Chapter 14.4, Problem 55E

Applications

53–56. Ideal flow A two-dimensional vector field describes ideal flow if it has both zero curl and zero divergence on a simply connected region (excluding the origin if necessary).

a.    Verify that the curl and divergence of the given field is zero.

b.    Find a potential function φ and a stream function ψ for the field.

c.    Verify that φ and ψ satisfy Laplace’s equation ${\phi }_{xx}+{\phi }_{yy}={\psi }_{xx}+{\psi }_{yy}=0$.

55.     $F=\langle {\tan }^{-1}\frac{y}{x},\frac{1}{2}\ln \left(x^2+y^2\right)\rangle$