Let $\mathbf F =F_1(x,y)\,\mathbf i+F_2(x,y)\,\mathbf j$ be a continuous vector field defined on a smooth curve $C$ given by a vector function $\mathbf r (t)$, with $a\leq t\leq b$.
If $\mathbf T$ is the unit tangent vector to the curve $C$, then the line integral of $\mathbf F$ along $C$ is
On the other hand, if $\mathbf n$ is the unit normal vector to the curve $C$ (obtained from the unit tangent vector $\mathbf T$), then the line integral of $\mathbf F$ across $C$ is
If the curve $C$ is closed, then we have that
This simulation shows approximate values of the line integral $\int_C\mathbf F \cdot \mathbf n\, ds$ for four different curves.
Things to try:
Note 1: The red arrows represent unit vectors normal to the curve.
Note 2: The simulation might show unexpected values of the integral, if the vector field has singularities.
This simulation shows approximate values of the line integral $\int_C\mathbf F \cdot \mathbf T\, ds$ for four different curves.
Things to try:
Note 1: The red arrows represent unit vectors tangent to the curve.
Note 2: The simulation might show unexpected values of the integral, if the vector field has singularities.
The following file contains activities associated with the simulations.