Function of several real variables

A function that assigns a real number to each input tuple of real numbers in its domain. Its input may be represented as a point in a multidimensional real space.

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Cartesian product: A construction that combines sets into a set of ordered tuples. A function's input tuples are drawn from products of real-number sets.
Partial derivative: A derivative measuring change with respect to one input coordinate while holding others fixed. It isolates the function's local change along each coordinate direction.
Scalar field: A function assigning a scalar value to each point in a space. A multivariable real function is a scalar field when its domain is spatial.
Single-variable function: A function whose input is one real number and whose output is a value. Its domain is one-dimensional, unlike the tuple-valued inputs considered here.
Euclidean space: A real vector space equipped with the standard notion of distance and angle. The domain is often viewed as a region in finite-dimensional Euclidean space.
Gradient: A vector of partial derivatives pointing in the direction of steepest increase. The gradient summarizes the function's first-order change at a point.
Constrained optimization: The search for extrema of an objective subject to restrictions on its variables. Many multivariable functions serve as objectives over constrained domains.
Directional continuity: Continuity tested only along specified paths or directions approaching a point. Continuity along every straight line can still fail to guarantee full continuity.
Domain of a function: The set of inputs for which a function is defined. Restrictions on valid input tuples determine where the function exists.
Directional derivative: The rate of change of a function at a point along a specified direction. It measures local change beyond the coordinate directions used by partial derivatives.