# chap4 - Chapter 4 Hints and Solutions Exercise 4.1 All the...

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Chapter 4: Hints and Solutions Exercise 4.1 All the functions are smooth. So try and find points where ∇ f vanishes and then figure out which level sets of f contain these points. (a) First note that f − 1 ( c ) is the empty set if c < 0, so f − 1 ( c ) is not an n-surface. Next, since ∇ f ( x 1 , . . . , x n +1 ) = 2( x 1 , . . . , x n +1 ), then, ∇ f ( x 1 , . . . , x n +1 ) = (0 , . . . , 0) if and only if ( x 1 , . . . , x n +1 ) = (0 , . . . , 0). Noting that the origin (0 , . . . , 0) is on the level set f − 1 (0) (in fact, it’s equal to { (0 , . . . , 0) } ). So c = 0 does not give an n-surface either. Now suppose c > 0 then f − 1 ( c ) negationslash = ∅ and if p ∈ f − 1 ( c ) then p negationslash = and ∇ f ( p ) negationslash = . We conclude that f − 1 ( c ) is an n-surface if and only if c > 0. (b) ∇ f ( x 1 , . . . , x n +1 ) = 2( x 1 , . . . , x n , − x n +1 ) so ∇ f ( x 1 , . . . , x n +1 ) = (0 , . . . , 0) if and only if ( x 1 , . . . , x n , − x n +1 ) = (0 , . . . , 0) if and only if ( x 1 , . . . , x n +1 ) = (0 , . . . , 0). Since f (0 , . . . , 0) = 0 then f − 1 (0) is not an n-surface. For any other c , i.e., for c negationslash = 0, f − 1 ( c ) is an n-surface in R n +1 . (c) ∂ ∂x i f ( x 1 , . . . , x n +1 ) = x 1 x 2 ··· x i − 1 x i +1 ··· x n +1 if and only if x j = 0 for some j negationslash = i . This shows that ∇ f ( x 1 , . . . , x n +1 ) = (0 , . . . , 0) if and only if x i = x j = 0 for for at least two indices i negationslash = j . The level set f − 1 (1) consists of points for which x 1 x 2 ··· x n +1 = 0 and so contains points at which ∇ f vanishes, which means it cannot be an n-surface. On the other hand, if c negationslash = 1, f − 1 (...
View Full Document

{[ snackBarMessage ]}

### Page1 / 6

chap4 - Chapter 4 Hints and Solutions Exercise 4.1 All the...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online