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Not Simultaneously Zero

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1

$begingroup$

I am reviewing Chapter 4, Section 6.1, Theorem 6 of D. Widder's Advanced Calculus. It states

$1. f(x,y,alpha),g(alpha),h(alpha) in C^1$

$2. f_1^2+f_2^2 ne 0$

$3. (g')^2+(h')^2 ne 0$

$4. f(g(alpha),h(alpha),alpha) equiv 0$

$5. f_3(g(alpha),h(alpha),alpha)) equiv 0$

$implies$ The family $f(x,y,alpha)=0$ has the curve $x=g(alpha),y=h(alpha)$ as an envelope.

Questions

1. Does (2) above mean that $f_1$ and $f_2$ can't be simultaneously zero? (i.e. $f_1$ and $f_2$ must be smooth functions)




2. Same question as 1 for (3) above.




3. Why do we sum-square? (e.g. why $f_1^2+f_2^2 ne 0$ instead of $f_1+f_2 ne 0$?)

calculus smooth-functions

share|cite|improve this question

edited Mar 12 at 21:29

A. Hendry

asked Mar 12 at 21:10

A. HendryA. Hendry

276

$endgroup$

add a comment | 

1

$begingroup$

I am reviewing Chapter 4, Section 6.1, Theorem 6 of D. Widder's Advanced Calculus. It states

$1. f(x,y,alpha),g(alpha),h(alpha) in C^1$

$2. f_1^2+f_2^2 ne 0$

$3. (g')^2+(h')^2 ne 0$

$4. f(g(alpha),h(alpha),alpha) equiv 0$

$5. f_3(g(alpha),h(alpha),alpha)) equiv 0$

$implies$ The family $f(x,y,alpha)=0$ has the curve $x=g(alpha),y=h(alpha)$ as an envelope.

Questions

1. Does (2) above mean that $f_1$ and $f_2$ can't be simultaneously zero? (i.e. $f_1$ and $f_2$ must be smooth functions)




2. Same question as 1 for (3) above.




3. Why do we sum-square? (e.g. why $f_1^2+f_2^2 ne 0$ instead of $f_1+f_2 ne 0$?)

calculus smooth-functions

share|cite|improve this question

edited Mar 12 at 21:29

A. Hendry

asked Mar 12 at 21:10

A. HendryA. Hendry

276

$endgroup$

add a comment | 

$begingroup$

I am reviewing Chapter 4, Section 6.1, Theorem 6 of D. Widder's Advanced Calculus. It states

$1. f(x,y,alpha),g(alpha),h(alpha) in C^1$

$2. f_1^2+f_2^2 ne 0$

$3. (g')^2+(h')^2 ne 0$

$4. f(g(alpha),h(alpha),alpha) equiv 0$

$5. f_3(g(alpha),h(alpha),alpha)) equiv 0$

$implies$ The family $f(x,y,alpha)=0$ has the curve $x=g(alpha),y=h(alpha)$ as an envelope.

Questions

1. Does (2) above mean that $f_1$ and $f_2$ can't be simultaneously zero? (i.e. $f_1$ and $f_2$ must be smooth functions)




2. Same question as 1 for (3) above.




3. Why do we sum-square? (e.g. why $f_1^2+f_2^2 ne 0$ instead of $f_1+f_2 ne 0$?)

calculus smooth-functions

share|cite|improve this question

edited Mar 12 at 21:29

A. Hendry

asked Mar 12 at 21:10

A. HendryA. Hendry

276

$endgroup$

share|cite|improve this question

edited Mar 12 at 21:29

A. Hendry

asked Mar 12 at 21:10

A. HendryA. Hendry

276

share|cite|improve this question

edited Mar 12 at 21:29

A. Hendry

asked Mar 12 at 21:10

A. HendryA. Hendry

276

asked Mar 12 at 21:10

A. HendryA. Hendry

276

asked Mar 12 at 21:10

A. HendryA. Hendry

276

1 Answer
1

active

oldest

votes

1

$begingroup$

1) Yes, assuming we are talking about real-valued functions, $f_1^2 + f_2^2 ne 0$ if and only if $f_1$ and $f_2$ are not both $0$.

2) Same question, same answer.

3) $f_1 + f_2$ would be $0$ if $f_2 = -f_1$, where both can be nonzero.

share|cite|improve this answer

answered Mar 12 at 21:18

Robert IsraelRobert Israel

327k23216469

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Ok, excellent. This is what I thought. I see the sum-square notation in many different math books and always assumed they were trying to say "non-simultaneously zero", but never had any definite confirmation that this was the case. Thank you!
  $endgroup$
  – A. Hendry
  Mar 12 at 21:29
  

  
  
  
  

add a comment | 

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1

$begingroup$

1) Yes, assuming we are talking about real-valued functions, $f_1^2 + f_2^2 ne 0$ if and only if $f_1$ and $f_2$ are not both $0$.

2) Same question, same answer.

3) $f_1 + f_2$ would be $0$ if $f_2 = -f_1$, where both can be nonzero.

share|cite|improve this answer

answered Mar 12 at 21:18

Robert IsraelRobert Israel

327k23216469

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Ok, excellent. This is what I thought. I see the sum-square notation in many different math books and always assumed they were trying to say "non-simultaneously zero", but never had any definite confirmation that this was the case. Thank you!
  $endgroup$
  – A. Hendry
  Mar 12 at 21:29
  

  
  
  
  

add a comment | 

1

$begingroup$

1) Yes, assuming we are talking about real-valued functions, $f_1^2 + f_2^2 ne 0$ if and only if $f_1$ and $f_2$ are not both $0$.

2) Same question, same answer.

3) $f_1 + f_2$ would be $0$ if $f_2 = -f_1$, where both can be nonzero.

share|cite|improve this answer

answered Mar 12 at 21:18

Robert IsraelRobert Israel

327k23216469

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Ok, excellent. This is what I thought. I see the sum-square notation in many different math books and always assumed they were trying to say "non-simultaneously zero", but never had any definite confirmation that this was the case. Thank you!
  $endgroup$
  – A. Hendry
  Mar 12 at 21:29
  

  
  
  
  

add a comment | 

$begingroup$

1) Yes, assuming we are talking about real-valued functions, $f_1^2 + f_2^2 ne 0$ if and only if $f_1$ and $f_2$ are not both $0$.

2) Same question, same answer.

3) $f_1 + f_2$ would be $0$ if $f_2 = -f_1$, where both can be nonzero.

share|cite|improve this answer

answered Mar 12 at 21:18

Robert IsraelRobert Israel

327k23216469

$endgroup$

share|cite|improve this answer

answered Mar 12 at 21:18

Robert IsraelRobert Israel

327k23216469

answered Mar 12 at 21:18

Robert IsraelRobert Israel

327k23216469

answered Mar 12 at 21:18

Robert IsraelRobert Israel

327k23216469