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(b) Let f: R2 → R be a rational function. Show that fay(a, b) = fyæ (a, b) for all (a, b) E D(f).

(b) Let f: R2 → R be a rational function. Show that fay(a, b) = fyæ (a, b) for all (a, b) E D(f).

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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Needed b part to be solved
Q3. In Unit 4, we learned about Clairaut's Theorem, which states that if f is C2 in a neighbour- hood of (a, b), then fry (a, b) = fyæ(a, b). In this exercise, you will prove Clairaut's Theorem in the special case of a rational function. (a) Let P(x, y) = Eo<ij<n@ija*y³ be a polynomial in x and y.' Show that Pay (a, b) = Pya (a, b) for all (a, b) E R². (b) Let f: R2 →R be a rational function. Show that fry(a, b) = fyæ (a, b) for all (a, b) E D(f). P(x, y) Q(x, y) [Hint: Write f(x, y) where P and Q are polynomials.]
Transcribed Image Text:Q3. In Unit 4, we learned about Clairaut's Theorem, which states that if f is C2 in a neighbour- hood of (a, b), then fry (a, b) = fyæ(a, b). In this exercise, you will prove Clairaut's Theorem in the special case of a rational function. (a) Let P(x, y) = Eo<ij<n@ija*y³ be a polynomial in x and y.' Show that Pay (a, b) = Pya (a, b) for all (a, b) E R². (b) Let f: R2 →R be a rational function. Show that fry(a, b) = fyæ (a, b) for all (a, b) E D(f). P(x, y) Q(x, y) [Hint: Write f(x, y) where P and Q are polynomials.]
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