Multivariable Calculus

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, rather than just one.

A study of limits and continuity in multivariable calculus yields many counter-intuitive results not demonstrated by single-variable functions. For example, there are scalar functions of two variables with points in their domain which give a particular limit when approached along any arbitrary line, yet give a different limit when approached along a parabola.

Indeed, the function

approaches zero along any line through the origin. However, when the origin is approached along a parabola ${\displaystyle y=x^{2}}$, it has a limit of 0.5. Since taking different paths toward the same point yields different values for the limit, the limit does not exist.

Continuity in each argument is not sufficient for multivariate continuity:. For instance, in the case of a real-valued function with two real-valued parameters, ${\displaystyle f(x,y)}$, continuity of ${\displaystyle f}$ in ${\displaystyle x}$ for fixed ${\displaystyle y}$ and continuity of ${\displaystyle f}$ in ${\displaystyle y}$ for fixed ${\displaystyle x}$ does not imply continuity of ${\displaystyle f}$.

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