Unformatted text preview: ρ and ﬂux ~ f based on a statement of the conservation of the total charge Q . 5. The sineGordon equation for u ( x, t ) arises in relativistic ﬁeld theories, nonlinear optics: ∂ 2 u ∂t 2∂ 2 u ∂x 2 + sin u = 0 Rewrite this equation for u ( q, s ), in terms of the new independent variables q = x + t and s = xt . What can you say about solutions to this equation using these new variables? 6. Solve the following PDEs for u ( x, t ) by ﬁnding a general solution: a) ∂ 2 u ∂t 2 + cos 2 ( x ) u = 0 b) ∂ 3 u ∂x 2 ∂y14 ∂u ∂y = y 7. Given that B = ∇ × A , show that R S B · ˆ n dA = 0 over any closed surface S ....
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 Fall '06
 BELMONTE
 Math, Calculus, Vector Calculus, Line integral, Stokes' theorem, total charge Q., new independent variables

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