Fundamental Theorem of Line Integrals

This article needs to be linked to other articles. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding these links. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{MissingLinks}} from the code.

This article, or a section of it, needs explaining. In particular: Explicate domain of validity, etc. etc. etc. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code.

Theorem

Let $\CC$ be a smooth curve given by the vector function $\map {\mathbf r} t$ for $a \le t \le b$.

Let $f$ be a differentiable function of two or three variables whose gradient vector $\nabla f$ is continuous on $\CC$.

Then:

$\ds \int_\CC \nabla f \cdot d \mathbf r = \map f {\map {\mathbf r} b} - \map f {\map {\mathbf r} a}$

Proof

\(\ds \int_\CC \nabla f \cdot \rd \mathbf r\)

\(=\)

\(\ds \int_a^b \nabla f \cdot \map {\mathbf r'} t \rd t\)

\(\ds \)

\(=\)

\(\ds \int_a^b \frac {\partial f} {\partial x} \frac {\rd x} {\rd t} + \frac {\partial f} {\partial y} \frac {\rd y} {\rd t} + \frac {\partial f} {\partial z} \frac {\rd z} {\rd t} \rd t\)

\(\ds \)

\(=\)

\(\ds \int_a^b \frac \rd {\rd t} \map f {\map {\mathbf r} t} \rd t\)

Chain Rule for Real-Valued Functions

\(\ds \)

\(=\)

\(\ds \map f {\map {\mathbf r} b} - \map f {\map {\mathbf r} a}\)

Fundamental Theorem of Calculus

$\blacksquare$