![]() The following code checks E = K + V for n, l,m = 7, 3, 1. The general form can be expressed as the product of a radial wavefunctionR and a spherical harmonic Y. Where L is a Laguerre polynomial, P is a Legendre polynomial and l and mare integers such that The Laplacian operator in sphericalcoordinates is Where n is an integer representing the quantization of total energy and ris the radial distance of the electron. Hydrogen wavefunctions ψ are solutions to the differential equation The symbolsf and h are used as temporary variables. The following Eigenmath code computes the surface integral. By the right handrule, crossing x into y yields n pointing upwards hence Note that the surface intersects the xy plane in a circle. Where F = xy2zi− 2x3j+ yz2k, S is the surface z = 1−x2− y2, x2 + y2 ≤ 1and n is upper.2 (In this model, the z direction points downwind.)By the properties of the cross product we have the following for the unitnormal n and for dA.įor example, evaluate the surface integral∫∫SF Let S be the surface of the sailparameterized by x and y. Theintegral sums over the entire area of the sail.
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