Fundamental Theorem of Line Integral

Fundamental Theorem of Line Integral

Theorem 1. Let $C$ be a smooth curve joining the point $A$ to the point $B$ in the plane or in space and parametrized by $\mathbf{r}(t)$. Let $f$ be a differentiable function with a continuous gradient vector $\mathbf{F} = \nabla f$ on a domain $D$ containing $C$. Then $$\int_C \mathbf{F} \cdot d \mathbf{r} = f(B) - f(A).$$

Proof. Suppose that $\mathbf{r}(t) = \langle x(t), y(t), z(t) \rangle, a \leq t \leq b$ and $\mathbf{r}(a)$ and $\mathbf{r}(b)$ represent $A$ and $B$, respectively. Then $$\int_C \mathbf{F} \cdot d \mathbf{r} = \int_C \nabla f \cdot d \mathbf{r} = \int_C df = \int_a^b \frac{d}{dt} f(\mathbf{r}(t)) dt \\ = f(\mathbf{r}(b)) - f(\mathbf{r}(a)) = f(B) - f(A).  \blacksquare$$

$\nabla f$가 일변수에서 $f'$과 비슷한 역할을 수행한다고 이해하면, 사실상 Fundamental Theorem of Calculus와 동일한 내용의 정리이다.