3.I.3A

Vector Calculus
Part IA, 2002

Determine whether each of the following is the exact differential of a function, and if so, find such a function: (a) (coshθ+sinhθcosϕ)dθ+(coshθsinϕ+cosϕ)dϕ(\cosh \theta+\sinh \theta \cos \phi) d \theta+(\cosh \theta \sin \phi+\cos \phi) d \phi, (b) 3x2(y2+1)dx+2(yx3z2)dy4yzdz3 x^{2}\left(y^{2}+1\right) d x+2\left(y x^{3}-z^{2}\right) d y-4 y z d z.