 
Summary: 18.024ESG Problem Set 4
Pramod N. Achar
Spring 2000
Thursday
1. Let f, g : A R3
be two differentiable functions, where A R. Prove
the product rule for cross product; viz.,
D(f × g) = Df × g + f × Dg.
Be careful when manipulating this formula: remember that the cross prod
uct, unlike multiplication in R, is not commutative.
2. Exercises 1 and 6 in Section 14.4 of Apostol, Volume I.
3. Exercises 8 and 12 in Section 14.4 of Apostol, Volume I. For Exercise 12,
recall that the hyperbolic cosine function is defined by the formula
cosh x =
ex
+ ex
2
.
4. (Optional) Let f and g be two matrixvalued functions defined on some
domain A R. We can define the notions of limit, derivative, and integral
