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Assume that f is a scalar, vector, or tensor field defined on a surface S. To find an explicit formula for the surface integral of f over S, we need to parameterize S by defining a system of curvilinear coordinates on S, like the latitude and longitude on a sphere. Let such a parameterization be r(s, t), where (s, t) varies in some region T in ...
Integration by parts is a heuristic rather than a purely mechanical process for solving integrals; given a single function to integrate, the typical strategy is to carefully separate this single function into a product of two functions u(x)v(x) such that the residual integral from the integration by parts formula is easier to evaluate than the ...
His formula showed that the Gaussian curvature could be calculated near a point as the limit of area over angle excess for geodesic triangles shrinking to the point. Since any closed surface can be decomposed up into geodesic triangles, the formula could also be used to compute the integral of the curvature over the whole surface.
On this chart, it may be pulled back to an n-form on an open subset of R n. Here, the form has a well-defined Riemann or Lebesgue integral as before. The change of variables formula and the assumption that the chart is positively oriented together ensure that the integral of ω is independent of the chosen chart.
The classical Stokes' theorem relates the surface integral of the curl of a vector field over a surface in Euclidean three-space to the line integral of the vector field over its boundary. It is a special case of the general Stokes theorem (with n = 2 {\displaystyle n=2} ) once we identify a vector field with a 1-form using the metric on ...
The use of a single sum, rather than two one dimensional schemes from discretizing the θ and φ integrals individually, leads to more efficient procedure: fewer total grid points are required to obtain similar accuracy. A competing factor is the computational speedup available when using the direct product of two one-dimensional grids.
It equates the surface integral of the curl of a vector field to the above line integral taken around the boundary of the surface. Another way one can define the curl vector of a function F at a point is explicitly as the limiting value of a vector-valued surface integral around a shell enclosing p divided by the volume enclosed, as the shell ...
The "remarkable", and surprising, feature of this theorem is that although the definition of the Gaussian curvature of a surface S in R 3 certainly depends on the way in which the surface is located in space, the end result, the Gaussian curvature itself, is determined by the intrinsic metric of the surface without any further reference to the ...