Mathematica Cookbook by Mangano Sal
Author:Mangano, Sal [Sal Mangano]
Language: eng
Format: epub
Tags: COMPUTERS / Mathematical & Statistical Software
ISBN: 9781449390761
Publisher: O'Reilly Media
Published: 2010-04-01T16:00:00+00:00
If we use a constant function (uniform density), we get the surface area of the half sphere as expected (surface area of an entire sphere is 4 πr2).
In[97]:= g2[{x_, y_, z_}] := 1 surfaceIntegralScalar[g2, f1, {ϕ, 0, Pi/2}, {θ, 0, 2 Pi}] Out[98]= 2 π
For a vector field, there is a similar equation using Dot in place of scalar multiplication by the norm. The traditional way to visualize the vector surface interval is to consider a fluid flowing through a surface where there is a vector function F describing the velocity of the fluid at various points on the surface. The surface integral is then the flux, or the quantity of fluid flowing through the surface in unit time.
In[99]:= surfaceIntegralVector[F_, f_, {v1_, v1a_, v1b_}, {v2_, v2a_, v2b_}] := Integrate[Dot[F[f[v1, v2]], Cross[D[f[v1, v2], v1], D[f[v1, v2], v2]]], {v1, v1a, v1b}, {v2, v2a, v2b}]
Here is the solution to the flux described by {3 y, -z, x^2} through a surface described parametrically as {s t, s + t, (s^2 - t^2)/2}.
In[100]:= f[s_, t_] := {s t, s + t, (s^2 - t^2) /2} F[{x_, y_, z_}] := {3 y, -z, x^2} surfaceIntegralVector[F, f, {s, 0, 1}, {t, 0, 3}] Out[102]= -15
A standard result from electrostatics is that the net flux out of a unit sphere, for a field that is everywhere normal, is zero. We can verify this as follows:
In[103]:= F2[{x_, y_, z_}] := {1, 1, 1}/(x^2 + y^2 + z^2) In[104]:= f2[θ_, ϕ_] := {Sin[ϕ] Cos[θ], Sin[ϕ] Sin[θ], Cos[ϕ]} In[105]:= surfaceIntegralVector[F2, f2, {θ, 0, 2 Pi}, {ϕ, 0, Pi}] Out[105]= 0
Download
This site does not store any files on its server. We only index and link to content provided by other sites. Please contact the content providers to delete copyright contents if any and email us, we'll remove relevant links or contents immediately.
Deep Learning with Python by François Chollet(12609)
Hello! Python by Anthony Briggs(9931)
OCA Java SE 8 Programmer I Certification Guide by Mala Gupta(9806)
The Mikado Method by Ola Ellnestam Daniel Brolund(9797)
Dependency Injection in .NET by Mark Seemann(9354)
Algorithms of the Intelligent Web by Haralambos Marmanis;Dmitry Babenko(8317)
Test-Driven iOS Development with Swift 4 by Dominik Hauser(7777)
Grails in Action by Glen Smith Peter Ledbrook(7710)
The Well-Grounded Java Developer by Benjamin J. Evans Martijn Verburg(7573)
Becoming a Dynamics 365 Finance and Supply Chain Solution Architect by Brent Dawson(7242)
Microservices with Go by Alexander Shuiskov(7006)
Practical Design Patterns for Java Developers by Miroslav Wengner(6916)
Test Automation Engineering Handbook by Manikandan Sambamurthy(6869)
Secrets of the JavaScript Ninja by John Resig Bear Bibeault(6431)
Angular Projects - Third Edition by Aristeidis Bampakos(6283)
The Art of Crafting User Stories by The Art of Crafting User Stories(5801)
NetSuite for Consultants - Second Edition by Peter Ries(5730)
Demystifying Cryptography with OpenSSL 3.0 by Alexei Khlebnikov(5551)
Kotlin in Action by Dmitry Jemerov(5078)
