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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 Surface integral ： ウィキペディア英語版 Surface integral In mathematics, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analog of the line integral. Given a surface, one may integrate over its scalar fields (that is, functions which return scalars as values), and vector fields (that is, functions which return vectors as values). Surface integrals have applications in physics, particularly with the theories of classical electromagnetism. == Surface integrals of scalar fields == To find an explicit formula for the surface integral, we need to parameterize the surface of interest, ''S'', by considering a system of curvilinear coordinates on ''S'', like the latitude and longitude on a sphere. Let such a parameterization be x(''s'', ''t''), where (''s'', ''t'') varies in some region ''T'' in the plane. Then, the surface integral is given by :$$ \iint_ f \,\mathrm dS = \iint_ f(\mathbf(s, t)) \left\|\times \right\| \mathrm ds\, \mathrm dt where the expression between bars on the right-hand side is the magnitude of the cross product of the partial derivatives of x(''s'', ''t''), and is known as the surface element. For example, if we want to find the surface area of the graph of some scalar function, say $z=f\backslash ,(x,y)$, we have :$$ A = \iint_S \,\mathrm dS = \iint_T \left\|\times \right\| \mathrm dx\, \mathrm dy where $\backslash mathbf=(x,\; y,\; z)=(x,\; y,\; f(x,y))$. So that $=(1,\; 0,\; f\_x(x,y))$, and $=(0,\; 1,\; f\_y(x,y))$. So, :$\backslash begin$ A &\right)\right\| \mathrm dx\, \mathrm dy \\ &, 1\right)\right\| \mathrm dx\, \mathrm dy \\ &\right)^2+1}\, \, \mathrm dx\, \mathrm dy \end which is the standard formula for the area of a surface described this way. One can recognize the vector in the second line above as the normal vector to the surface. Note that because of the presence of the cross product, the above formulas only work for surfaces embedded in three-dimensional space. This can be seen as integrating a Riemannian volume form on the parameterized surface, where the metric tensor is given by the first fundamental form of the surface. == Surface integrals of vector fields == Consider a vector field v on ''S'', that is, for each x in ''S'', v(x) is a vector. The surface integral can be defined component-wise according to the definition of the surface integral of a scalar field; the result is a vector. This applies for example in the expression of the electric field at some fixed point due to an electrically charged surface, or the gravity at some fixed point due to a sheet of material. Alternatively, if we integrate the normal component of the vector field, the result is a scalar. Imagine that we have a fluid flowing through ''S'', such that v(x) determines the velocity of the fluid at x. The flux is defined as the quantity of fluid flowing through ''S'' per unit time. This illustration implies that if the vector field is tangent to ''S'' at each point, then the flux is zero, because the fluid just flows in parallel to ''S'', and neither in nor out. This also implies that if v does not just flow along ''S'', that is, if v has both a tangential and a normal component, then only the normal component contributes to the flux. Based on this reasoning, to find the flux, we need to take the dot product of v with the unit surface normal n to ''S'' at each point, which will give us a scalar field, and integrate the obtained field as above. We find the formula :$\backslash begin$ \iint_S \cdot\mathrm d\cdot \right)\,\mathrm dS\\ &(s, t)) \cdot \times \right) \over \left\|\left(\times \right)\right\|}\right) \left\|\left(\times \right)\right\| \mathrm ds\, \mathrm dt\\ &(s, t))\cdot \left(\times \right) \mathrm ds\, \mathrm dt. \end The cross product on the right-hand side of this expression is a (not necessarily unital) surface normal determined by the parametrization. This formula ''defines'' the integral on the left (note the dot and the vector notation for the surface element). We may also interpret this as a special case of integrating 2-forms, where we identify the vector field with a 1-form, and then integrate its Hodge dual over the surface. This is equivalent to integrating $\backslash langle\; \backslash mathbf,\; \backslash mathbf\; \backslash rangle\; \backslash ;\backslash mathrm\; dS$ over the immersed surface, where $\backslash mathrm\; dS$ is the induced volume form on the surface, obtained by interior multiplication of the Riemannian metric of the ambient space with the outward normal of the surface. 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「Surface integral」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.