The Fundamental Theorem for Line Integrals

A line integral describes the accumulated contribution of a vector field along a curve connecting two points. It is used to evaluate how the direction and magnitude of a vector field interact with the direction of motion along a path. In certain cases, this calculation can be greatly simplified by identifying whether the vector field is associated with a potential function.

Let F be a vector field in two or three dimensions. If there exists a scalar function g such that

\begin{equation*}\mathbf{F} = \nabla g\end{equation*}

Then F is called a gradient field, and g is called a potential function. The gradient vector points in the direction of the greatest increase of the potential function and gives the rate of that increase.

Fundamental Theorem for Line Integrals

For a smooth curve C beginning at point A and ending at point B, the Fundamental Theorem for Line Integrals states that

\begin{equation*}\int_C \mathbf{F} \cdot d\mathbf{r} = g(B) - g(A)\end{equation*}

This means that the line integral of a gradient field can be evaluated using only the values of the potential function at the endpoints of the curve.

Path Independence

An important consequence of the theorem is path independence. If two smooth curves, C₁ and C₂, connect the same points A and B, then

\begin{equation*}\int_{C_1} \mathbf{F} \cdot d\mathbf{r} =\int_{C_2} \mathbf{F} \cdot d\mathbf{r}\end{equation*}

So, the integral does not depend on the shape, length, or direction changes of the path, but only on the starting and ending locations.