be the curve . Find

We know that gradient is perpendicular to level curve, so . Therefore, .

Gradient Vector Field

When the vector field is the gradient of another function (i.e., ), we can apply the fundamental theorem of calculus for line integrals to compute the line integral:

See Also

• surface integral
• Green’s theorem: converting line integral in 2D to double integral
• contour integral
• conservative vector field