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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Total derivative ： ウィキペディア英語版 Total derivative In the mathematical field of differential calculus, a total derivative or full derivative of a function $f$ of several variables, e.g., $t$, $x$, $y$, etc., with respect to an exogenous argument, e.g., $t$, is the limiting ratio of the change in the function's value to the change in the exogenous argument's value (for arbitrarily small changes), taking into account the exogenous argument's direct effect as well as its indirect effects via the other arguments of the function. The total derivative of a function is different from its corresponding partial derivative ($\backslash partial$). Calculation of the total derivative of $f$ with respect to $t$ does not assume that the other arguments are constant while $t$ varies; instead, it allows the other arguments to depend on $t$. The total derivative adds in these ''indirect dependencies'' to find the overall dependency of $f$ on $t.$〔Chiang, Alpha C. ''Fundamental Methods of Mathematical Economics'', McGraw-Hill, third edition, 1984.〕 For example, the total derivative of $f(t,x,y)$ with respect to $t$ is :$\backslash frac=\backslash frac\; \backslash frac\; +\; \backslash frac\; \backslash frac\; +\; \backslash frac\; \backslash frac$ which simplifies to :$\backslash frac=\backslash frac\; +\; \backslash frac\; \backslash frac\; +\; \backslash frac\; \backslash frac.$ Consider multiplying both sides of the equation by the differential $\backslash operatorname\; dt$: :$=\backslash frac\backslash operatorname\; dt\; +\; \backslash frac\; \backslash operatorname\; dx\; +\; \backslash frac\; \backslash operatorname\; dy.$ The result is the differential change $\backslash operatorname\; df$ in, or total differential of, the function $f$. Because $f$ depends on $t$, some of that change will be due to the partial derivative of $f$ with respect to $t$. However, some of that change will also be due to the partial derivatives of $f$ with respect to the variables $x$ and $y$. So, the differential $\backslash operatorname\; dt$ is applied to the total derivatives of $x$ and $y$ to find differentials $\backslash operatorname\; dx$ and $\backslash operatorname\; dy$, which can then be used to find the contribution to $\backslash operatorname\; df$. "Total derivative" is sometimes also used as a synonym for the material derivative, $\backslash frac,$ in fluid mechanics. ==Differentiation with indirect dependencies== Suppose that ''f'' is a function of two variables, ''x'' and ''y''. Normally these variables are assumed to be independent. However, in some situations they may be dependent on each other. For example ''y'' could be a function of ''x'', constraining the domain of ''f'' to a curve in $\backslash mathbb^2$. In this case the partial derivative of ''f'' with respect to ''x'' does not give the true rate of change of ''f'' with respect to changing ''x'' because changing ''x'' necessarily changes ''y''. The total derivative takes such dependencies into account. For example, suppose :$f(x,y)=xy$. The rate of change of ''f'' with respect to ''x'' is usually the partial derivative of ''f'' with respect to ''x''; in this case, :$\backslash frac\; =\; y$. However, if ''y'' depends on ''x'', the partial derivative does not give the true rate of change of ''f'' as ''x'' changes because it holds ''y'' fixed. Suppose we are constrained to the line :$y=x;$ then :$f(x,y)\; =\; f(x,x)\; =\; x^2$. In that case, the total derivative of ''f'' with respect to ''x'' is :$\backslash frac\; =\; 2\; x$. Instead of immediately substituting for ''y'' in terms of ''x'', this can be found equivalently using the chain rule: :$\backslash frac=\; \backslash frac\; +\; \backslash frac\backslash frac\; =\; y+x\; \backslash cdot\; 1\; =\; x+y.$ Notice that this is not equal to the partial derivative: :$\backslash frac\; =\; 2\; x\; \backslash neq\; \backslash frac\; =\; y\; =\; x$. While one can often perform substitutions to eliminate indirect dependencies, the chain rule provides for a more efficient and general technique. Suppose ''M''(''t'', ''p''1, ..., ''pn'') is a function of time ''t'' and ''n'' variables $p\_i$ which themselves depend on time. Then, the total time derivative of ''M'' is :$t\}\; =\; \backslash frac\; M\; \backslash bigl(t,\; p\_1(t),\; \backslash ldots,\; p\_n(t)\backslash bigr).$ The chain rule for differentiating a function of several variables implies that :$t\}$ = \frac + \sum_^n \frac\frac = \biggl(\frac + \sum_^n \frac\frac\biggr)(M). This expression is often used in physics for a gauge transformation of the Lagrangian, as two Lagrangians that differ only by the total time derivative of a function of time and the ''n'' generalized coordinates lead to the same equations of motion. An interesting example concerns the resolution of causality concerning the Wheeler–Feynman time-symmetric theory. The operator in brackets (in the final expression) is also called the total derivative operator (with respect to ''t''). For example, the total derivative of ''f''(''x''(''t''), ''y''(''t'')) is :$\backslash frac\; =\; +.$ Here there is no ∂''f'' / ∂''t'' term since ''f'' itself does not depend on the independent variable ''t'' directly. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Total derivative」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.