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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:

In the above text from Tu we read the following:

" ... ... In case \(\displaystyle n = 1\) and \(\displaystyle p = 0\), this lemma says that

\(\displaystyle f(x) = f(0) + x g_1(x)\) ... ... "Now Tu seems to put \(\displaystyle n= 1\) in the equation in the lemma but does not change \(\displaystyle \mathbb{R}^n\) to \(\displaystyle \mathbb{R}^1\) and does not change \(\displaystyle x = (x^1, x^2, \ ... \ ... \ x^n)\) to \(\displaystyle x = (x^1)\) ... ... How can this be valid?

" ... ... Applying the lemma repeatedly gives \(\displaystyle g_i(x) = g_i(0) + x g_{ i + 1 } (x)\) ... ... "How exactly does Tu arrive at the above equation ... I take it he puts \(\displaystyle f = g_i\) and he pits p = 0 ... but how does he get \(\displaystyle x g_{ i + 1 } (x)\) out of the summation term .. ? ( ... note that it is the i + 1 term in g_{ i + 1 } that I find puzzling ... )

... ... but mind you he only seems to show this for \(\displaystyle p= 0\)?

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 \(\displaystyle n = 1\) and \(\displaystyle p = 0\), this lemma says that

\(\displaystyle f(x) = f(0) + x g_1(x)\) ... ... "Now Tu seems to put \(\displaystyle n= 1\) in the equation in the lemma but does not change \(\displaystyle \mathbb{R}^n\) to \(\displaystyle \mathbb{R}^1\) and does not change \(\displaystyle x = (x^1, x^2, \ ... \ ... \ x^n)\) to \(\displaystyle x = (x^1)\) ... ... How can this be valid?

__In the above text from Tu we read the following:__**Question 2**" ... ... Applying the lemma repeatedly gives \(\displaystyle g_i(x) = g_i(0) + x g_{ i + 1 } (x)\) ... ... "How exactly does Tu arrive at the above equation ... I take it he puts \(\displaystyle f = g_i\) and he pits p = 0 ... but how does he get \(\displaystyle x g_{ i + 1 } (x)\) out of the summation term .. ? ( ... note that it is the i + 1 term in g_{ i + 1 } that I find puzzling ... )

__I 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 ...?__**Question 3**... ... but mind you he only seems to show this for \(\displaystyle p= 0\)?

Hope someone can help ...?

Peter