How to find out the direction of the gradient?

Question

Given a function $f(x, y) = -x^2 - y^2$, does the gradient point to the origin everywhere in $\mathbb{R}^2 \setminus \mathbf{0}$?

I tried using a plotter und got the following result:

But I'm not sure how to interpret this. How can I mathematically identify the gradient's direction?

Answer 1

The gradient is given in coordinates as the partial derivatives of your function. Here,

$$\nabla f_{(x,y)}=(-2x,-2y),$$

and indeed this vector is colinear to the one linking the origin and $(x,y)$, which is $(x,y)-(0,0)=(x,y)$.