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Concept
Multivariable calculus
Summary
Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one variable to calculus with functions of several variables: the differentiation and integration of functions involving multiple variables (multivariate), rather than just one.
Multivariable calculus may be thought of as an elementary part of advanced calculus. For advanced calculus, see calculus on Euclidean space. The special case of calculus in three dimensional space is often called vector calculus.
A study of limits and continuity in multivariable calculus yields many counterintuitive results not demonstrated by single-variable functions. For example, there are scalar functions of two variables with points in their domain which give different limits when approached along different paths. E.g., the function.
approaches zero whenever the point is approached along lines through the origin (). However, when the origin is approached along a parabola , the function value has a limit of . Since taking different paths toward the same point yields different limit values, a general limit does not exist there.
Continuity in each argument not being sufficient for multivariate continuity can also be seen from the following example. In particular, for a real-valued function with two real-valued parameters, , continuity of in for fixed and continuity of in for fixed does not imply continuity of .
Consider
It is easy to verify that this function is zero by definition on the boundary and outside of the quadrangle . Furthermore, the functions defined for constant and and by
and
are continuous. Specifically,
for all x and y.
However, the sequence (for natural ) converges to , rendering the function as discontinuous at . Approaching the origin not along parallels to the - and -axis reveals this discontinuity.
If is continuous at and is a single variable function continuous at then the composite function defined by is continuous at
For examples, and
If and are both continuous at then
(i) are continuous at
(ii) is continuous at for any constant c.
Official source
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