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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 ( multivariate ), rather than just one. [1]
The rectangular region at the bottom of the body is the domain of integration, while the surface is the graph of the two-variable function to be integrated. In mathematics (specifically multivariable calculus ), a multiple integral is a definite integral of a function of several real variables, for instance, f(x, y) or f(x, y, z) .
Miscellanea. v. t. e. In mathematics, matrix calculus is a specialized notation for doing multivariable calculus, especially over spaces of matrices. It collects the various partial derivatives of a single function with respect to many variables, and/or of a multivariate function with respect to a single variable, into vectors and matrices that ...
Vector calculus or vector analysis is a branch of mathematics concerned with the differentiation and integration of vector fields, primarily in three-dimensional Euclidean space, . The term vector calculus is sometimes used as a synonym for the broader subject of multivariable calculus , which spans vector calculus as well as partial ...
Calculus on Manifolds is a brief monograph on the theory of vector-valued functions of several real variables (f : R n →R m) and differentiable manifolds in Euclidean space. . In addition to extending the concepts of differentiation (including the inverse and implicit function theorems) and Riemann integration (including Fubini's theorem) to functions of several variables, the book treats ...
Second partial derivative test. The Hessian approximates the function at a critical point with a second-degree polynomial. In mathematics, the second partial derivative test is a method in multivariable calculus used to determine if a critical point of a function is a local minimum, maximum or saddle point .
t. e. A directional derivative is a concept in multivariable calculus that measures the rate at which a function changes in a particular direction at a given point. [citation needed] The directional derivative of a multivariable differentiable (scalar) function along a given vector v at a given point x intuitively represents the instantaneous ...
For functions of a single variable, the theorem states that if is a continuously differentiable function with nonzero derivative at the point ; then is injective (or bijective onto the image) in a neighborhood of , the inverse is continuously differentiable near = (), and the derivative of the inverse function at is the reciprocal of the derivative of at : ′ = ′ = ′ (()).
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