Multivariable calculus
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.
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.
A study of limits and continuity in multivariable calculus yields many counterintuitive results not demonstrated by single-variable functions.
can hence be defined as Note that the value of this limit can be dependent on the form of
, i.e. the path chosen, not just the point which the limit approaches.
) is chosen, then the limit becomes: Since taking different paths towards the same point yields different values, a general limit at the point
From the concept of limit along a path, we can then derive the definition for multivariate continuity in the same manner, that is: we say for a function
Consider It is easy to verify that this function is zero by definition on the boundary and outside of the quadrangle
The derivative of a single-variable function is defined as Using the extension of limits discussed above, one can then extend the definition of the derivative to a scalar-valued function
: Unlike limits, for which the value depends on the exact form of the path
is a function of one variable), we can write the Taylor expansion of
Substituting these two conditions into 12, whose limit depends only on
It is not possible to define a unique scalar derivative without a direction; it is clear for example that
A partial derivative of a multivariable function is a derivative with respect to one variable with all other variables held constant.
) is used to define the concepts of gradient, divergence, and curl 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 which directly varies from point to point in the domain of the function.
Differential equations containing partial derivatives are called partial differential equations or PDEs.
These equations are generally more difficult to solve than ordinary differential equations, which contain derivatives with respect to only one variable.
Double and triple integrals may be used to calculate areas and volumes of regions in the plane and in space.
The link between the derivative and the integral in multivariable calculus is embodied by the integral theorems of vector calculus:[1]: 543ff 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.
[2] 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.
