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MATH 423/673 (Spring 2008) Homework on Classical Vector Calculus (1) Stewart: 17.7: 10,24; 17.8: 10,14,20a; 17.9: 19,21,28,29 (2) Let F be a vector field in the plane. Prove that the circulation density of F at (x, y) is equal to the value of (∇ × F) · k at (x, y). (Use a similar argument to the one we used in class to show that the flux density of F is equal to ∇ · F.) (3) Prove that the flux form of Green’s Theorem implies the circulation form of Green’s Theorem. That is, suppose that for all vector fields F on R2 we know that Z ZZ F · n ds = ∇ · F dA, R ∂R where R is a domain in the plane, ∂R is the boundary of R with the induced orientation, and n is the outward unit normal vector field to ∂R. Then show that for all vector fields F on R2 Z ZZ F · T ds = (∇ × F) · k dA, ∂R R where T is the positive unit tangent vector to the oriented curve ∂R. Hint: Use the fact that T is a 90◦ rotation of n. (4) In the wave theory of light, light is regarded as a pair of time-dependent vector fields in space: The electric field E and the magnetic field B. Maxwell’s equations for the propagation of light in a vacuum state that ∇·E=0 ∇·B=0 ∂B ∂t ∂E ∇ × B = µ0 0 , ∂t ∇×E=− where µ0 and 0 are constants. Use the physical meanings of the curl and divergence we discussed in class to provide physical interpretations for each of Maxwell’s equations. (See http://en.wikipedia.org/wiki/Maxwell’s equations for additional background.)

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