A difficulty in understanding a part of a paragraph in P.41 in Guillemin & Pollack (2)












0












$begingroup$


The paragraph is given below:



enter image description here



But I do not understand:



1-In the forth line why we can not have the case $df_{x} =$ constant other than 0, could anyone explain this for me please?



2-In the sixth line how f is simply the first coordinate function, could anyone give me a concrete example for describing this please?



3-In the tenth line I could not understand why the authors said "But if $f(x)$ ia an extreme value, then obviously $f$ can not be a coordinate function near x" , could anyone explain this statement for me please may be by a concrete example?



thank!






general-topology differential-geometry algebraic-topology differential-topology







share|cite|improve this question









$endgroup$












  • $begingroup$
    1. $df_x$ is a linear map of vector spaces, so how many such constant maps could it be?
    $endgroup$
    – Randall
    Dec 5 '18 at 3:42










  • $begingroup$
    you mean $f$ is a linear map of vector spaces or $df$? @Randall
    $endgroup$
    – hopefully
    Dec 5 '18 at 3:46












  • $begingroup$
    Check the definitions: given $f: X to Y$ a smooth map, $df_x$ is a linear transformation $df_x: T_xM to T_{f(x)}N$.
    $endgroup$
    – Randall
    Dec 5 '18 at 3:48










  • $begingroup$
    So by this definition since the differential(derivative) of a linear map is the linear map itself .... I do not know the answer for your first question .... I am confused.@Randall
    $endgroup$
    – hopefully
    Dec 5 '18 at 3:52






  • 1




    $begingroup$
    You are confusing the concept of linear function from elementary mathematics with the concept of a linear transformation of vector spaces that occurs in linear algebra. There is a connection between the two, but they are not the same thing.
    $endgroup$
    – Justin Young
    Dec 5 '18 at 15:01
















0












$begingroup$


The paragraph is given below:



enter image description here



But I do not understand:



1-In the forth line why we can not have the case $df_{x} =$ constant other than 0, could anyone explain this for me please?



2-In the sixth line how f is simply the first coordinate function, could anyone give me a concrete example for describing this please?



3-In the tenth line I could not understand why the authors said "But if $f(x)$ ia an extreme value, then obviously $f$ can not be a coordinate function near x" , could anyone explain this statement for me please may be by a concrete example?



thank!






general-topology differential-geometry algebraic-topology differential-topology







share|cite|improve this question









$endgroup$












  • $begingroup$
    1. $df_x$ is a linear map of vector spaces, so how many such constant maps could it be?
    $endgroup$
    – Randall
    Dec 5 '18 at 3:42










  • $begingroup$
    you mean $f$ is a linear map of vector spaces or $df$? @Randall
    $endgroup$
    – hopefully
    Dec 5 '18 at 3:46












  • $begingroup$
    Check the definitions: given $f: X to Y$ a smooth map, $df_x$ is a linear transformation $df_x: T_xM to T_{f(x)}N$.
    $endgroup$
    – Randall
    Dec 5 '18 at 3:48










  • $begingroup$
    So by this definition since the differential(derivative) of a linear map is the linear map itself .... I do not know the answer for your first question .... I am confused.@Randall
    $endgroup$
    – hopefully
    Dec 5 '18 at 3:52






  • 1




    $begingroup$
    You are confusing the concept of linear function from elementary mathematics with the concept of a linear transformation of vector spaces that occurs in linear algebra. There is a connection between the two, but they are not the same thing.
    $endgroup$
    – Justin Young
    Dec 5 '18 at 15:01














0












0








0





$begingroup$


The paragraph is given below:



enter image description here



But I do not understand:



1-In the forth line why we can not have the case $df_{x} =$ constant other than 0, could anyone explain this for me please?



2-In the sixth line how f is simply the first coordinate function, could anyone give me a concrete example for describing this please?



3-In the tenth line I could not understand why the authors said "But if $f(x)$ ia an extreme value, then obviously $f$ can not be a coordinate function near x" , could anyone explain this statement for me please may be by a concrete example?



thank!






general-topology differential-geometry algebraic-topology differential-topology







share|cite|improve this question









$endgroup$




The paragraph is given below:



enter image description here



But I do not understand:



1-In the forth line why we can not have the case $df_{x} =$ constant other than 0, could anyone explain this for me please?



2-In the sixth line how f is simply the first coordinate function, could anyone give me a concrete example for describing this please?



3-In the tenth line I could not understand why the authors said "But if $f(x)$ ia an extreme value, then obviously $f$ can not be a coordinate function near x" , could anyone explain this statement for me please may be by a concrete example?



thank!






general-topology differential-geometry algebraic-topology differential-topology




general-topology differential-geometry algebraic-topology differential-topology






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Dec 5 '18 at 3:33









hopefullyhopefully

317113




317113












  • $begingroup$
    1. $df_x$ is a linear map of vector spaces, so how many such constant maps could it be?
    $endgroup$
    – Randall
    Dec 5 '18 at 3:42










  • $begingroup$
    you mean $f$ is a linear map of vector spaces or $df$? @Randall
    $endgroup$
    – hopefully
    Dec 5 '18 at 3:46












  • $begingroup$
    Check the definitions: given $f: X to Y$ a smooth map, $df_x$ is a linear transformation $df_x: T_xM to T_{f(x)}N$.
    $endgroup$
    – Randall
    Dec 5 '18 at 3:48










  • $begingroup$
    So by this definition since the differential(derivative) of a linear map is the linear map itself .... I do not know the answer for your first question .... I am confused.@Randall
    $endgroup$
    – hopefully
    Dec 5 '18 at 3:52






  • 1




    $begingroup$
    You are confusing the concept of linear function from elementary mathematics with the concept of a linear transformation of vector spaces that occurs in linear algebra. There is a connection between the two, but they are not the same thing.
    $endgroup$
    – Justin Young
    Dec 5 '18 at 15:01


















  • $begingroup$
    1. $df_x$ is a linear map of vector spaces, so how many such constant maps could it be?
    $endgroup$
    – Randall
    Dec 5 '18 at 3:42










  • $begingroup$
    you mean $f$ is a linear map of vector spaces or $df$? @Randall
    $endgroup$
    – hopefully
    Dec 5 '18 at 3:46












  • $begingroup$
    Check the definitions: given $f: X to Y$ a smooth map, $df_x$ is a linear transformation $df_x: T_xM to T_{f(x)}N$.
    $endgroup$
    – Randall
    Dec 5 '18 at 3:48










  • $begingroup$
    So by this definition since the differential(derivative) of a linear map is the linear map itself .... I do not know the answer for your first question .... I am confused.@Randall
    $endgroup$
    – hopefully
    Dec 5 '18 at 3:52






  • 1




    $begingroup$
    You are confusing the concept of linear function from elementary mathematics with the concept of a linear transformation of vector spaces that occurs in linear algebra. There is a connection between the two, but they are not the same thing.
    $endgroup$
    – Justin Young
    Dec 5 '18 at 15:01
















$begingroup$
1. $df_x$ is a linear map of vector spaces, so how many such constant maps could it be?
$endgroup$
– Randall
Dec 5 '18 at 3:42




$begingroup$
1. $df_x$ is a linear map of vector spaces, so how many such constant maps could it be?
$endgroup$
– Randall
Dec 5 '18 at 3:42












$begingroup$
you mean $f$ is a linear map of vector spaces or $df$? @Randall
$endgroup$
– hopefully
Dec 5 '18 at 3:46






$begingroup$
you mean $f$ is a linear map of vector spaces or $df$? @Randall
$endgroup$
– hopefully
Dec 5 '18 at 3:46














$begingroup$
Check the definitions: given $f: X to Y$ a smooth map, $df_x$ is a linear transformation $df_x: T_xM to T_{f(x)}N$.
$endgroup$
– Randall
Dec 5 '18 at 3:48




$begingroup$
Check the definitions: given $f: X to Y$ a smooth map, $df_x$ is a linear transformation $df_x: T_xM to T_{f(x)}N$.
$endgroup$
– Randall
Dec 5 '18 at 3:48












$begingroup$
So by this definition since the differential(derivative) of a linear map is the linear map itself .... I do not know the answer for your first question .... I am confused.@Randall
$endgroup$
– hopefully
Dec 5 '18 at 3:52




$begingroup$
So by this definition since the differential(derivative) of a linear map is the linear map itself .... I do not know the answer for your first question .... I am confused.@Randall
$endgroup$
– hopefully
Dec 5 '18 at 3:52




1




1




$begingroup$
You are confusing the concept of linear function from elementary mathematics with the concept of a linear transformation of vector spaces that occurs in linear algebra. There is a connection between the two, but they are not the same thing.
$endgroup$
– Justin Young
Dec 5 '18 at 15:01




$begingroup$
You are confusing the concept of linear function from elementary mathematics with the concept of a linear transformation of vector spaces that occurs in linear algebra. There is a connection between the two, but they are not the same thing.
$endgroup$
– Justin Young
Dec 5 '18 at 15:01










1 Answer
1






active

oldest

votes


















1












$begingroup$

(1) The only constant linear map is the zero map. (2) The claim in the sixth line is essentially the implicit function theorem. (3) Consider $f(x)=x^2$. No change of coordinates will turn this into $x$ near $0$.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    why the only constant linear map is the zero map?
    $endgroup$
    – hopefully
    Dec 5 '18 at 6:30










  • $begingroup$
    How it is the implicit function theorem?
    $endgroup$
    – hopefully
    Dec 5 '18 at 6:43










  • $begingroup$
    do not understand the answer of (3)
    $endgroup$
    – hopefully
    Dec 5 '18 at 6:51











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1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









1












$begingroup$

(1) The only constant linear map is the zero map. (2) The claim in the sixth line is essentially the implicit function theorem. (3) Consider $f(x)=x^2$. No change of coordinates will turn this into $x$ near $0$.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    why the only constant linear map is the zero map?
    $endgroup$
    – hopefully
    Dec 5 '18 at 6:30










  • $begingroup$
    How it is the implicit function theorem?
    $endgroup$
    – hopefully
    Dec 5 '18 at 6:43










  • $begingroup$
    do not understand the answer of (3)
    $endgroup$
    – hopefully
    Dec 5 '18 at 6:51
















1












$begingroup$

(1) The only constant linear map is the zero map. (2) The claim in the sixth line is essentially the implicit function theorem. (3) Consider $f(x)=x^2$. No change of coordinates will turn this into $x$ near $0$.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    why the only constant linear map is the zero map?
    $endgroup$
    – hopefully
    Dec 5 '18 at 6:30










  • $begingroup$
    How it is the implicit function theorem?
    $endgroup$
    – hopefully
    Dec 5 '18 at 6:43










  • $begingroup$
    do not understand the answer of (3)
    $endgroup$
    – hopefully
    Dec 5 '18 at 6:51














1












1








1





$begingroup$

(1) The only constant linear map is the zero map. (2) The claim in the sixth line is essentially the implicit function theorem. (3) Consider $f(x)=x^2$. No change of coordinates will turn this into $x$ near $0$.






share|cite|improve this answer









$endgroup$



(1) The only constant linear map is the zero map. (2) The claim in the sixth line is essentially the implicit function theorem. (3) Consider $f(x)=x^2$. No change of coordinates will turn this into $x$ near $0$.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Dec 5 '18 at 5:38









Kevin CarlsonKevin Carlson

32.8k23372




32.8k23372












  • $begingroup$
    why the only constant linear map is the zero map?
    $endgroup$
    – hopefully
    Dec 5 '18 at 6:30










  • $begingroup$
    How it is the implicit function theorem?
    $endgroup$
    – hopefully
    Dec 5 '18 at 6:43










  • $begingroup$
    do not understand the answer of (3)
    $endgroup$
    – hopefully
    Dec 5 '18 at 6:51


















  • $begingroup$
    why the only constant linear map is the zero map?
    $endgroup$
    – hopefully
    Dec 5 '18 at 6:30










  • $begingroup$
    How it is the implicit function theorem?
    $endgroup$
    – hopefully
    Dec 5 '18 at 6:43










  • $begingroup$
    do not understand the answer of (3)
    $endgroup$
    – hopefully
    Dec 5 '18 at 6:51
















$begingroup$
why the only constant linear map is the zero map?
$endgroup$
– hopefully
Dec 5 '18 at 6:30




$begingroup$
why the only constant linear map is the zero map?
$endgroup$
– hopefully
Dec 5 '18 at 6:30












$begingroup$
How it is the implicit function theorem?
$endgroup$
– hopefully
Dec 5 '18 at 6:43




$begingroup$
How it is the implicit function theorem?
$endgroup$
– hopefully
Dec 5 '18 at 6:43












$begingroup$
do not understand the answer of (3)
$endgroup$
– hopefully
Dec 5 '18 at 6:51




$begingroup$
do not understand the answer of (3)
$endgroup$
– hopefully
Dec 5 '18 at 6:51


















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