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A difficulty in understanding 2 examples in Petrovic.

0

$begingroup$

Those are the two examples:

But it is not clear for me how example 1 shows if $f$ is injective then $Df$ is injective, could anyone explain for me how this is shown please?

Also in example 2:

1-It is not clear for me why he wrote in DF(a) an $x$, what is this $x$ multiplied by the given matrix, from where it comes?

2-Also I do not understand how he proved that the injectivity of the derivatives implies the injectivity of $**f**$? more precisely I do not understand why $DF(a)(x_1) = DF(a)(x_2)$ iff $x_1 = x_2$ and also what is the importance of saying $**f**(t_1) = **f**(t_2)$ implies $t_1 = t_2$, is there a relation between $t_1$ and $x_1$, and $t_2$ and $x_2$ ?

3-It is not clear for me also why the inverse is incorrect i.e. injectivity of $f$ does not imply injectivity of the derivative, could anyone clarify this for me please?

Could anyone clarify this points for me please?

real-analysis calculus analysis multivariable-calculus implicit-function-theorem

share|cite|improve this question

edited Apr 1 at 1:20

hopefully

asked Mar 31 at 22:48

hopefullyhopefully

190215

$endgroup$

add a comment | 

0

$begingroup$

Those are the two examples:

But it is not clear for me how example 1 shows if $f$ is injective then $Df$ is injective, could anyone explain for me how this is shown please?

Also in example 2:

1-It is not clear for me why he wrote in DF(a) an $x$, what is this $x$ multiplied by the given matrix, from where it comes?

2-Also I do not understand how he proved that the injectivity of the derivatives implies the injectivity of $**f**$? more precisely I do not understand why $DF(a)(x_1) = DF(a)(x_2)$ iff $x_1 = x_2$ and also what is the importance of saying $**f**(t_1) = **f**(t_2)$ implies $t_1 = t_2$, is there a relation between $t_1$ and $x_1$, and $t_2$ and $x_2$ ?

3-It is not clear for me also why the inverse is incorrect i.e. injectivity of $f$ does not imply injectivity of the derivative, could anyone clarify this for me please?

Could anyone clarify this points for me please?

real-analysis calculus analysis multivariable-calculus implicit-function-theorem

share|cite|improve this question

edited Apr 1 at 1:20

hopefully

asked Mar 31 at 22:48

hopefullyhopefully

190215

$endgroup$

add a comment | 

$begingroup$

Those are the two examples:

But it is not clear for me how example 1 shows if $f$ is injective then $Df$ is injective, could anyone explain for me how this is shown please?

Also in example 2:

1-It is not clear for me why he wrote in DF(a) an $x$, what is this $x$ multiplied by the given matrix, from where it comes?

2-Also I do not understand how he proved that the injectivity of the derivatives implies the injectivity of $**f**$? more precisely I do not understand why $DF(a)(x_1) = DF(a)(x_2)$ iff $x_1 = x_2$ and also what is the importance of saying $**f**(t_1) = **f**(t_2)$ implies $t_1 = t_2$, is there a relation between $t_1$ and $x_1$, and $t_2$ and $x_2$ ?

3-It is not clear for me also why the inverse is incorrect i.e. injectivity of $f$ does not imply injectivity of the derivative, could anyone clarify this for me please?

Could anyone clarify this points for me please?

real-analysis calculus analysis multivariable-calculus implicit-function-theorem

share|cite|improve this question

edited Apr 1 at 1:20

hopefully

asked Mar 31 at 22:48

hopefullyhopefully

190215

$endgroup$

share|cite|improve this question

edited Apr 1 at 1:20

hopefully

asked Mar 31 at 22:48

hopefullyhopefully

190215

share|cite|improve this question

edited Apr 1 at 1:20

hopefully

asked Mar 31 at 22:48

hopefullyhopefully

190215

asked Mar 31 at 22:48

hopefullyhopefully

190215

asked Mar 31 at 22:48

hopefullyhopefully

190215