Vector fields

Let F = (Fx,Fy,Fz) be a vector field. The divergence of F is ∂F ∂F ∂F ∇ · F = x + y + z , ∂x ∂y ∂z while the curl of F is ∂F ∂F ∂F ∂F ∂F ∂F  ∇ × F = z − y , x − z , y − x . ∂y ∂z ∂z ∂x ∂x ∂y

A vector field F is conservative if there is a scalar field φ such that F = ∇φ. Such a φ is called a potential function for F. The potential function is unique up to addition of a constant.

1. Consider r = (x, y, z). (a) Show that ∇ · r = 3 and ∇ × r = 0. (b) Find a potential function φ for which ∇φ = r. 2. Let a and b be constant vectors. Compute the divergence and curl of

a(r · b) and a × r.

3. Define 1 1 φ(r) = = . krk px2 + y2 + z2 (a) Let E = −∇φ. Show that

x r 1 E(r) = = y . krk3 2 2 2 3/2   (x + y + z ) z

(b) Compute ∇ · E and ∇ × E. 4. Define A(r) = −2 log(krk) zˆ = (0, 0, − log(x2 + y2)). (a) Let B = ∇ × A. Show that

2  2y 2x  B(r) = θˆ = − , , 0 . krk x2 + y2 x2 + y2

(b) Compute ∇ · B and ∇ × B.

5. A two-dimensional vector field F = (Fx,Fy) can also be regarded as a three-dimensional vector field F = (Fx,Fy, 0) restricted to the xy-plane. In this setting, compute ∇ · F and ∇ × F in terms of Fx and Fy.

1 6. (a) Let φ be a scalar field. Show that ∇ × ∇φ = 0. Remark: a vector field is “irrotational” if it has curl 0. This problem shows that any conservative vector field is irrotational. In a simply connected domain, the two are equivalent. (b) Let A be a vector field. Show that ∇ · (∇ × A) = 0. Remark: a vector field is “solenoidal” if it has divergence 0. This problems shows that any vector field which is the curl of another is solenoidal. When the domain is R3, the two are equivalent. 7. Verify directly that the vector field

u(r) = (ex(x cos y + cos y − y sin y), ex(−x sin y − sin y − y cos y), 0)

is irrotational and express it as the gradient of a scalar field φ. Also check that u is solenoidal and show that it can be written as the curl of the vector field v = (0, 0, ψ) for some function ψ that you should determine.

8. Consider the two-dimensional vector field  y x  F(x, y) = − , x2 + y2 x2 + y2

defined on D = {(x, y) | (x, y) 6= (0, 0)}. (a) Let f(x, y) = arctan(y/x). Show that ∇f = F on {(x, y) | x > 0}. (b) (?) Explain why there is no (differentiable) function g : D → R such that ∇g = F on all of D. Hint: using f, define g on {(x, y) | x > 0} and {(x, y) | x < 0}.

9. (?, Maxwell’s equations) In electromagnetism, Maxwell’s equations in a vacuum relate the electric field E and magnetic field B to the electric scalar potential φ and magnetic vector potential A by

∇ · E = 0 ∇ · B = 0 1 ∂B 1 ∂E ∇ × E = − ∇ × B = . c ∂t c ∂t Let 1 c U = (E · E + B · B) and S = (E × B). 8π 4π Using these definitions together with Maxwell’s equations, give a proof of the energy conservation law ∂U + ∇ · S = 0. ∂t

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