Partiellpartial

Partiellpartial is a term found mainly in informal mathematical discussions about functions of several variables. It is used to describe a notion of differentiability that applies selectively to some variables, distinguishing between differentiability with respect to individual coordinates and joint behavior across multiple coordinates. The concept is not a standard, universally adopted definition in mainstream texts, but it can serve as a convenient shorthand in expository contexts.

Definition. Let f: U ⊆ R^n → R be defined on an open set U, and let S be a subset of {1, …, n}. The function is said to be partiellpartial with respect to the index set S if, for every i in S, the partial derivative ∂f/∂x_i exists. In addition, some discussion of partiellpartial involves the existence (or nonexistence) of mixed partial derivatives ∂^2f/∂x_i∂x_j for i, j in S, which may exist for some pairs but not for others, or may fail to be equal when they do exist. The emphasis is on the distinction between differentiability in individual variables and the behavior of derivatives taken with respect to multiple variables in combination.

Properties and implications. Existence of first-order partial derivatives with respect to a subset of variables does not guarantee continuity of those derivatives, nor does it guarantee the equality of mixed partials. Clairaut’s theorem and related results illustrate that joint differentiability requires stronger smoothness assumptions than partial differentiability alone.

Examples and context. Functions can have partial derivatives with respect to certain variables while lacking well-defined mixed partials across those variables. In teaching and analysis, partiellpartial is a way to highlight that local smoothness can be uneven across coordinates.

See also. Partial derivative, Mixed partial derivative, Clairaut’s theorem, Differentiability, Multivariable calculus.