Vector Line Integral

#Math

$\displaystyle\int_\mathscr{C}\vec F\cdot\vec t\space ds=\int_\mathscr{C}\vec F\cdot d\vec r=\int_\mathscr{C}(F_1dx+F_2dy+F_3dz)=\int_{t_0}^{t_f}\vec F(\vec r(t))\cdot\vec r'(t)\space dt$

• $\vec t$ 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
• $dx=x'(t)dt,\space dy=y'(t)dt,\space dz=z'(t)dt$, where $\vec r'(t)=\left\langle x'(t),y'(t),z'(t)\right\rangle$

$\int_\mathscr{C}\nabla f\cdot d\vec r=f(S_f)-f(S_0)$

• Fundamental Theorem of Vector Integrals