Multivariable calculus

Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one variable to functions of several variables: the differentiation and integration of functions involving multiple variables (multivariate), rather than just one. Multivariable calculus may be thought of as an elementary part of calculus on Euclidean space. The special case of calculus in three dimensional space is often called vector calculus.

Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one variable to functions of several variables: the differentiation and integration of functions involving multiple variables (multivariate), rather than just one. Multivariable calculus may be thought of as an elementary part of calculus on Euclidean space. The special case of calculus in three dimensional space is often called vector calculus.

Introduction

In single-variable calculus, operations like differentiation and integration are made to functions of a single variable. In multivariate calculus, it is required to generalize these to multiple variables, and the domain is therefore multi-dimensional. Care is therefore required in these generalizations, because of two key differences between 1D and higher dimensional spaces: There are infinite ways to approach a single point in higher dimensions, as opposed to two (from the positive and negative direction) in 1D; There are multiple extended objects associated with the dimension; for example, a 1D function is represented as a curve on the 2D Cartesian plane, but a scalar-valued function of two variables is a surface in 3D, while curves can also live in 3D space. The consequence of the first difference is the difference in the definition of the limits and continuity. Directional limits and derivatives define the limit and differential along a 1D parametrized curve, reducing the problem to the 1D case. Further higher-dimensional objects can be constructed from these operators. The consequence of the second difference is the existence of multiple types of integration, including line integrals, surface integrals and volume integrals. Due to the non-uniqueness of these integrals, an antiderivative or indefinite integral cannot be properly defined. {{NumBlk|:| \lim_{\overrightarrow{x} \to s(t_0)} f(\overrightarrow{x}) = \lim_{t \to t_0} f(s(t)) | }} Note that the value of this limit can be dependent on the form of s(t) , i.e. the path chosen, not just the point which the limit approaches. For example, consider the function f(x,y) = \frac{x^2y}{x^4+y^2}. If the point (0,0) is approached through the line y=kx , or in parametric form: Then the limit along the path will be: On the other hand, if the path y=\pm x^2 (or parametrically, x(t)=t,\, y(t)=\pm t^2 ) is chosen, then the limit becomes: Since taking different paths towards the same point yields different values, a general limit at the point (0,0) cannot be defined for the function. A general limit can be defined if the limits to a point along all possible paths converge to the same value, i.e. we say for a function f: \mathbb{R}^n \to \mathbb{R}^m that the limit of f to some point x_0 \in \mathbb{R}^n is L, if and only if for all continuous functions s(t): \mathbb{R} \to \mathbb{R}^n such that s(t_0)=x_0 . for all continuous functions s(t): \mathbb{R} \to \mathbb{R}^n such that s(t_0)=x_0 . As with limits, being continuous along one path s(t) does not imply multivariate continuity. Continuity in each argument not being sufficient for multivariate continuity can also be seen from the following example. For example, for a real-valued function f: \mathbb{R}^2 \to \mathbb{R} with two real-valued parameters, f(x,y) , continuity of f in x for fixed y and continuity of f in y for fixed x does not imply continuity of f . Consider f(x,y)= \begin{cases} \frac{y}{x}-y & \text{if}\quad 0 \leq y It is easy to verify that this function is zero by definition on the boundary and outside of the quadrangle (0,1)\times (0,1) . Furthermore, the functions defined for constant x and y and 0 \le a \le 1 by g_a(x) = f(x,a)\quad and \quad h_a(y) = f(a,y)\quad are continuous. Specifically, g_0(x) = f(x,0) = h_0(0,y) = f(0,y) = 0 for all and . Therefore, f(0,0)=0 and moreover, along the coordinate axes, \lim_{x \to 0} f(x,0) = 0 and \lim_{y \to 0} f(0,y) = 0 . Therefore the function is continuous along both individual arguments. However, consider the parametric path x(t) = t,\, y(t) = t . The parametric function becomes Therefore, It is hence clear that the function is not multivariate continuous, despite being continuous in both coordinates. Hence, for every \alpha > 0 , choose \delta = \frac{\alpha}{K} ; there exists an \epsilon > 0 such that for all t satisfying |t-t_0| , |s(t)-s(t_0)| , and |f(s(t)) - f(s(t_0))| \leq K|s(t)-s(t_0)| . Hence \lim_{t \to t_0} f(s(t)) converges to f(s(t_0)) regardless of the precise form of s(t) . Note also that given the continuity of s'(t) , s'(\tau) = s'(t_0)+O(h) as h \to 0 . Substituting these two conditions into , whose limit depends only on s'(t_0) as the dominant term. It is therefore possible to generalize the definition of the directional derivative as follows: The directional derivative of a scalar-valued function f:\mathbb{R}^n \to \mathbb{R} along the unit vector \hat{\bold{u}} at some point x_0 \in \mathbb{R}^n is {{NumBlk|:| \nabla_{\hat{\bold{u}}} f(x_0) = \lim_{t \to 0} \frac{f(x_0+\hat{\bold{u}} t) - f(x_0)}{t} | }} or, when expressed in terms of ordinary differentiation, {{NumBlk|:| \nabla_{\hat{\bold{u}}} f(x_0) = \left . \frac{df(x_0+\hat{\bold{u}}t)}{dt} \right |_{t=0} | }} which is a well defined expression because f(x_0+\hat{\bold{u}}t) is a scalar function with one variable in t . It is not possible to define a unique scalar derivative without a direction; it is clear for example that \nabla_{\hat{\bold{u}}}f(x_0) = - \nabla_{-\hat{\bold{u}}}f(x_0) . It is also possible for directional derivatives to exist for some directions but not for others.

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 integral theorems of vector calculus: Gradient theorem Stokes' theorem Divergence theorem Green's theorem. 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 material world. In particular, Multivariable calculus can be applied to analyze deterministic systems that have multiple degrees of freedom. Functions with independent variables 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. Multivariate calculus is used in the optimal control of continuous time dynamic systems. It is used in regression analysis to derive formulas for estimating relationships among various sets of empirical data. 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. In economics, for example, consumer choice over a variety of goods, and producer choice over various inputs to use and outputs to produce, are modeled with multivariate calculus. Non-deterministic, or stochastic systems can be studied using a different kind of mathematics, such as stochastic calculus.

References

External links

MIT video lectures on Multivariable Calculus, Fall 2007 Multivariable Calculus: A free online textbook by George Cain and James Herod Multivariable Calculus Online: A free online textbook by Jeff Knisley Multivariable Calculus – A Very Quick Review, Prof. Blair Perot, University of Massachusetts Amherst Multivariable Calculus, Online text by Dr. Jerry Shurman

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