Gradient
Undergraduate
Consider a differentiable function $f : \reals^n \to \reals$. The gradient of $f$ is the vector of partial derivatives
\[\nabla f(x) = \left( \frac{\partial f}{\partial x_1}(x), \dots, \frac{\partial f}{\partial x_n}(x) \right).\]
Graduate (Riemannian manifolds)
Consider a smooth function $f$ on a Riemannian manifold, with inner product $\ip{\cdot}{\cdot}$ on each tangent space. The gradient of $f$ at $x$ is the vector $\nabla f(x)$ such that
\[\ip{\nabla f(x)}{v} = \mathrm{d}f_x(v) \quad \text{for every tangent vector } v.\]Examples
Worked examples
- $f(x, y, z) = x^2 + y^2$ : $\nabla f(x, y, z) = (2x, 2y, 0)$ Undergraduate
-
$f(x, y) = x^2 + 3y$
: $\nabla f(x, y) = (2x, 3)$
Undergraduate
A vector.
Properties
- If $\nabla f(x) \neq 0$, then it points in the direction of steepest increase of $f$, and its length is the rate of increase in that direction.
- The gradient is perpendicular to the level sets of $f$.
- The gradient depends on the choice of inner product (Riemannian metric); the differential $\mathrm{d}f$ does not.
Distinctions involving this term
- Types of multivariable derivatives: The directional derivative, the Jacobian, the gradient, the differential, and the Lie derivative all package the partial derivatives of a function differently: as a vector (gradient), a matrix (Jacobian), a linear map (differential), or a single number in a chosen direction (directional derivative) or along a vector field (Lie derivative).