Line integrals of vector fields

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

Flux

This simulation shows approximate values of the line integral $\int_C\mathbf F \cdot \mathbf n\, ds$ for four different curves.

Things to try:

• Drag the points A and B, or type specific coordinates, to change the position of the curve.
• Change the components of the vector field. Adjust the Scale, if needed.
• Change the curve to analyse the values of the integral.

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.

Work and Circulation

This simulation shows approximate values of the line integral $\int_C\mathbf F \cdot \mathbf T\, ds$ for four different curves.

Things to try:

• Drag the points A and B, or type specific coordinates, to change the position of the curve.
• Change the components of the vector field. Adjust the Scale, if needed.
• Change the curve to analyse the values of the integral.

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.

Worksheet exemplar

The following file contains activities associated with the simulations.

• Line integrals of vector fields (pdf)
• Line integrals of vector fields (tex)