this lesson is PDE partial differential equation Consider the wave equationu = Utt- Urz = 0,u(t, x).(1)Given the (Lorentz) transform from t, x to ť', x': (t, x) → (ť', x')1t = y(t – Bx), a' = r(x – Bt),(2)- 32where B: real constant, 0 < B < 1, [better to use: u(t, x) = u(t(t', x'), x(t', x')) = u(t', x')instead of w(t', x') in the notation of the textbook].a) Write the differential operators 8/ot and ở/dx interms of ť and r' using the Chainrule: 8/dt = (at /t)() + (x/ət)(0/x'), 0/dx = (dt/dx)(0/0t') + (ar'/dx)(8/dx').b) Write this wave equation interms of the new variables t', r'. [You should concludethat this wave equation is covariant/invariant (not change its form) under this transform.]

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Consider the wave equation = Uu – Uzz = 0, u(t,x). (1) Given the (Lorentz) transform from t, r to ť', x': (t, x) → (t', x') 1 t = y(t – Bx), r' = y(x – Bt), y = - V1 - 32 (2) where B: real constant, 0 < B < 1, [better to use: u(t, r) = u(t(t', x'), x(t', x')) = u(t', x') instead of w(t', x') in the notation of the textbook). a) Write the differential operators 8/at and 8/dx interms of ť' and a' using the Chain rule: 8/dt = (t'/at)(8/t')+ (dx' /8t)(8/dx'), 8/dx = (dť /dx)(8/dt') + (dx' /dx)(8/dx'). b) Write this wave equation interms of the new variables ť', ď'. [You should conclude that this wave equation is covariant/invariant (not change its form) under this transform.]

Consider the wave equation = Uu – Uzz = 0, u(t,x). (1) Given the (Lorentz) transform from t, r to ť', x': (t, x) → (t', x') 1 t = y(t – Bx), r' = y(x – Bt), y = - V1 - 32 (2) where B: real constant, 0 < B < 1, [better to use: u(t, r) = u(t(t', x'), x(t', x')) = u(t', x') instead of w(t', x') in the notation of the textbook). a) Write the differential operators 8/at and 8/dx interms of ť' and a' using the Chain rule: 8/dt = (t'/at)(8/t')+ (dx' /8t)(8/dx'), 8/dx = (dť /dx)(8/dt') + (dx' /dx)(8/dx'). b) Write this wave equation interms of the new variables ť', ď'. [You should conclude that this wave equation is covariant/invariant (not change its form) under this transform.]

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter6: The Trigonometric Functions

Section6.6: Additional Trigonometric Graphs

Problem 78E

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