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f(x, y) = xy 0. x² - y² x² + y²¹ (x, y) = (0,0) (x, y) = (0,0) Compute fa and fy for all points in R². Compute fay and fy for all points in R2. Show that fay and fy are not continuous at (0, 0).

f(x, y) = xy 0. x² - y² x² + y²¹ (x, y) = (0,0) (x, y) = (0,0) Compute fa and fy for all points in R². Compute fay and fy for all points in R2. Show that fay and fy are not continuous at (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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2. We have mentioned the Clairaut's theorem in lecture: 2 Theorem 1 (Clairaut's Theorem). Suppose ƒ : R² → R for which the mixed partial derivatives fry and fyr are both continuous on an open disk D that contains the point (xo, yo), then fxy(xo, Yo) = fyx (xo, Yo) In this exercise, you are supposed to explore a counter-example when the assumption of the theorem fails. To this end, consider the following example. (a) (b) (c) f(x, y): = xy 0, x² y² x² + y²¹ (x, y) = (0,0) (x, y) = (0,0) Compute fx and fy for all points in R². Compute fay and fyr for all points in R2. yx Show that fay and fyx are not continuous at (0, 0).
Transcribed Image Text:2. We have mentioned the Clairaut's theorem in lecture: 2 Theorem 1 (Clairaut's Theorem). Suppose ƒ : R² → R for which the mixed partial derivatives fry and fyr are both continuous on an open disk D that contains the point (xo, yo), then fxy(xo, Yo) = fyx (xo, Yo) In this exercise, you are supposed to explore a counter-example when the assumption of the theorem fails. To this end, consider the following example. (a) (b) (c) f(x, y): = xy 0, x² y² x² + y²¹ (x, y) = (0,0) (x, y) = (0,0) Compute fx and fy for all points in R². Compute fay and fyr for all points in R2. yx Show that fay and fyx are not continuous at (0, 0).
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