SampleQ_postM2 - ρ and flux ~ f based on a statement of...

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MATH 405: Sample Questions (Material Since 2nd Midterm) 1. Evaluate the line integral C B · dr along the curve C defined by y = 2 x 3 between x = 0 and 4, for the vector field B = ( yx, x ). 2. a) Given the functions F ( x, y ) and G ( x, y ) which are bounded and differentiable every- where inside some circle of radius R and on its boundary, write out Green’s Theorem for these functions integrated along the boundary. b) Using this theorem, write out the integral over the interior of the circle for the case F = G = ∂ψ ∂x + ∂ψ ∂y . 3. Show that the following integral is or is not exact: a) C e xy 2 ln x dx + e 2 xy ln x dy If exact, find the potential φ . How will the value of the integral depend on φ ? 4. Derive the conservation equation ∂ρ ∂t + ∇ · f = 0 for charge density
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Unformatted text preview: ρ and flux ~ f based on a statement of the conservation of the total charge Q . 5. The sine-Gordon 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 = x-t . 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 ∂y-14 ∂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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