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I am reading Loring W.Tu's book: "An Introduction to Manifolds" (Second Edition) ...
I need help in order to fully understand the proof of Tu's Lemma 1.4: Taylor's Theorem with Remainder ...
Lemma 1.4 reads as follows:View attachment 8631
View attachment 8632
My questions are as follows:Question 1
In the above text from Tu we read the following:
" ... ... In case $$n = 1$$ and $$p = 0$$, this lemma says that
$$f(x) = f(0) + x g_1(x)$$ ... ... "Now Tu seems to put $$n= 1$$ in the equation in the lemma but does not change $$\mathbb{R}^n$$ to $$\mathbb{R}^1$$ and does not change $$x = (x^1, x^2, \ ... \ ... \ x^n)$$ to $$x = (x^1)$$ ... ... How can this be valid?Question 2In the above text from Tu we read the following:
" ... ... Applying the lemma repeatedly gives $$g_i(x) = g_i(0) + x g_{ i + 1 } (x)$$ ... ... "How exactly does Tu arrive at the above equation ... I take it he puts $$f = g_i$$ and he pits p = 0 ... but how does he get $$x g_{ i + 1 } (x)$$ out of the summation term .. ? ( ... note that it is the i + 1 term in g_{ i + 1 } that I find puzzling ... )
Question 3I must say that generally I am having trouble following the overall 'strategy' of the proof ... can it be summarised as transforming the equations of the lemma into a valid Taylor series ...?
... ... but mind you he only seems to show this for $$p= 0$$?
Hope someone can help ...?
Peter
I need help in order to fully understand the proof of Tu's Lemma 1.4: Taylor's Theorem with Remainder ...
Lemma 1.4 reads as follows:View attachment 8631
View attachment 8632
My questions are as follows:Question 1
In the above text from Tu we read the following:
" ... ... In case $$n = 1$$ and $$p = 0$$, this lemma says that
$$f(x) = f(0) + x g_1(x)$$ ... ... "Now Tu seems to put $$n= 1$$ in the equation in the lemma but does not change $$\mathbb{R}^n$$ to $$\mathbb{R}^1$$ and does not change $$x = (x^1, x^2, \ ... \ ... \ x^n)$$ to $$x = (x^1)$$ ... ... How can this be valid?Question 2In the above text from Tu we read the following:
" ... ... Applying the lemma repeatedly gives $$g_i(x) = g_i(0) + x g_{ i + 1 } (x)$$ ... ... "How exactly does Tu arrive at the above equation ... I take it he puts $$f = g_i$$ and he pits p = 0 ... but how does he get $$x g_{ i + 1 } (x)$$ out of the summation term .. ? ( ... note that it is the i + 1 term in g_{ i + 1 } that I find puzzling ... )
Question 3I must say that generally I am having trouble following the overall 'strategy' of the proof ... can it be summarised as transforming the equations of the lemma into a valid Taylor series ...?
... ... but mind you he only seems to show this for $$p= 0$$?
Hope someone can help ...?
Peter