The Chain Rule in n Dimensions .... Browder Theorem 8.15

In summary: C \cdot \frac{|k(h)|}{|h|}Since C is a constant and \frac{|k(h)|}{|h|} remains bounded as h approaches 0, we can conclude that \frac{ r_g ( k(h) ) }{ |h| } \to 0 as h \to 0.I hope this helps to clarify the reasoning behind the statement in the proof of Theorem 8.15. Please let me know if you need any further explanation.In summary, the proof of Theorem 8.15 shows that the map g is
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I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ...

I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.2 Differentials ... ...

I need some help in order to fully understand the proof of Theorem 8.15 ...

Theorem 8.15 and its proof read as follows:

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View attachment 9413

In the above proof by Browder we read the following:" ... ... Then \(\displaystyle |k| \leq C |h|\) for \(\displaystyle |h|\) sufficiently small, if \(\displaystyle C \gt \| T \|\), by Proposition 8.13; it follows that

\(\displaystyle \frac{ r_g ( k(h) ) }{ |h| } \to 0\) as \(\displaystyle h \to 0\). ... ... "
My question is as follows:Can someone demonstrate formally and rigorously how/why \(\displaystyle \frac{ r_g ( k(h) ) }{ |h| } \to 0\) as \(\displaystyle h \to 0\). ... ...
Help will be much appreciated ...Peter==============================================================================The above post mentions Proposition 8.13 ... Proposition 8.13 reads as follows:
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View attachment 9415
Hope that helps ...

Peter
 

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Dear Peter,

Thank you for reaching out for help with the proof of Theorem 8.15 in Andrew Browder's book. I will try my best to explain the formal and rigorous reasoning behind the statement \frac{ r_g ( k(h) ) }{ |h| } \to 0 as h \to 0.

First, let's recall the definitions of the terms involved in the statement. The function k(h) represents the differential of the map g at the point a, denoted by dg_a. This means that k(h) is the linear approximation of g at the point a, which can be written as:

g(a+h) = g(a) + k(h) + r_g(h)

where r_g(h) is the remainder term, which satisfies the following property:

\lim_{h \to 0} \frac{r_g(h)}{|h|} = 0

This means that as h approaches 0, the remainder term becomes smaller and smaller compared to the size of h.

Now, let's look at the expression \frac{ r_g ( k(h) ) }{ |h| }. This can be rewritten as:

\frac{ r_g ( k(h) ) }{ |h| } = \frac{r_g(h)}{|h|} \cdot \frac{|k(h)|}{|h|}

Since we know that \lim_{h \to 0} \frac{r_g(h)}{|h|} = 0, we just need to show that \frac{|k(h)|}{|h|} remains bounded as h approaches 0. This is where Proposition 8.13 comes in. Proposition 8.13 states that if the differential of a map is bounded by some constant C, then the map itself is Lipschitz continuous with Lipschitz constant C.

In our case, we know that |k(h)| \leq C |h| for |h| sufficiently small, if C \gt \| T \|, which means that the differential k(h) is bounded by some constant C. Therefore, by Proposition 8.13, we can conclude that the map g is Lipschitz continuous with Lipschitz constant C. This means that \frac{|k(h)|}{|h|} is also bounded by C.

Putting everything together, we have:

\frac{ r_g ( k(h) ) }{ |h|
 

FAQ: The Chain Rule in n Dimensions .... Browder Theorem 8.15

1. What is the chain rule in n dimensions?

The chain rule in n dimensions is a mathematical concept that allows us to find the derivative of a multivariable function by breaking it down into simpler functions and applying the derivative of each function separately.

2. How does the chain rule work in n dimensions?

The chain rule in n dimensions works by using partial derivatives to find the rate of change of a multivariable function in each direction. These partial derivatives are then multiplied together to find the total derivative of the function.

3. Why is the chain rule important in n dimensions?

The chain rule is important in n dimensions because it allows us to find the derivative of complex multivariable functions, which are frequently encountered in real-world applications. It also helps us understand the relationship between different variables in a function.

4. Can the chain rule be applied to any multivariable function?

Yes, the chain rule can be applied to any multivariable function as long as it is differentiable. This means that it must have continuous partial derivatives in all directions.

5. How does Browder Theorem 8.15 relate to the chain rule in n dimensions?

Browder Theorem 8.15 is a generalization of the chain rule in n dimensions. It states that if a function is differentiable at a point, then it is also continuous at that point. This theorem helps us understand the conditions under which the chain rule can be applied to a multivariable function.

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