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.