Let Z = f(u,v) where u=xy^3 + e^3 and v = (x^2)y.

Question

Obtain expressions for Zx and Zy in terms of fu and fv.

I don't understand the "in terms of" part. How do I do this question? thank you

Answer 1

dZ = Zx dx + Zy dy (1)

Also:

dZ = df = fu du + fv dv (2)

Express du and dv in terms of dx and dy:

du = ux dx + uy dy (3)

dv = vx dx + vy dy (4)

Substitute (3) and (4) in (2):

dZ = fu (ux dx + uy dy ) + fv (vx dx + vy dy) =

(fu ux + fv vx) dx + (fu uy + fv vy)dy

Comparing this to (1) then gives:

Zx = fu ux + fv vx

Zy = fu uy + fv vy