A river flows across the xy-plane in the positive

Question

Question: (a) A river flows across the xy-plane in the positive x-direction and around a circular rock of radius 1 centered at the origin. The velocity of the river can be modeled using the potential function φ = x + (x/(x² + y²)). Compute the velocity vector field, v = grad φ.

(b) Show that div v = 0.

(c) Show that the flow of v is tangent to the circle x² + y² = 1. This means that no water crosses the circle. The water on the outside must therefore all flow around the circle.

(d) Use a computer to sketch the vector field v in the region outside the unit circle.