Multivariable calculus (also known as multivariate calculus) is the extension of calculus Calculus is a branch in mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem of calculus. Calculus is the study of change, in in one variable A variable is a symbol that stands for a value that may vary; the term usually occurs in opposition to constant, which is a symbol for a non-varying value, i.e. completely fixed or fixed in the context of use. The concepts of constants and variables are fundamental to all modern mathematics, science, engineering, and computer programming to calculus in several variables: the functions which are differentiated and integrated involve several variables rather than one variable.
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Typical operations
Limits and continuity
A study of limits and continuity In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous". An intuitive though imprecise idea of continuity is in multiple dimensions yields many counter-intuitive and pathological results not demonstrated by single-variable functions. There exist, for example, scalar functions of two variables having points in their domain which, when approached along any arbitrary line, give a particular limit, yet give a different limit when approached along a parabola. The function
approaches zero along any line through the origin. However, when the origin is approached along a parabola y = x2, 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.
Partial differentiation
Main article: Partial derivative In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant . Partial derivatives are used in vector calculus and differential geometryThe partial derivative generalizes the notion of the derivative to higher dimensions. A partial derivative of a multivariable function is a derivative with respect to one variable with all other variables held constant.
Partial derivatives may be combined in interesting ways to create more complicated expressions of the derivative. In vector calculus Vector calculus is a branch of mathematics concerned with differentiation and integration of vector fields. The term "vector calculus" is sometimes used as a synonym for the broader subject of multivariable calculus, which includes vector calculus as well as partial differentiation and multiple integration. Vector calculus plays an, the del In vector calculus, del is a vector differential operator, usually represented by the nabla symbol ∇. When applied to a function defined on a one-dimensional domain, it denotes its standard derivative as defined in calculus. When applied to a field , del may denote the gradient (slope) of a scalar field, the divergence of a vector field, or the operator () is used to define the concepts of gradient In vector calculus, the gradient of a scalar field is a vector field which points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change, divergence In vector calculus, divergence is an operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point. For example, consider air as it is heated, and curl In vector calculus, the curl is a vector operator that describes the rotation of a vector field. At every point in the field, the curl is represented by a vector. The attributes of this vector (length and direction) characterize the rotation at that point in terms of partial derivatives. A matrix of partial derivatives, the Jacobian matrix, may be used to represent the derivative of a function between two spaces of arbitrary dimension. The derivative can thus be understood as a linear transformation In mathematics, a linear map is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. The expression "linear operator" is commonly used for linear maps from a vector space to itself (endomorphisms). In advanced mathematics, the definition of linear function coincides with the which varies from point to point in the domain of the function.
Differential equations A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. Differential equations play a prominent role in engineering, physics, economics, and other disciplines containing partial derivatives are called partial differential equations In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function (or functions) of several independent variables and their partial derivatives with respect to those variables. Partial differential equations are used to formulate, and thus aid the solution of, problems involving or PDEs. These equations are generally more difficult to solve than ordinary differential equations, which contain derivatives with respect to only one variable.
Multiple integration
Main article: Multiple integral The multiple integral is a type of definite integral extended to functions of more than one real variable, for example, ƒ or ƒ(x, y, z)The multiple integral expands the concept of the integral to functions of many variables. Double and triple integrals may be used to calculate areas and volumes of regions in the plane and in space. Fubini's theorem In mathematical analysis, Fubini's theorem, named after Guido Fubini, is a result which gives conditions under which it is possible to change the order of integration guarantees that a multiple integral may be evaluated as a repeated integral.
The surface integral In mathematics, a surface integral is a definite integral taken over a surface ; it can be thought of as the double integral analog of the line integral. Given a surface, one may integrate over its scalar fields (that is, functions which return numbers as values), and vector fields (that is, functions which return vectors as values) and the line integral In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve are used to integrate over curved manifolds In mathematics, more specifically in differential geometry and topology, a manifold is a mathematical space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold. Thus a line and a circle are one-dimensional manifolds, a plane and sphere are two-dimensional manifolds, and so forth such as surfaces In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 — for example, the surface of a ball. On the other hand, there are surfaces, such as the Klein bottle, that cannot be and curves.
Fundamental theorem of calculus in multiple dimensions
In single-variable calculus, the fundamental theorem of calculus establishes a link between the derivative and the integral. The link between the derivative and the integral in multivariable calculus is embodied by the famous integral theorems of vector calculus:
- Gradient theorem
- Stokes' theorem In differential geometry, Stokes' theorem is a statement about the integration of differential forms on manifolds, which generalizes several theorems from vector calculus. William Thomson first discovered the result and communicated it to George Stokes in July 1850. Stokes set the theorem as a question on the 1854 Smith's Prize exam, which led to
- Divergence theorem In vector calculus, the divergence theorem, also known as Gauss' theorem , Ostrogradsky's theorem (Mikhail Vasilievich Ostrogradsky), or Gauss–Ostrogradsky theorem is a result that relates the flow (that is, flux) of a vector field through a surface to the behavior of the vector field inside the surface
- Green's theorem In physics and mathematics, Green's theorem gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C. It is the two-dimensional special case of the more general Stokes' theorem, and is named after British mathematician George Green
In a more advanced study of multivariable calculus, it is seen that these four theorems are specific incarnations of a more general theorem, the generalized Stokes' theorem, which applies to the integration of differential forms over manifolds.
Applications and uses
Techniques of multivariable calculus are used to study many objects of interest in the physical world. In particular,
Multivariable calculus can be applied to analyze deterministic Determinism is the philosophical view that every event, including human cognition, behaviour, decision, and action is causally determined (completely predictable) by previous events systems that have multiple degrees of freedom Degrees of freedom is a general term used in explaining dependence on parameters, and implying the possibility of counting the number of those parameters. In mathematical terms, the degrees of freedom are the dimensions of a phase space. Functions with independent variables The terms "dependent variable" and "independent variable" are used in similar but subtly different ways in mathematics and statistics as part of the standard terminology in those subjects. They are used to distinguish between two types of quantities being considered, separating them into those available at the start of a corresponding to each of the degrees of freedom are often used to model these systems, and multivariable calculus provides tools for characterizing the system dynamics System dynamics is an approach to understanding the behaviour of complex systems over time. It deals with internal feedback loops and time delays that affect the behaviour of the entire system. What makes using system dynamics different from other approaches to studying complex systems is the use of feedback loops and stocks and flows. These.
Multivariable calculus is used in many fields of natural and social science and engineering to model and study high-dimensional systems that exhibit deterministic behavior. Non-deterministic, or stochastic In probability theory, a stochastic process, or sometimes random process, is the counterpart to a deterministic process . Instead of dealing with only one possible reality of how the process might evolve under time (as is the case, for example, for solutions of an ordinary differential equation), in a stochastic or random process there is some systems can be studied using a different kind of mathematics, such as stochastic calculus Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes. It is used to model systems that behave randomly.
See also
- List of multivariable calculus topics
- Multivariate statistics Multivariate statistics is a form of statistics encompassing the simultaneous observation and analysis of more than one statistical variable. The application of multivariate statistics is multivariate analysis. Methods of bivariate statistics, for example ANOVA and correlation, are special cases of multivariate statistics in which two variables
External links
- UC Berkeley Video Course - Multivariable Calculus
- Multivariable Calculus: A free online textbook by George Cain and James Herod
- Multivariable Calculus Online: A free online textbook by Jeff Knisley
Categories: Multivariable calculus
Vadnais Heights Press
In addition to his research, he took a multivariable calculus course at the University of Minnesota. While his path to becoming a physics major has been ...
Cate W.
Mon, 29 Mar 2010 04:14:54 GM
Unlike my previous math courses (single and . multivariable calculus. ), this one is really abstract and definitely requires a deeper sense of mathematical maturity. Topics include row operations, linear transformations, subspaces, ...
