Mathematical Tools for Physics

9—Vector Calculus 1 280

and graph the potential functionV(r)for this limiting case. This violates Eq. ( 42 ). Why?
(c) Compute the area mass density,σ=dM/dA, in this limiting case and find the relationship between the
discontinuity indV/drand the value ofσ.

9.35 Evaluate

δijijk, mjknjk, ∂ixi, ∂ixj, ijkijk, δijvj

9.36 Verify the identities for arbitraryA~,

(

A~.∇

)

~r=A~ or Ai∂ixj=Aj

∇.∇×~v= 0 or ∂iijk∂jvk= 0

∇×∇f= 0 or ijk∂j∂kf= 0

∇.

(

fA~

)

=

(

∇f

)

.A~+f

(

∇.A~

)

or ∂i(fAi) = (∂if)Ai+f∂iAi

You can try proving all these in the standard vector notation, but use the index notation instead. It’s a lot easier.

9.37 Prove∇×∇×~v=∇(∇.~v)−∇^2 ~v. First translate it into index notation and see what identity you have
to prove about’s.

9.38 Is∇×~vperpendicular to~v? Either prove it’s true or give an explicit example for which it’s false.

9.39 If for arbitraryAiand arbitraryBjit is known thataijAiBj= 0, prove then that all theaijare zero.

9.40 Compute the divergences of
Axxˆ+By^2 yˆ+Cˆzin rectangular coordinates.
Arˆr+Bθ^2 ˆθ+Cφˆin spherical coordinates.
How do the pictures of these vector fields correspond to the results of these calculations?