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Matrix of first partial derivatives of a multivariable map
A Jacobian matrix of a map (with ) at a point is the matrix
provided these partial derivatives exist.
When is differentiable at , the Fréchet derivative is a linear map, and is the matrix that represents in the standard bases of and . For scalar-valued , is closely related (up to transpose conventions) to the gradient.
A partial derivative of a map (with ) at with respect to the th coordinate is the limit
when it exists (for vector-valued , this limit is taken in ).
Partial derivatives are one-coordinate versions of the limit at a point and are the entries used to build the Jacobian matrix. Existence of all partial derivatives at does not by itself guarantee that is a differentiable map at .
A differentiable map at a point is a map (with ) for which there exists a linear map such that
where is the Euclidean norm.
In this case is the Fréchet derivative of at , and (when it exists) it is represented in coordinates by the Jacobian matrix. Maps that are differentiable at every point of their domain are the basic objects of multivariable differentiability in higher dimensions, and higher smoothness is recorded by C^k maps.
A Fréchet derivative of a function (with ) at a point is a linear map such that
where is the Euclidean norm.
If such a map exists, it is unique; it is the linear part of the first-order approximation . When has partial derivatives, is represented by the Jacobian matrix at , and existence of is the defining condition for a differentiable map at .
A gradient of a differentiable function (with ) 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.