Introduction of Multivariable calculus
The Multivariable calculus (also termed as multivariate calculus) is the extension of calculus in one variable to calculus in more than one variable: the differentiated and integrated functions include multiple variables, instead of merely one.
Typical operations
Limits and continuity:
The study of limits and continuity in multivariable calculus results many counter-intuitive outcomes not elaborated by single-variable functions. For illustration, there are scalar functions of two variables with points in their domain that give a specific limit whenever approached along any random line, though give a dissimilar limit whenever approached along a parabola. For example, the function is as follows:
F(x,y)= x^{2}y/ x^{4} + y^{2}
Approaches zero along any line via the origin. Whenever origin is approaching along a parabola y = x^{2}, it has a limit of 0.5. As taking various paths toward the same point results various values for the limit, and that limit does not exist.
Continuity in each argument does not imply the multivariate continuity: For illustration, in the case of a real-valued function with two real-valued parameter i.e., f(x,y) continuity of f in x for fixed y and continuity of f in y for fixed x does not imply the continuity of f. As an illustration, consider
Applications and uses
Domain/Range
Applicable techniques
Curves
Lengths of curves, line integrals, and curvature.
Surfaces
Areas of surfaces, surface integrals, flux through surfaces, and curvature.
Scalar fields
Maxima and minima, Lagrange multipliers, directional derivatives.
Vector fields
Any of the operations of vector calculus including gradient, divergence, and curl.
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