Student handout: Vector Integrals (Contour Map)

Students explore path integrals using a vector field map and thinking about integration as chop-multiply-add.
What students learn
  • Recall the relationship between the sign of the dot product and the orientation of the vectors.
  • Use graphical methods to estimate the value of a vector line integral.
  • Lay the groundwork for thinking about conservative and non-conservative vector fields.

The Sign of Work: For the force vector shown above, draw three paths such that the work done by \(\vec{F}\) is positive, zero, and negative for small displacements along each path.

Estimate Vector Line Integral: For each segment of the path on the vector field \(\vec{F}\) shown below, estimate the value of the integral:

\[ \int_{\textrm{path}} \vec{F} \cdot d\vec{r} \]

where the side of each square is 1 cm and the length of the longest arrow is 10 units (appropriate for the field \(\vec{F})\).

E&M Path integrals
Learning Outcomes