Gradients of a function

Question

Here's the question: Find the most general function f:R^2 --> R such that ∇f(p)=p for each p∈R^2.

 

So, I'm guessing this means that I just need to find a function of x and y such that the derivatives with respect to x and y are the same. Does that mean it should just be a constant so that the partials would all be 0? If not, how would I go about this?

Answer 1

∇f(x,y) = ⟨x,y⟩

 

∂f/∂x = x ⇒ f(x,y) = x²/2+ g(y)

 

∂f/∂y = y ⇒ f(x,y) = y²/2+ h(x)

 

f(x,y) = (x² + y²)/2 + C