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. 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.:19–22 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 ( 0 , 0 ) {displaystyle (0,0)} is approached along lines through the origin ( y = k x {displaystyle y=kx} ). However, when the origin is approached along a parabola y = ± x 2 {displaystyle y=pm x^{2}} , the function value has a limit of ± 0.5 {displaystyle pm 0.5} . 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.:17–19 In particular, for a real-valued function with two real-valued parameters, f ( x , y ) {displaystyle f(x,y)} , continuity of f {displaystyle f} in x {displaystyle x} for fixed y {displaystyle y} and continuity of f {displaystyle f} in y {displaystyle y} for fixed x {displaystyle x} does not imply continuity of f {displaystyle f} .