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Remembering that z = xiy and w = uiv, we note that there is a very useful test to determine whether a function w = f(z) is analytic at a point This is provided by the CauchyRiemann equations These state that w = f(z) is differentiable at a point z = z 0 if, and only if, ∂u ∂x =Conversely, if F and Gare of class C2, then udefined by u(x,t) = F(x ct) G(x−ct) is a classical solution of (61) T The families of lines x−ct= constant and xct= constant, 42 on (x,t) plane are called the characteristics of the wave equation These are straight lines with slopes ±1/c The function G(x− ct) represents aSolutions for Review Problems Math 2850 Sec 4 page 4 of 14 (b) In general, the normal vector for the tangent plane to the level surface of F(x,y,z) = k at the point (a,b,c) is ∇F(a,b,c) 2 ƒ~ƒZƒX "¯Œ^ 50'ã ƒ{ƒu
