06 - Path Independency of Line Integrals

Path Independency of Line Integrals

Consider a given vector field $\mathbf{v}$ for which there exists a scalar function $f$ s.t.

$$\mathbf{v}=\nabla f$$

Then

$${\int }_A^B\nabla f\cdot d\mathbf{r}=f(B)-f(A)$$

where $A,B$ are the start and end points along the curve

Note that it is path independent, isn’t affected by the curve

Proof

$$\begin{array}{rl}{\int }_A^B\nabla f\cdot d\mathbf{r}& ={\int }_A^B\left(\nabla f\cdot \frac{d\mathbf{r}}{dt}\right)dt\\ & ={\int }_A^B\left(\frac{\partial f}{\partial x}\frac{\partial x}{\partial t}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial t}+\frac{\partial f}{\partial z}\frac{\partial z}{\partial t}\right)dt\\ & ={\int }_A^B\frac{df}{dt}dt\\ & =f(B)-f(A)\end{array}$$