Vector Line Integral

${\int }_{\mathcal{C}}\vec{F}\cdot \vec{t}ds={\int }_{\mathcal{C}}\vec{F}\cdot d\vec{r}={\int }_{\mathcal{C}}(F_{1}dx+F_{2}dy+F_{3}dz)={\int }_{t_0}^{t_f}\vec{F}(\vec{r}(t))\cdot {\vec{r}}^{\prime }(t)dt$

• $\vec{t}$ represents the unit tangent vector of the path
• If $\vec{F}$ represents a vector field for some force, then the vector line integral represents the work done by the force
• $dx=x^{\prime }(t)dt,dy=y^{\prime }(t)dt,dz=z^{\prime }(t)dt$, where ${\vec{r}}^{\prime }(t)=⟨x^{\prime }(t),y^{\prime }(t),z^{\prime }(t)⟩$

${\int }_{\mathcal{C}}\nabla f\cdot d\vec{r}=f(S_f)-f(S_0)$

• Fundamental Theorem of Vector Integrals