“surface integral”的概念、定义、翻译、参考文献-科学参考

surface integral

Physics

An integral taken over a surface that can involve vectors or scalars. If V(x,y,z) is a vector function defined in a region that contains the surface S and the unit vector $\hat{n}$ is defined as the (outward) normal to that surface, the surface integral is defined by $\int_{S} V \cdot \hat{n} d S = & \int V \cdot d S .$ This integral is the (outward) flux of V through S. Surface integrals for flux occur in electricity and magnetism, an example being Gauss’ law. It is also possible to define the surface integral ∫‎V×dS and for the scalar function ϕ‎(x,y,z) the surface integral ∫‎_Sϕ‎dS.

Mathematics

For a smooth surface Σ‎, and continuous (see continuous function) scalar field f on Σ‎ and a continuous vector field F on Σ‎, a surface integral may take either of the forms $\int_{Σ} f d S$ or $\int_{Σ} F \cdot d S$ . If f is identically 1, then the integral $\int_{Σ} f d S$ defines the surface area of Σ‎. The integral $\int_{Σ} F \cdot d S$ defines the flux of F through Σ‎. If r(u,v), where (u,v) ε‎ U ⊆ ℝ², is a parametrization of Σ‎, then the surface integrals are defined by

$\begin{array}{rcl} \int_{Σ} f d S & = {\int\int}_{U} f (r (u, v)) | \frac{\partial r}{\partial u} \times \frac{\partial r}{\partial v} | d u d v, \\ \int_{Σ} F \cdot d S & = {\int\int}_{U} F (r (u, v)) \cdot (\frac{\partial r}{\partial u} \times \frac{\partial r}{\partial v}) d u d v, \end{array}$

where × denotes the vector product. See divergence theorem, Stokes’ theorem.

Electronics and Electrical Engineering

The integration of a function f(x,y,z) over a surface S.
$\iint_{S}^{} f (x, y, z) d S$
There are different formulations for the surface integral in terms of double integrals, depending on how S is defined. If S is defined as g(x,y) over a region D, then we obtain
$\iint_{D}^{} f (x, y, g (x, y) \sqrt{{(\frac{\partial g}{\partial x})}^{2} + {(\frac{\partial g}{\partial y})}^{2} + 1 d A}$

单词	surface integral
释义	surface integral Physics An integral taken over a surface that can involve vectors or scalars. If V(x,y,z) is a vector function defined in a region that contains the surface S and the unit vector $\hat{n}$ is defined as the (outward) normal to that surface, the surface integral is defined by $\int_{S} V \cdot \hat{n} d S = & \int V \cdot d S .$ This integral is the (outward) flux of V through S. Surface integrals for flux occur in electricity and magnetism, an example being Gauss’ law. It is also possible to define the surface integral ∫‎V×dS and for the scalar function ϕ‎(x,y,z) the surface integral ∫‎_Sϕ‎dS. Mathematics For a smooth surface Σ‎, and continuous (see continuous function) scalar field f on Σ‎ and a continuous vector field F on Σ‎, a surface integral may take either of the forms $\int_{Σ} f d S$ or $\int_{Σ} F \cdot d S$ . If f is identically 1, then the integral $\int_{Σ} f d S$ defines the surface area of Σ‎. The integral $\int_{Σ} F \cdot d S$ defines the flux of F through Σ‎. If r(u,v), where (u,v) ε‎ U ⊆ ℝ², is a parametrization of Σ‎, then the surface integrals are defined by $\begin{array}{rcl} \int_{Σ} f d S & = {\int\int}_{U} f (r (u, v)) \| \frac{\partial r}{\partial u} \times \frac{\partial r}{\partial v} \| d u d v, \\ \int_{Σ} F \cdot d S & = {\int\int}_{U} F (r (u, v)) \cdot (\frac{\partial r}{\partial u} \times \frac{\partial r}{\partial v}) d u d v, \end{array}$ where × denotes the vector product. See divergence theorem, Stokes’ theorem. Electronics and Electrical Engineering The integration of a function f(x,y,z) over a surface S. $\iint_{S}^{} f (x, y, z) d S$ There are different formulations for the surface integral in terms of double integrals, depending on how S is defined. If S is defined as g(x,y) over a region D, then we obtain $\iint_{D}^{} f (x, y, g (x, y) \sqrt{{(\frac{\partial g}{\partial x})}^{2} + {(\frac{\partial g}{\partial y})}^{2} + 1 d A}$
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