Recall that if $z = f(x, y)$ is a two variable real-valued function, then the gradient of $f$ is given by the following formula:

(1)
\begin{align} \quad \nabla f(x, y) = \frac{\partial f}{\partial x} \vec{i} + \frac{\partial f}{\partial y} \vec{j} \end{align}

Thus we can see that $\nabla f(x, y)$ is a vector field on $\mathbb{R}^2$. Similarly, if $w = f(x, y, z)$ is a three variable real-valued function, then the gradient of $f$ is given by:

(2)
\begin{align} \quad \nabla f(x, y, z) = \frac{\partial f}{\partial x} \vec{i} + \frac{\partial f}{\partial y} \vec{j} + \frac{\partial f}{\partial z} \vec{k} \end{align}

So we can see that $\nabla f(x, y, z)$ is a vector field on $\mathbb{R}^3$. These vector fields are given an important name which we define below.

 Definition: If $z = f(x, y)$ is a two variable real-valued function then the Gradient Vector Field of $f$ on $\mathbb{R}^2$ is the vector function $\nabla f (x, y) = \frac{\partial f}{\partial x} \vec{i} + \frac{\partial f}{\partial y} \vec{j}$. If $w = f(x, y, z)$ is a three variable real-valued function then the Gradient Vector Field of $f$ on $\mathbb{R}^3$ is the vector function $\nabla f(x, y, z) = \frac{\partial f}{\partial x} \vec{i} + \frac{\partial f}{\partial y} \vec{j} + \frac{\partial f}{\partial z} \vec{k}$.

For example, consider the function $f(x, y) = (x - 5)(y - 2) - 2$. The gradient of $f$ is $\nabla f(x, y) = (y - 2)\vec{i} + (x - 5) \vec{j}$ and the graph of the gradient vector field is given below alongside some of the level curves of $f$:

There are two important properties of gradient fields we should address. Firstly, notice that the plotted vectors are perpendicular to their respective level curves. This is as a result of what we saw from The Perpendicularity of The Gradient at a Point on a Level Curve page. Secondly, notice that the length of the vectors increases where the level curves and closer together. Note that $\| \nabla f \|$ is equal to the value of the direction derivative at a point.