 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2020/paper-1/19/c/solution

Compute the flux directly using the outward vector area element, as required. Restrict the vector field to the surface parametric equation and take the dot product:

F (R) \cdot d S = [\frac{b}{a} r^{2} (cos^{2} θ + sin^{2} θ) + b r (1 - \frac{r}{a})] d r d θ = b r d r d θ .

The terms quadratic in

r

cancel. Therefore the surface integral is

\int_{C} F \cdot d S = \int_{0}^{2 π} \int_{0}^{a} b r d r d θ = π a^{2} b .

 New to topics? Read the docs here!