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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.)