利用者:ShuBraque/sandbox/多変数微積分

偏微分

詳細は「en:Partial derivative」を参照

The partial derivative generalizes the notion of the derivative to higher dimensions. 多変数関数の偏導関数は他の変数を定数であるとおいたうえでの一つの変数に関する導関数である。

偏導関数は他の手法と組み合わせ、より複雑な表示を得ることが可能である。ベクトル解析においては、del operator ($\nabla$)は勾配、発散、回転 という概念を偏導関数に関して定義するために用いられる。偏導関数の行列であるヤコビ行列は二つの任意の次元の空間の間の関数の導関数を表すために用いることができる。このため導関数は関数の定義域において直接的に点から点へ変化する線型写像と理解することができる。

偏導関数を含む微分方程式は偏微分方程式もしくはPDEs（Partial Differential Equations）と呼ばれる。これらの等式は一つの変数のみに関する導関数を含み、一般的に常微分方程式より解くのが難しい。

Multiple integration

詳細は「Multiple integral」を参照

The multiple integral expands the concept of the integral to functions of any number of variables. Double and triple integrals may be used to calculate areas and volumes of regions in the plane and in space. Fubini's theorem guarantees that a multiple integral may be evaluated as a repeated integral or iterated integral as long as the integrand is continuous throughout the domain of integration.

The surface integral and the line integral are used to integrate over curved manifolds such as surfaces 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 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,

		Domain/Codomain	Applicable techniques
Curves		$f:\mathbb {R} \to \mathbb {R} ^{n}$	Lengths of curves, line integrals, and curvature.
Surfaces		$f:\mathbb {R} ^{2}\to \mathbb {R} ^{n}$	Areas of surfaces, surface integrals, flux through surfaces, and curvature.
Scalar fields		$f:\mathbb {R} ^{n}\to \mathbb {R}$	Maxima and minima, Lagrange multipliers, directional derivatives.
Vector fields		$f:\mathbb {R} ^{m}\to \mathbb {R} ^{n}$	Any of the operations of vector calculus including gradient, divergence, and curl.

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. Quantitative analysts in finance also often use multivariate calculus to predict future trends in the stock market.

Non-deterministic, or stochastic systems can be studied using a different kind of mathematics, such as stochastic calculus.