A line integral is an integral where the function to be integrated is evaluated along a curve. There are two main types of line integrals we consider in vector calculus:

Scalar Field Line Integral

For a scalar field $f(x, y, z)$ and a curve $C$, the line integral is:

Here, $ds$ represents an infinitesimal arc length along the curve $C$. This computes the area of a "curtain" of height $f$ following the path $C$.

Vector Field Line Integral

More common in physics is the line integral of a vector field $\vec{F}(x, y, z)$ along a curve $C$. It is defined as:

Where $d\vec{r}$ is an infinitesimal vector displacement along the path.

The most important physical interpretation of a vector line integral is the concept of mechanical work. The work done $W$ by a force field $\vec{F}$ on a particle moving along a path $C$ is precisely the line integral of the force along that path.

To compute a line integral, we typically parameterize the curve $C$ using a parameter $t$. Let the curve be given by $\vec{r}(t) = x(t)\hat{i} + y(t)\hat{j} + z(t)\hat{k}$, for $a \leq t \leq b$.

We can express the differential displacement as:

Then the line integral transforms into a standard single-variable integral over $t$:

A vector field $\vec{F}$ is called conservative if it can be expressed as the gradient of a scalar potential function $\phi(x, y, z)$:

Fundamental Theorem of Line Integrals

For a conservative field, the line integral is path-independent. It depends only on the start point $A$ and end point $B$, not on the specific curve $C$ connecting them:

Closed Loop Integral

An immediate consequence is that the line integral of a conservative field around any closed loop is zero (since start and end points are the same, $A = B$):