Differentiability of Multivariable Vector-Valued Functions .... ....

In summary, Lafontaine and Shifrin have slightly different definitions of the derivative of a multivariable vector-valued function. However, they can be shown to be equivalent through formal and rigorous proof. This involves manipulating Lafontaine's use of the $o(h)$ notation and showing that it is essentially the same as Shifrin's definition using the limit notation. Ultimately, both definitions lead to the same condition for differentiability.
  • #1
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In Theodore Shifrin's book: Multivariable Mathematics, he defines the derivative of a multivariable vector-valued function as follows:View attachment 8503
Lafontaine in his book: An Introduction to Differential Manifolds, defines the derivative of a multivariable vector-valued function slightly differently as follows:View attachment 8504
Although intuitively the definitions look similar, I need help to show (formally and rigorously) that Shifrin's definition implies Lafontaine's definition and vice versa ...

Can someone please (formally and rigorously) demonstrate that the two definitions are equivalent ...
Hope that someone can help ...

Peter
 

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  • #2
Lafontaine explains that his usage of the notation $o(h)$ means that $\lim_{h\to0}\dfrac{\|o(h)\|_1}{\|h\|_2} = 0$. If you apply that to his Definition 1.1, he is saying that the condition for differentiability is \[\lim_{h\to0}\frac{\|f(a+h) - f(a) - L\cdot h\|}{\|h\|} = 0.\] But it is true in any normed space that the conditions $\lim_{x\to0}F(x) = 0$ and $\lim_{x\to0}\|F(x)\| = 0$ are equivalent. So Lafontaine's definition is equivalent to \[\lim_{h\to0}\frac{f(a+h) - f(a) - L\cdot h}{\|h\|} = 0.\] That is exactly the definition used by Shifrin, except for the difference in notation where Lafontaine calls the derivative $L$ whereas Shifrin uses the notation $Df(a)$.
 
  • #3
Opalg said:
Lafontaine explains that his usage of the notation $o(h)$ means that $\lim_{h\to0}\dfrac{\|o(h)\|_1}{\|h\|_2} = 0$. If you apply that to his Definition 1.1, he is saying that the condition for differentiability is \[\lim_{h\to0}\frac{\|f(a+h) - f(a) - L\cdot h\|}{\|h\|} = 0.\] But it is true in any normed space that the conditions $\lim_{x\to0}F(x) = 0$ and $\lim_{x\to0}\|F(x)\| = 0$ are equivalent. So Lafontaine's definition is equivalent to \[\lim_{h\to0}\frac{f(a+h) - f(a) - L\cdot h}{\|h\|} = 0.\] That is exactly the definition used by Shifrin, except for the difference in notation where Lafontaine calls the derivative $L$ whereas Shifrin uses the notation $Df(a)$.

Hi Opalg,

Thanks for the help ... but just a minor point ...

I can see that if we treat \(\displaystyle o(h)\) as an algebraic element or variable, then we can manipulate Lafontaine's equation as follows

\(\displaystyle f(a+h) = f(a) + L \cdot h+ o(h)\)\(\displaystyle \Longrightarrow o(h) = f(a+h) - f(a) - L \cdot h \)\(\displaystyle \Longrightarrow \frac{o(h) }{ \| h \| } = \frac{ f(a+h) - f(a) - L \cdot h }{ \| h \| }\)Therefore ... \(\displaystyle \lim_{ h \to 0 } \frac{ \| o(h) \| }{ \| h \| } = 0 \) ... ...... implies that ... \(\displaystyle \lim_{ h \to 0 } \frac{ \| f(a+h) - f(a) - L \cdot h \| }{ \| h \| } = 0\)
... and ... as we can see ... \(\displaystyle \lim_{ h \to 0 } \frac{ \| f(a+h) - f(a) - L \cdot h \| }{ \| h \| } = 0\) ...

... is essentially Shifrin's definition ... ... BUT ...... how do we justify treating o(h) as an algebraic variable in the above manipulations ...
Hope someone can help ...

Peter
 
  • #4
Peter said:
... how do we justify treating o(h) as an algebraic variable in the above manipulations ...
Your comment illustrates why many people dislike the use of the $o(h)$ notation. It is sometimes hard to reconcile its use with the "formal and rigorous" approach to mathematics that you favour.

In essence, $X = o(h)$ means $\lim_{h\to0}X = 0$. That usage is then broadened so that (for example) $X = Y + o(h)$ is interpreted to mean $X-Y = o(h)$.
 
  • #5
Opalg said:
Your comment illustrates why many people dislike the use of the $o(h)$ notation. It is sometimes hard to reconcile its use with the "formal and rigorous" approach to mathematics that you favour.

In essence, $X = o(h)$ means $\lim_{h\to0}X = 0$. That usage is then broadened so that (for example) $X = Y + o(h)$ is interpreted to mean $X-Y = o(h)$.
Thanks Opalg ...

Appreciate your help...

Peter
 

FAQ: Differentiability of Multivariable Vector-Valued Functions .... ....

1. What does it mean for a multivariable vector-valued function to be differentiable?

Differentiability of a multivariable vector-valued function means that the function has a defined derivative at each point in its domain. This means that as the input values change, the output values also change in a predictable and smooth manner.

2. How is differentiability of multivariable vector-valued functions different from single-variable functions?

In single-variable functions, the derivative represents the slope of a tangent line at a specific point on the function's graph. In multivariable vector-valued functions, the derivative represents the slope of a tangent plane at a specific point in space. This is because the function takes multiple inputs and produces multiple outputs.

3. What is the relationship between continuity and differentiability of multivariable vector-valued functions?

A function must be continuous in order to be differentiable. This means that the function must be defined and have a limit at each point in its domain. However, a function can be continuous without being differentiable, as there may be sharp turns or discontinuities in the function that prevent the existence of a derivative.

4. How is the differentiability of multivariable vector-valued functions determined?

The differentiability of a multivariable vector-valued function is determined by calculating the partial derivatives with respect to each input variable and checking for their existence and continuity. If all partial derivatives exist and are continuous, then the function is differentiable.

5. What are the practical applications of differentiability of multivariable vector-valued functions?

The concept of differentiability is crucial in many fields such as physics, engineering, and economics. It allows us to make predictions and analyze the behavior of complex systems by understanding how small changes in inputs affect the outputs. Some examples of its applications include optimization, curve fitting, and the study of motion and fluid dynamics.

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