Question
Asked Nov 16, 2019
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Please note that f denotes a vector function defined by the set of vectors V in E (one dimensional Euclidean space).

swedot
7.53 If f satisfies the conditions of the inverse function theorem on an open set V in E, show that
J(f-1)J (f) = 1 where f is 1-1.
bol
elt od
re2.
Xo
14ob y
Le. sin cale. + (o cos dea about
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swedot 7.53 If f satisfies the conditions of the inverse function theorem on an open set V in E, show that J(f-1)J (f) = 1 where f is 1-1. bol elt od re2. Xo 14ob y Le. sin cale. + (o cos dea about

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Expert Answer

Step 1

Inverse function theorem-

Let, any mapf R" -R" is continous ly differentiable on some open set
which contains a, and consider Jf(a # 0.
Then there exist some open set V containing a and an open set W
which contains f (a) such that f V->W has a continuous inverse
fWVwhich is differentiable Vy e W
help_outline

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Let, any mapf R" -R" is continous ly differentiable on some open set which contains a, and consider Jf(a # 0. Then there exist some open set V containing a and an open set W which contains f (a) such that f V->W has a continuous inverse fWVwhich is differentiable Vy e W

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Step 2

Jacobi matrix satisfies composable differentiable function

Let f:R R
then f:R> R"
m
fof=I (Identity function)
J)J(x)), (x)
fof
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Let f:R R then f:R> R" m fof=I (Identity function) J)J(x)), (x) fof

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Step 3

Put the values in the r...

J()J((x)), (x)
J(x)=J((x)(x)
1=J, ((x)(x)
fof
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J()J((x)), (x) J(x)=J((x)(x) 1=J, ((x)(x) fof

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