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10.Assume that f: R2 R has continuous first order partial derivatives on R2. Let E R be given. Consider the function S(a + h,b+k)-f(a,b)-hf-(a, b) – k fy(a, b) V + k2 E(h, k): %3D (a) Prove that there are e1,02 E (0,1) such that |Elbk) s If.(a + 0,h,b + k) - f.(a,b)| + If,(a,b+ ů,k) - f,(a,b)|. Where are not 0 (b) Prove that lim E (h, k)| = 0. ><0,0>

10.Assume that f: R2 R has continuous first order partial derivatives on R2. Let E R be given. Consider the function S(a + h,b+k)-f(a,b)-hf-(a, b) – k fy(a, b) V + k2 E(h, k): %3D (a) Prove that there are e1,02 E (0,1) such that |Elbk) s If.(a + 0,h,b + k) - f.(a,b)| + If,(a,b+ ů,k) - f,(a,b)|. Where are not 0 (b) Prove that lim E (h, k)| = 0. ><0,0>

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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10a and b

10.Assume that f: R2 R has continuous first order partial derivatives on R2. Let <a,b> E R be given. Consider the function S(a + h,b+k) - f (a, b) – hf-(a, b) – k f,(a, b) E(h, k) = (a) Prove that there are 01,02 E (0,1) such that |Elb.k) s If.(a + 0,h,b + k)- f.(a, b)| + If,(a,b+ 8,k) – fi(a,b)|. Where <h,k> are not o (b) Prove that lim |E (h, k)| = 0. <h,k>><0,0>
Transcribed Image Text:10.Assume that f: R2 R has continuous first order partial derivatives on R2. Let <a,b> E R be given. Consider the function S(a + h,b+k) - f (a, b) – hf-(a, b) – k f,(a, b) E(h, k) = (a) Prove that there are 01,02 E (0,1) such that |Elb.k) s If.(a + 0,h,b + k)- f.(a, b)| + If,(a,b+ 8,k) – fi(a,b)|. Where <h,k> are not o (b) Prove that lim |E (h, k)| = 0. <h,k>><0,0>
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