Independence of Path

# Independence of Path

Recall from The Fundamental Theorem for Line Integrals page that if $C$ is a smooth curve defined parametrically by the vector equation $\vec{r}(t)$ for $a ≤ t ≤ b$ and $f$ is a differentiable function and $\nabla f$ is continuous on the path $C$ then:

(1)
\begin{align} \quad \int_C \nabla f \cdot \: d \vec{r} = f(\vec{r}(b)) - f(\vec{r}(a)) \end{align}

Note that thus the line integral of $C$ of the vector field $\nabla f$ along the curve $C$ depends ONLY on the values of $\vec{r}(t)$ at the endpoints of the interval $[a, b]$. Now suppose that we have two smooth curves $C_1$ and $C_2$ whose endpoints are the same and such that $\nabla f$ is continuous on both $C_1$ and $C_2$:

Then since $C_1$ and $C_2$ have the same initial and terminal points, we have that from the Fundamental Theorem for Line Integrals:

(2)
\begin{align} \quad \int_{C_1} \nabla f \cdot \: d \vec{r} = \int_{C_2} \nabla f \cdot \: d \vec{r} \end{align}

Note that thus if $\mathbf{F}$ is a conservative vector field, then there exists a potential function $\phi$ such that $\mathbf{F} = \nabla \phi$, and so if $\mathbf{F}$ is continuous on the curves $C_1$ and $C_2$ which have the same initial and terminal points, then $\int_{C_2} \mathbf{F} \cdot \: d \vec{r} = \int_{C_2} \mathbf{F} \cdot \: d \vec{r}$. Therefore, the line integrals of a conservative vector field $\mathbf{F}$ depends only on the initial and terminal points of a curve. We generalize this concept in the following definition.

 Definition: Let $\mathbf{F}$ is a continuous vector field on the domain $D$. Then the line integral $\int_C \mathbf{F} \cdot \: d \vec{r}$ is Independent of Path if for any two curves $C_1$ and $C_2$ in $D$ whose initial and terminal points coincide, we have that $\int_{C_1} \mathbf{F} \cdot \: d \vec{r} = \int_{C_2} \mathbf{F} \cdot \: d \vec{r}$.

From the definition above, we have that if $\mathbf{F}$ is a conservative vector field then the line integrals of $\mathbf{F}$ are independent of path.