1.6. REFERENCES 13
The next exercises provide some practice with the multiindex notation
introduced in Appendix A.
3. Prove the Multinomial Theorem:
(x1 + . . . +
, α! = α1!α2! . . . αn!, and
. . . xnn
. The sum
is taken over all multiindices α = (α1, . . . , αn) with |α| = k.
4. Prove Leibniz’s formula:
where u, v :
→ R are smooth,
, and β ≤ α means
βi ≤ αi (i = 1, . . . , n).
5. Assume that f : Rn → R is smooth. Prove
as x → 0
for each k = 1, 2, . . . . This is Taylor’s formula in multiindex notation.
(Hint: Fix x ∈
and consider the function of one variable g(t) :=
Klainerman’s article [Kl] is a nice modern overview of the ﬁeld of partial
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.