> [!NOTE] Lemma > Let $F:U \subset \mathbb{R}^{3}\to \mathbb{R}^{3}$ be a [[Conservative Vector Field on Subset of Real 3-Space|conservative vector field]]. For any two points, the [[Line Integral of Vector Field on Subset of Real n-Space|line integral]] of $\underline{F}$ along a path $C$ joining them is the same. **Proof**. By definition, there exists $f\in C^{1}$ so that $\underline{F}=\nabla f.$ Thus $\begin{align} \int \underline{F} \cdot \, d\underline{r} &= \int_{a}^{b} \nabla f \cdot \frac{d\underline{r}}{dt} \, dt \\ & = \int_{a}^{b} \begin{pmatrix}\frac{ \partial f }{ \partial x } \\ \frac{ \partial f }{ \partial y } \\ \frac{ \partial f }{ \partial z } \end{pmatrix} \cdot \begin{pmatrix} \frac{dx}{dt} \\ \frac{dy}{dt} \\ \frac{dz}{dt} \end{pmatrix} \, dt \\ & = \int_{a}^{b} \left[ \frac{ \partial f }{ \partial x } \frac{dx}{dt } + \frac{ \partial f }{ \partial y } \frac{dy}{dt} + \frac{ \partial f }{ \partial z } \frac{dz}{dt} \right] \, dt \tag{*} \\ &=\int_{a}^{b} \frac{df}{dt} \, dt \\ &= f(\underline{x}_{b}) -f(\underline{x}_{a}) \end{align}$using the [[Chain rule for derivative|chain rule]] to rewrite $(*)$ and [[Fundamental theorem of calculus|FTC II]] to gain the last line. # Applications **Corollaries**: Note that if $a=b,$ the paths are closed, then the line integral of the conservative vector field along is always zero.