The Gradient Points Uphill
The gradient packs all the partial derivatives into one vector that points in the direction of fastest increase.
The gradient vector
The gradient of f is the vector whose components are the partial derivatives, written nabla f. For the bowl-shaped function x squared plus y squared at the point (1, 1), the gradient is (2, 2), pointing away from the origin. Each point of a surface gets its own gradient, so it is a vector field rather than a single arrow.
\nabla f = \langle f_x,\, f_y \rangleDirectional derivative
The rate of change of f along a unit vector u is the dot product of the gradient with u, called the directional derivative. Continuing the example, along the diagonal direction (1 over root 2, 1 over root 2) at (1, 1), the dot product gives 2 root 2. Any direction with a smaller alignment with the gradient gives a smaller rate.
D_{\mathbf{u}} f = \nabla f \cdot \mathbf{u}Slopes and level curves
The largest possible directional derivative at a point is the magnitude of the gradient, achieved by moving exactly along it. The gradient is also perpendicular to the level curve through that point, which is why contours and steepest paths cross at right angles. When the gradient is the zero vector, every direction has slope zero, marking a potential peak, valley, or saddle.
Flashcard
Vector collecting all first partial derivatives
Check your understanding
For the function f(x, y) = xy, what is the gradient at the point (2, 3)?