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If f has continuous second partial derivatives, then mixed partials commute
Let be open and let be of class .
Corollary: For all and all indices ,
Connection to parent theorem: Apply Schwarz's theorem at each point . The hypothesis guarantees the mixed partial derivatives exist in a neighborhood and are continuous, which is exactly the hypothesis of Schwarz's theorem.
An open set in a topological space is a subset such that .
Open sets are the basic “observable” sets in topology: they define neighborhoods of points and determine operations like interior and closure. They also control continuity through preimages.
A class map is a function defined on an open set such that all partial derivatives of of total order at most exist on and are continuous on (with order meaning itself).
When , this is the standard “continuously differentiable” hypothesis in multivariable calculus: the derivative is encoded by the Jacobian matrix, and regularity is closely aligned with having a continuous Fréchet derivative. Higher regularity interacts with mixed derivatives via results like the Schwarz–Clairaut theorem.
A mixed partial derivative of a scalar function (with ) at is a second-order partial derivative of the form
provided the relevant partial derivatives exist.
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 neighborhood of a point in a topological space is a subset such that there exists an open set with .
Let be a function between metric spaces, and let . We say is continuous on if is continuous at every point (equivalently, the restriction is continuous).
Spelled out: for every and every , there exists such that for all ,