Line Integrals over Vector Fields

$$\begin{gathered} {\int }_C{F}_{T}ds \\ ={\int }_C\vec{F}\cdot d\vec{r} \\ ={\int }_a^b\vec{F}(\vec{r}(t))\cdot {\vec{r}}^{\prime }(t)dt \end{gathered}$$

Fundamental Theorem

• If
  1. $\vec{F}$ is conservative in E
  2. $C$ lies in E and goes from point $A$ to $B$
• Then
  • ${\int }_C\vec{F}\cdot d\vec{r}={\int }_{C^{\prime }}\vec{F}\cdot d\vec{r}=f(B)-f(A)$
  • Means the line integral does not depend on path in conservative fields