1. The Vector Crowned: Definition and Notation
Definition — The Gradient Vector
Given a differentiable scalar function $f$ of two or three variables, the gradient of $f$ is the vector of all its partial derivatives:
$$ \nabla f(x,y) = \left\langle \frac{\partial f}{\partial x},\; \frac{\partial f}{\partial y} \right\rangle \qquad \text{(two variables)} $$ $$ \nabla f(x,y,z) = \left\langle \frac{\partial f}{\partial x},\; \frac{\partial f}{\partial y},\; \frac{\partial f}{\partial z} \right\rangle \qquad \text{(three variables)} $$The symbol $\nabla$ is called nabla (or del). It is read aloud as "nabla $f$" or "del $f$." The name nabla comes from the shape of an ancient Egyptian/Assyrian harp.
Evaluated at a Point vs. as a Vector Field
There are two complementary ways to think about the gradient:
- At a single point: $\nabla f(a,b)$ is a specific vector — a single arrow rooted at $(a,b)$. It tells you the direction and rate of fastest increase of $f$ right there.
- Over the whole domain (a vector field): If you evaluate $\nabla f$ at every point in the domain, you get an arrow at every point. This collection of arrows is a vector field — the first one you meet in this course. Vocabulary preview: vector fields are the central objects of the vector-analysis block ahead (divergence, curl, flux, circulation). The gradient field is the gentlest entry point.
Picture it: Imagine a topographic map of a mountain. At every point on the map, draw an arrow pointing uphill with length proportional to the steepness. That carpet of arrows is the gradient field $\nabla f$.
Interactive Visualization — Gradient Vector Field
Interactive visualization — available in the full lesson
Gradient field of $f(x,y)=x^2+y^2$. Each arrow is $\nabla f = \langle 2x,2y\rangle$, pointing radially outward (uphill on the paraboloid). Longer arrows = steeper slope.