Gradient

f — direction of steepest ascent, perpendicular to level sets.

The idea

The gradient of a function $f(x, y)$ collects its two partial derivatives into a single vector: $\nabla f = \left( \frac{\partial f}{\partial x},\ \frac{\partial f}{\partial y} \right).$ We read it aloud as grad $f$. In $n$ variables, with the variables named $x_{1}, \dots, x_{n}$, it is $\nabla f = \left( \frac{\partial f}{\partial x_{1}}, \dots, \frac{\partial f}{\partial x_{n}} \right)$, one slot per variable. Note what kind of object this is: $f$ takes points to numbers, but $\nabla f$ takes points to vectors, and the vector is generally different at every point.

Each partial derivative alone reports the rate of change of $f$ in one axis direction, and a point of the plane has infinitely many other directions leading away from it. Collected into a vector, the partials carry all the local information about $f$: the rate of change in an arbitrary direction, the direction in which $f$ climbs fastest, and the shape of the curves along which $f$ stays constant are all recoverable from $\nabla f$.

Ways to work on it

• Walkthrough. Build the gradient from partial derivatives and see what its direction and length mean.
• Practice. Compute components and magnitude of f at a point.
• Hardest. Directional derivative of f at a point in a given (non-unit) direction.

Not sure where to start? Take the ten-question placement test.