If f is a function of x and y, find ║∇f║^2 and ∇f and in polar coordinates r and θ.

Answer 1

f = f(x,y)
∇f = df/dx i + df/fy j = gradient of f
|∇f|^2 = (df/dx)^2 + (df/dy)^2
for the polar form see:
https://math.mit.edu/classes/18.013A/HTML/chapter09/section04.html