Line Integrals

Tangent Vectors

Normal Vectors

Circulation over Curves

$\int _C \vec{F}(x,y) * \vec {T} ds = \int _c \vec{f}(x,y) * d \vec{r}$

• $F$ is a vector field, which can represent quantities like velocity, force, or an electric field
  • $\vec T ds = dr$
• $d \vec {r}$ is an infinitesimal displacement vector along the curve $C$
• The dot product $F \cdot d \vec {r}$ measures the component of $F$ that is tangential to the curve at each point

Computation:

1. Find parameterization $r(t)$
2. Find $d \vec r$, which is just $\vec T ds$
   1. Derive the parameterization wrt $t$
3. Compute the integral

Flux Integrals

$\int _C \vec {F} * \vec {n} ds$

• Same idea as circulation but we use the normal vector instead of the tangential
• If we split $C$ into pieces, how much vector field goes through $C$?

Surface Integrals of Scalar Functions

$\iint _S f(x,y,z) ds$

• Where $ds$ is the area of a parallelogram

  $$ ds = \left| \begin{array}{c} \frac{\partial \mathbf{r}}{\partial u} \times \frac{\partial \mathbf{r}}{\partial v} \end{array} \right| \, du \, dv $$