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2. Let f: DR where DCR is open, and suppose that f € C²(D). Let xo E D and suppose that B(xo, e) CD for some > 0 (this is guaranteed by the fact that D is open). (a) Prove that 1 8² f (xo) = lim lim (f(xo + tek + sej) — f(xo + tek) — f(xo + sej) + f(xo) Әхкдх ј + f(x))] t-0 8-0 ts and that 8² f дхјдхк 1 -(xo) = lim lim (f(xo + sej + tek) — ƒ (xo + se;) − ƒ (xo + ter) + f(xo); -f(xo))]. 8-0 t-0 ts

2. Let f: DR where DCR is open, and suppose that f € C²(D). Let xo E D and suppose that B(xo, e) CD for some > 0 (this is guaranteed by the fact that D is open). (a) Prove that 1 8² f (xo) = lim lim (f(xo + tek + sej) — f(xo + tek) — f(xo + sej) + f(xo) Әхкдх ј + f(x))] t-0 8-0 ts and that 8² f дхјдхк 1 -(xo) = lim lim (f(xo + sej + tek) — ƒ (xo + se;) − ƒ (xo + ter) + f(xo); -f(xo))]. 8-0 t-0 ts

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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2. Let f: DR where DCR is open, and suppose that f € C² (D). Let xo E D and suppose that B(xo, e) C D for some € > 0 (this is guaranteed by the fact that D is open). (a) Prove that 8² f əxkxj and that 8² f əxjxk -(xo) -(xo) = = 1 lim lim (f(xo + tek + sej)-f(xo + tek)-f(xo + sej) + t-0 s 0 ts f(xa))] 1 lim lim (f(xo + se; + tek) - f(xo + sej) - f(xo + tex) + f(xo + f(xo))]. 8-0 t-0 ts (Hint: If you can prove one of the identities, the other one follows by the identical argument, so you only need to prove one of them.)
Transcribed Image Text:2. Let f: DR where DCR is open, and suppose that f € C² (D). Let xo E D and suppose that B(xo, e) C D for some € > 0 (this is guaranteed by the fact that D is open). (a) Prove that 8² f əxkxj and that 8² f əxjxk -(xo) -(xo) = = 1 lim lim (f(xo + tek + sej)-f(xo + tek)-f(xo + sej) + t-0 s 0 ts f(xa))] 1 lim lim (f(xo + se; + tek) - f(xo + sej) - f(xo + tex) + f(xo + f(xo))]. 8-0 t-0 ts (Hint: If you can prove one of the identities, the other one follows by the identical argument, so you only need to prove one of them.)
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