Surface_Integral_Summary
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This document provides key information for calculating surface integrals: (1) Surface integrals can be found by parameterizing the surface S as r(s,t) and integrating over the parameter domain T. (2) Simple surfaces allow using the surface geometry and outward normal vector n. (3) Examples are given for calculating fluxes through a graph surface, cylindrical surface, and spherical surface by parameterizing each and computing the appropriate integrand containing the outward normal and Jacobian.
Surface_Integral_Summary
- 1. SURFACE INTEGRAL SUMMARY PORAMATE (TOM) PRANAYANUNTANA Key For Surface Integrals dAS = (rs × rt) |J| = rs × rt hidden here dsdt dAT , s t x y r θ z θ φ θ u v ... ... There are only 2 ways to find S F dAS : (1) By parameterization: F dAS S:r(s,t),(s,t)∈T in st-plane = T F (rs × rt) dsdt dAT . (2) By Geometry (for simple surfaces): S F dAS ˆnSdAS = S F ˆnS dAS |J|dAT . Observe that dAS = ˆnS|J| (rs × rt) dsdt dAT = ˆnS dAS |J| dsdt dAT . Coordinates used rs × rt (orientation of S) S : r(x, y) = x y f(x, y) rx × ry = −fx −fy 1 (up) S : r(z, θ) = R cos θ R sin θ z where r = R is a constant radius from z-axis rθ × rz = x y 0 (away from z-axis) S : r(φ, θ) = R sin φ cos θ R sin φ sin θ R cos φ where ρ = R is a constant radius from origin (0, 0, 0) rφ × rθ = x y z R R2 sin φ (away from (0, 0, 0)) Date: June 28, 2015.
- 2. Surface Integral Summary Poramate (Tom) Pranayanuntana To find AS, AS = S dAS = dAS S:r(s,t),(s,t)∈T = T dAS rs × rt |J| dsdt dAT . The flux through a graph of z = f(x, y) above a region R in the xy-plane, oriented upward, is S F dAS = R F1(x, y, f(x, y)) F2(x, y, f(x, y)) F3(x, y, f(x, y)) −fx −fy 1 nSR dxdy dAR = R F1(x, y, f(x, y)) F2(x, y, f(x, y)) F3(x, y, f(x, y)) 1 1 + f2 x + f2 y −fx −fy 1 ˆnS 1 + f2 x + f2 y |J| dxdy dAR dAS . The flux through a cylindrical surface S of radius r = R and oriented away from the z-axis is S F dAS = T F1(R, θ, z) F2(R, θ, z) F3(R, θ, z) 1 R x y 0 nST ˆnS R |J| dzdθ dAT = T F1 F2 F3 cos θ sin θ 0 ˆnS R |J| dzdθ dAT , where T is the θz-region corresponding to S. The flux through a spherical surface S of radius ρ = R and oriented away from the origin is S F dAS = T F1(R, φ, θ) F2(R, φ, θ) F3(R, φ, θ) 1 R x y z ˆnS R2 sin φ dφdθ = T F1 F2 F3 sin φ cos θ sin φ sin θ cos φ ˆnS R2 sin φ |J| dφdθ dAT , where T is the θφ-region corresponding to S. June 28, 2015 Page 2 of 2