Question

1. (10 Points Exam Extra Credit): Let's revisit the problem of how to compute derivatives of basis vectors, which we did in Problem 1 of HWW2 (note: you will need to refer back to this HW at to do this problem). Consider the Laplacian operator, V2, in spherical coordinates. It looks like this, where the scalar (say V) goes into the 2) 10.2001 8801 VO - por l" or ) + 2gino do some do ) 320) sin20 382 Now imagine what happens when the Laplacian acts on a vector, A. Ă - Arf + As h + Add The entire Laplacion (all 3 terms) has to act on each component of the vector, so we have to insert, say, Ar finto the () and hence take derivatives of both Ar and f. And the same for the other components. In so doing, the expression for v2 A looks different for r, and than it does for a scalar How different? Show that the Laplacian of a vector, VĀ, can be written as the following in spherical coordinates: (V²A) = (v2A +(²Ao 2 a (Ao sin ) p2 sin 80 A 2DA, p2 sin²op2 a0 Ap2 A , - +- p2 sino p2 sin o p - +- 2 A 2 sin е до 2 cost DA, p2 sin²o de 2 cos e a Ag r sin dy (² Ao- - - - where v A, refers to the Laplacian of the scalar Ar, and the same with 0 and 0. How do you expect the A to look like in Cartesian coordinates? (note: there's no extensive computation required here - you should be able to answer based on what you know about Cartesian unit vectors).

2nd attached picture is problem 1 from HW 2

Answer 1

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