xercise II.2. let F(): R² R³ and G(): R² R² be given by (fi(x)) f2(x) f3(x)) F(x) = (iii) f3 and ere fi(), f2), f3(-) are as in Exercise II.1(a). The notation is x = (2₂) (1₂). Use the Chain Rule to calculate (FoG)'(x) for the various choices of x: = (-₁²) (b) x = (3) (c) x = (a) x = () (d) x = = (-¹). 21 X₂ (2y1+342) y² + y² and_G(y) = (² For reference F1,F2, F3 Exercise II.1. Consider the case n = 2. (a) Find the gradients of the following functions from R² to R: x1 (i) fi := x² + x² - 2 (ii) f₂ | := 4x+1 X₂ := 9(x₁ - 1)² + (2+1)²-2 (iv) f₁ 21 X₂ I1 I₂. := √(x₁ + 1)² + (x₂ - 2)²
xercise II.2. let F(): R² R³ and G(): R² R² be given by (fi(x)) f2(x) f3(x)) F(x) = (iii) f3 and ere fi(), f2), f3(-) are as in Exercise II.1(a). The notation is x = (2₂) (1₂). Use the Chain Rule to calculate (FoG)'(x) for the various choices of x: = (-₁²) (b) x = (3) (c) x = (a) x = () (d) x = = (-¹). 21 X₂ (2y1+342) y² + y² and_G(y) = (² For reference F1,F2, F3 Exercise II.1. Consider the case n = 2. (a) Find the gradients of the following functions from R² to R: x1 (i) fi := x² + x² - 2 (ii) f₂ | := 4x+1 X₂ := 9(x₁ - 1)² + (2+1)²-2 (iv) f₁ 21 X₂ I1 I₂. := √(x₁ + 1)² + (x₂ - 2)²
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:Exercise II.2. let F(.) : R² → R³ and G(): R² → R² be given by
(fi(x))
f2(x)
f3(x))
y =
9/₁
F(x) =
where f1(), f2), f3(-) are as in Exercise II.1(a). The notation is x =
(a) x =
and G(y) =
(2y1 + 3y2
y² + y²
21
I₂
Use the Chain Rule to calculate (FG)'(x) for the various choices of x:
- (₁²) (b) x = - (-³₁) (c) x =
- (4) (d) x =
For reference F1,F2, F3
Exercise II.1. Consider the case n = 2.
(a) Find the gradients of the following functions from R2 to R:
(i) f₁ (22) := x² + x² − 2
= 4x² +
(iii) f3
(ii) f₂
:=
= 9(x₁ − 1)² + (²+¹)² — 2 (iv) fa
x1
X2
(=-¹).
I1
In
x1
X₂
and
−1
== ((x₁ + 1)² + (x₂ −2)²
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