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.
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