# Multivariable Differentiation .... McInerney Definition 3.1.1

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In summary: That is, T_a(h)= (2, 3)h.In summary, the conversation is about understanding Definition 3.1.1 and using an example to help with this understanding. The example involves a linear transformation T_a from R^n to R^n, which can be written as T_a(h) = (2, 3)h. The conversation also discusses how to justify this notation and how to solve the problem at hand.
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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 ...

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

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• McInerney - Definition 3.1.1.png
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f(x)=(2x,3x+1) so f(a+h)=(2a+2h,3a+1+3h)=(2a,3a+1)+(2h,3h)=f(a)+Ta(h). Where Ta is a linear transformation which satisfies the limit zero.
If you consider the example f(x)=(x2,3x+1) then f(a+h)=(a2+2ah+h2,3a+1+3h)
so f(a+h)=(a2,3a+1)+(2ah,3h)+(h2,0)
f(a+h)=f(a)+Ta(h)+S(h). Where Ta(h)=(2ah,3h) is a linear transformation and
$\lim_{h \to 0} \frac{\parallel f(a+h)-f(a)-T_a(h) \parallel}{\parallel h \parallel}=\lim_{h \to 0} \frac{\parallel S(h) \parallel}{\parallel h \parallel}=0$

I am confused by the notation $$\displaystyle T_a(h)$$. From what was said before, we have a "linear transformation $T_a$ from $$R^n\to R^n$$". There is no dependence of "h" Was that just $$T_a$$ times h?

. If "$$T_a(h)$$" is simply $$T_a$$ times h, we can factor out ||h||: $$\frac{||(2h, 3h)- T_ah||}{||h||}= \frac{||h||||(2, 3)- T_a||}{||h||}= (2, 3)- T_a= 0$$ so $$T_a= (2, 3)$$.

## What is multivariable differentiation?

Multivariable differentiation is a mathematical concept that involves finding the rate of change of a function with respect to multiple variables. It is an extension of single variable differentiation, where the rate of change is only calculated with respect to one variable.

## What is the definition of multivariable differentiation according to McInerney?

According to McInerney, multivariable differentiation is defined as the process of finding the partial derivatives of a multivariable function with respect to each of its variables.

## What is the purpose of multivariable differentiation?

The purpose of multivariable differentiation is to understand how a function changes when multiple variables are changed simultaneously. This is useful in many fields such as physics, economics, and engineering.

## What are the different types of multivariable differentiation?

There are two types of multivariable differentiation: partial differentiation and total differentiation. Partial differentiation involves finding the rate of change of a function with respect to one variable while holding all other variables constant. Total differentiation involves finding the rate of change of a function with respect to all of its variables simultaneously.

## What are some applications of multivariable differentiation?

Multivariable differentiation has many applications in real-world problems, such as optimization, finding maximum and minimum values, and determining the direction of steepest ascent or descent. It is also used in vector calculus to calculate line and surface integrals.

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