翻訳と辞書 Words near each other ・ Gradowo, Warmian-Masurian Voivodeship ・ GrADS ・ Gradsect ・ Gradska ・ Gradska Plaža Stadium ・ Gradski SRC Slavija ・ Gradski stadion ・ Gradient enhanced NMR spectroscopy ・ Gradient method ・ Gradient multilayer nanofilm ・ Gradient network ・ Gradient noise ・ Gradient oven tester ・ Gradient pattern analysis ・ Gradient Salience Model ・ Gradient theorem ・ Gradient well-formedness ・ Gradient-index optics ・ Gradient-like vector field ・ Gradient-related ・ Gradiente Expert ・ Gradients of agreement scale ・ Gradignan ・ Gradimir Crnogorac ・ Gradimir Gojer ・ Gradimir Grujičić ・ Gradimir Milovanović ・ Gradimir Smudja ・ Gradin ・ Gradina Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Gradient theorem ： ウィキペディア英語版 Gradient theorem The gradient theorem, also known as the fundamental theorem of calculus for line integrals, says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve. Let $\backslash varphi\; :\; U\; \backslash subseteq\; \backslash mathbb^n\; \backslash to\; \backslash mathbb$. Then :$\backslash varphi\backslash left(\backslash mathbf\backslash right)-\backslash varphi\backslash left(\backslash mathbf\backslash right)\; =\; \backslash int\_"\; TITLE="\backslash mathbf,\backslash ,\backslash mathbf">)\}\; \backslash nabla\backslash varphi(\backslash mathbf)\backslash cdot\; d\backslash mathbf.$ It is a generalization of the fundamental theorem of calculus to any curve in a plane or space (generally ''n''-dimensional) rather than just the real line. The gradient theorem implies that line integrals through gradient fields are path independent. In physics this theorem is one of the ways of defining a "conservative" force. By placing φ as potential, ∇φ is a conservative field. Work done by conservative forces does not depend on the path followed by the object, but only the end points, as the above equation shows. The gradient theorem also has an interesting converse: any path-independent vector field can be expressed as the gradient of a scalar field. Just like the gradient theorem itself, this converse has many striking consequences and applications in both pure and applied mathematics. ==Proof== If φ is a differentiable function from some open subset ''U'' (of R''n'') to R, and if r is a differentiable function from some closed interval () to ''U'', then by the multivariate chain rule, the composite function φ ∘ r is differentiable on (''a'', ''b'') and :$\backslash frac(\backslash varphi\; \backslash circ\; \backslash mathbf)(t)=\backslash nabla\; \backslash varphi(\backslash mathbf(t))\; \backslash cdot\; \backslash mathbf\text{'}(t)$ for all ''t'' in (''a'', ''b''). Here the ⋅ denotes the usual inner product. Now suppose the domain ''U'' of φ contains the differentiable curve γ with endpoints p and q, (oriented in the direction from p to q). If r parametrizes γ for ''t'' in (''b'' ), then the above shows that 〔Williamson, Richard and Trotter, Hale. (2004). ''Multivariable Mathematics, Fourth Edition,'' p. 374. Pearson Education, Inc. 〕 :$\backslash begin$ \int_ \nabla\varphi(\mathbf) \cdot d\mathbf &=\int_a^b \nabla\varphi(\mathbf(t)) \cdot \mathbf'(t)dt \\ &=\int_a^b \frac\varphi(\mathbf(t))dt =\varphi(\mathbf(b))-\varphi(\mathbf(a))=\varphi\left(\mathbf\right)-\varphi\left(\mathbf\right) \end where the definition of the line integral is used in the first equality, and the fundamental theorem of calculus is used in the third equality. 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「Gradient theorem」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.