1.6. REFERENCES 13
The next exercises provide some practice with the multiindex notation
introduced in Appendix A.
3. Prove the Multinomial Theorem:
(x1 + . . . +
xn)k
=
|α|=k
|α|
α
xα,
where
(|α|)
α
:=
|α|!
α!
, α! = α1!α2! . . . αn!, and
xα
= x1
α1
. . . xnn
α
. The sum
is taken over all multiindices α = (α1, . . . , αn) with |α| = k.
4. Prove Leibniz’s formula:
Dα(uv)
=
β≤α
α
β
DβuDα−βv,
where u, v :
Rn
→ R are smooth,
(α)
β
:=
α!
β!(α−β)!
, and β ≤ α means
βi ≤ αi (i = 1, . . . , n).
5. Assume that f : Rn → R is smooth. Prove
f(x) =
|α|≤k
1
α!
Dαf(0)xα
+
O(|x|k+1)
as x → 0
for each k = 1, 2, . . . . This is Taylor’s formula in multiindex notation.
(Hint: Fix x ∈
Rn
and consider the function of one variable g(t) :=
f(tx).)
1.6. REFERENCES
Klainerman’s article [Kl] is a nice modern overview of the field of partial
differential equations.
Good general texts and monographs on PDE include Arnold [Ar2],
Courant–Hilbert [C-H], DiBenedetto [DB1], Folland [F1], Friedman [Fr2].
Garabedian [G], John [J2], Jost [Jo], McOwen [MO], Mikhailov [M], Petro-
vsky [Py], Rauch [R], Renardy–Rogers [R-R], Showalter [Sh], Smirnov
[Sm], Smoller [S], Strauss [St2], Taylor [Ta], Thoe–Zachmanoglou [T-Z],
Zauderer [Za], and many others. The prefaces to Arnold [Ar2] and to Bern-
stein [Bt] are particularly interesting reading. Zwillinger’s handbook [Zw]
on differential equations is a useful compendium of methods for PDE.
