xy where f(x, y) = U if (x, y) = (0,0) and f(x, y) = x² + y² if (x, y) = (0,0). (a) Prove that Duf(0,0) = 0 if u = (1,0). (b) Prove that Duf(0,0) = 0 if u = (0, 1). (c) Prove that Duf(0,0) does not exist if u = (a, b) where ab ‡ 0.
xy where f(x, y) = U if (x, y) = (0,0) and f(x, y) = x² + y² if (x, y) = (0,0). (a) Prove that Duf(0,0) = 0 if u = (1,0). (b) Prove that Duf(0,0) = 0 if u = (0, 1). (c) Prove that Duf(0,0) does not exist if u = (a, b) where ab ‡ 0.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![xy
1. Let f: R² → R where f(x, y) = 0 if (x, y) = (0,0) and f(x, y) = 2241²2
if (x, y) = (0,0).
(a) Prove that Duf(0, 0) = 0 if u = (1,0).
(b) Prove that Duf(0,0) = 0 if u = (0, 1).
(c) Prove that Duf(0, 0) does not exist if u = (a, b) where ab 0.
(d) Prove that Df(0, 0) does not exist (this can be done by proving
that f is not continuous at (0,0)).
4) Fix 230 chrose 8 = [E]. Assume Q = [7] ~ S. Then 11 + [F(t,0) - ((0,0)] - 011
2
19-07-011=0 < 2
citu = (1,0).
La = (0,0) (0) = 0 Lu√² = 0 | (0,0) 0
1) Fin 2-0. Choose S. Assume 0<|||< S. Then 11 = [f(0, 1) - f(0,0)]- 011
[00] =011=0 < E
c²t₁ = (0₁1)
La= (0,0)(1)-0 Life 01 (0,0) = 0
Sidemort: f(t,0) = + (0).
C
0
Q
f (0,1)=
( (0,₁1) - 01/10/= = = = 0
(1) 0
0²
c) sicut cata= (at, bt) f(at, bt) = _abte
45²
X² (₁²-b²) ₁²-6²
=](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F79599c56-a340-49a0-b0ff-829b3947a798%2Fdcdf7c57-cf61-4451-a9f7-876e6dddcaa7%2Fjic2p1_processed.jpeg&w=3840&q=75)
Transcribed Image Text:xy
1. Let f: R² → R where f(x, y) = 0 if (x, y) = (0,0) and f(x, y) = 2241²2
if (x, y) = (0,0).
(a) Prove that Duf(0, 0) = 0 if u = (1,0).
(b) Prove that Duf(0,0) = 0 if u = (0, 1).
(c) Prove that Duf(0, 0) does not exist if u = (a, b) where ab 0.
(d) Prove that Df(0, 0) does not exist (this can be done by proving
that f is not continuous at (0,0)).
4) Fix 230 chrose 8 = [E]. Assume Q = [7] ~ S. Then 11 + [F(t,0) - ((0,0)] - 011
2
19-07-011=0 < 2
citu = (1,0).
La = (0,0) (0) = 0 Lu√² = 0 | (0,0) 0
1) Fin 2-0. Choose S. Assume 0<|||< S. Then 11 = [f(0, 1) - f(0,0)]- 011
[00] =011=0 < E
c²t₁ = (0₁1)
La= (0,0)(1)-0 Life 01 (0,0) = 0
Sidemort: f(t,0) = + (0).
C
0
Q
f (0,1)=
( (0,₁1) - 01/10/= = = = 0
(1) 0
0²
c) sicut cata= (at, bt) f(at, bt) = _abte
45²
X² (₁²-b²) ₁²-6²
=
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