- #1

Math Amateur

Gold Member

MHB

- 3,998

- 48

I am reading Andrew McInerney's book: First Steps in Diofferential Geometry: Riemannian, Contact, Symplectic ... and I am focused on Chapter 3: Advanced Calculus ... and in particular on Section 3.1: The Derivative and Linear Approximation ...

I am trying to fully understand Definition 3.1.1 and need help with an example based on the definition ...

Definition 3.1.1 reads as follows:

View attachment 8913I constructed the following example ...

Let \(\displaystyle f: \mathbb{R} \to \mathbb{R}^2 \)

such that \(\displaystyle f = ( f^1, f^2 )\)

where \(\displaystyle f^1(x) = 2x\) and \(\displaystyle f^2(x) = 3x + 1\)

We wish to determine \(\displaystyle T_a(h)\) ... We have \(\displaystyle f(a + h) = ( f^1(a + h), f^2(a + h) )= (2a + 2h, 3a + 3h +1 )\)

and

\(\displaystyle f(a ) = ( f^1(a ), f^2(a ) ) = (2a , 3a +1 )\)

Now ... consider ... $\displaystyle \lim_{ \mid \mid h \mid \mid \to 0} \frac{ \mid \mid f(a + h) - f(a) - T_a(h) \mid \mid }{ \mid \mid h \mid \mid } $\(\displaystyle \Longrightarrow \displaystyle \lim_{ \mid \mid h \mid \mid \to 0} \frac{ \mid \mid (2a + 2h, 3a + 3h +1) - (2a, 3a + 1) - T_a(h) \mid \mid }{ \mid \mid h \mid \mid }\)\(\displaystyle \Longrightarrow \displaystyle \lim_{ \mid \mid h \mid \mid \to 0} \frac{ \mid \mid ( 2h, 3h ) - T_a(h) \mid \mid }{ \mid \mid h \mid \mid }\)... ... but how do I proceed from here ... ?

... can I take \(\displaystyle T_a (h) = T_a.h\) ... but how do I justify this?Hope someone can help ...

Peter

I am trying to fully understand Definition 3.1.1 and need help with an example based on the definition ...

Definition 3.1.1 reads as follows:

View attachment 8913I constructed the following example ...

Let \(\displaystyle f: \mathbb{R} \to \mathbb{R}^2 \)

such that \(\displaystyle f = ( f^1, f^2 )\)

where \(\displaystyle f^1(x) = 2x\) and \(\displaystyle f^2(x) = 3x + 1\)

We wish to determine \(\displaystyle T_a(h)\) ... We have \(\displaystyle f(a + h) = ( f^1(a + h), f^2(a + h) )= (2a + 2h, 3a + 3h +1 )\)

and

\(\displaystyle f(a ) = ( f^1(a ), f^2(a ) ) = (2a , 3a +1 )\)

Now ... consider ... $\displaystyle \lim_{ \mid \mid h \mid \mid \to 0} \frac{ \mid \mid f(a + h) - f(a) - T_a(h) \mid \mid }{ \mid \mid h \mid \mid } $\(\displaystyle \Longrightarrow \displaystyle \lim_{ \mid \mid h \mid \mid \to 0} \frac{ \mid \mid (2a + 2h, 3a + 3h +1) - (2a, 3a + 1) - T_a(h) \mid \mid }{ \mid \mid h \mid \mid }\)\(\displaystyle \Longrightarrow \displaystyle \lim_{ \mid \mid h \mid \mid \to 0} \frac{ \mid \mid ( 2h, 3h ) - T_a(h) \mid \mid }{ \mid \mid h \mid \mid }\)... ... but how do I proceed from here ... ?

... can I take \(\displaystyle T_a (h) = T_a.h\) ... but how do I justify this?Hope someone can help ...

Peter

#### Attachments

Last edited: