Given a multivariable function
The \nabla
symbol in Latex.
Intuitively, gradient measures the direction of “fastest increase” through a vector field.
Gradient in Coordinates
The most familiar definition is a list of partial derivative:
We can also understand it as a list of directional derivative across the coordinate axis.
This definition has two potential problems:
- Role of inner product is not clear
- No way to differentiate functions of functions F(f) since we don’t have a finite list of coordinates
Gradient As Fastest Increase
For an function
This interpretation of gradient leads naturally to the gradient descent algorithm, a fundamental optimization technique.
Gradient For Implicit Surface
For an implicitly defined surfaces
Relation with Multi-dimensional Derivative and Gradient
See: Multi-dimensional derivative, gradient, and directional Derivative