Find a concept
Start typing to search the mathematical index.
Vector of first partial derivatives of a scalar function
A gradient of a differentiable function , where is open, at a point is the vector
formed from the partial derivatives of .
With respect to the standard inner product on , the gradient encodes directional derivatives via whenever is differentiable at . Points where are critical points.
Let be open, let , and let . The map is differentiable at if there is a linear map such that
where is small enough that , and is the Euclidean norm.
The map , which is necessarily unique, is the Fréchet derivative . In standard coordinates it is represented by the Jacobian matrix.
Let be open, , and . The partial derivative of at with respect to the -th coordinate is
when this limit exists in .
Partial derivatives are the entries of the Jacobian matrix. Existence of every partial derivative at does not by itself imply differentiability at .
A directional derivative of a function , where is open, at in the direction is the limit
when it exists.
If is differentiable at , then exists for every and equals , where is the Fréchet derivative.
A critical point of a differentiable function (with ) is a point such that
equivalently, the Fréchet derivative is the zero linear map.