8. s.) Complete the Implicit Function Theorem for F: R4 R² where F(x) = (F1(x), F2(x)) and x = = (x1, x2, x3, x4): Let F: R¹ R2 be F = (0,0), and a(F₁, F2) (x3, x4) where F = (F1, F2), c = (C₁, C2, C3, C4) E R4, such that x3 = (0, 0) for all -(c) = det Then there exists a neighborhood N of (C₁, C2) and C¹ functions 03: → R. * →R 03(1, 2) and 4 = 4(x1, x2) solve F(x1, 2, 3, 4) = EN and 03 (C1, C2) = C3 and 04 (C1, C₂) = C4.
8. s.) Complete the Implicit Function Theorem for F: R4 R² where F(x) = (F1(x), F2(x)) and x = = (x1, x2, x3, x4): Let F: R¹ R2 be F = (0,0), and a(F₁, F2) (x3, x4) where F = (F1, F2), c = (C₁, C2, C3, C4) E R4, such that x3 = (0, 0) for all -(c) = det Then there exists a neighborhood N of (C₁, C2) and C¹ functions 03: → R. * →R 03(1, 2) and 4 = 4(x1, x2) solve F(x1, 2, 3, 4) = EN and 03 (C1, C2) = C3 and 04 (C1, C₂) = C4.
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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Transcribed Image Text:8.
s.) Complete the Implicit Function Theorem for F: R¹ R2
where F(x) = (F1(x), F2(x)) and x = (x1, x2, x3, x4):
Let F R¹ R2 be
F(
=
(0, 0), and
(F1, F₂)
(x3, x4)
where F = (F₁, F₂), c = (C₁, C2, C3, C4) € R¹,
-(c) = det
Then there exists a neighborhood N of (c₁, c₂) and C¹ functions
→ R.
03:
04:
such that x3 = 03(1, 2) and 4 = $4(1, 2) solve F(x1, x2, 3, 4)
(0, 0) for all
EN and 03 (C1, C2) = C3 and 4 (C₁, C2) =
= C4.
→R
Choices:
D4F2(c) 0 D4F1(c) N (x1, x2)
D3F1(c) N D3F2(c) C1
C
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