Line Integrals

Scalar vs Vector Line Integrals

A line integral accumulates a field along a curve $C$.

A scalar line integral integrates a scalar field $f$ along $C$:

$$\int_C f\,ds = \int_a^b f(\vec{r}(t))\,\|\vec{r}'(t)\|\,dt$$

A vector line integral integrates a vector field $\vec{F}$ along $C$, computing the work done:

$$\int_C \vec{F}\cdot d\vec{r} = \int_a^b \vec{F}(\vec{r}(t))\cdot\vec{r}'(t)\,dt$$

Conservative Fields & Path Independence

A field $\vec{F}$ is conservative if $\vec{F} = \nabla\phi$ for some potential $\phi$. Then the integral depends only on endpoints:

$$\int_C \vec{F}\cdot d\vec{r} = \phi(\vec{r}_B) - \phi(\vec{r}_A)$$

Test for conservativeness in $\mathbb{R}^{3}$: verify $\nabla\times\vec{F} = \vec{0}$.

Example: Gravity work moving from $h_1$ to $h_2$:

$$W = -mg(h_2-h_1)$$

This is path-independent because gravity is conservative with $\phi = mgh$.

Common pitfall: Path-independence requires a conservative field. Always verify $\nabla\times\vec{F}=\vec{0}$ on a simply connected domain before using endpoints.