# Derivative of a Real-Valued Function of Several Variables....

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In summary: Longrightarrow## ...(1)##f( x_0 + v) = f(x_0) + f'(x_0) \cdot v + \epsilon (v)## ... ... ... ... ... (1)... and (9.1a) ##\Longrightarrow## ... (1a)##\lim_{ v \rightarrow 0 } \frac{ \epsilon (v) }{ \| v \| } = 0##... ... ... ... ... (1a)... and
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I am reading Hugo D. Junghenn's book: "A Course in Real Analysis" ...

I am currently focused on Chapter 9: "Differentiation on \mathbb{R}^n"

I need some help with an aspect of Definition 9.1.3 ...

Definition 9.1.3 and the relevant accompanying text read as follows:

At the top of the above text, in Definition 9.1.3 we read the following text:

" ... ... there exists a vector ##f'(a)## in ##\mathbb{R}^n## ... ... "My question is as follows:

How (arguing from the definition of derivative) do we indicate\demonstrate\prove that ##f'(a) \in \mathbb{R}^n## ... ...?Hope someone can help ... ...

Peter

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<Error filled quote deleted>

Thanks Mark44 ...

Just now reflecting on what you have written ...

Struggling a bit ...

If ##f'(\vec a)## is a scalar ... that is ##f'(\vec a) \in \mathbb{R}## then how do we interpret the term ##f'(\vec a) \cdot \vec h## in equation (9.1) ... it looks as if to make sense of such an expression ##f'(\vec a)## has to be a vector in ##\mathbb{R}^n## ...

I am assuming that ## \vec h \in \mathbb{R}^n## and that ## \vec h = (h_1, h_2, \ ... \ ... \ , h_n) ##

Can you clarify ...Thanks again for your help ...

Peter

Last edited by a moderator:
The author uses the following notations for the derivative of the in ##\vec{x}=\vec{a}## differentiable function ##f \, : \,\mathbb{R}^n\longrightarrow \mathbb{R}\,##:
\begin{align*}
f\,'(a) &= df_a = (\nabla f)(a) = \nabla f (a) \\
&= df_{\vec{a}}= (\nabla f)(\vec{a}) = \nabla f(\vec{a}) = f\,'(\vec{a}) = \left. \dfrac{d}{d\vec{x}}\right|_{\vec{x}=\vec{a}} f(\vec{x})\\
&= (\partial_1f(a),\ldots ,\partial_n f(a)) \\
& = (\partial_1f(\vec{a}),\ldots ,\partial_n f(\vec{a})) = \left(\left. \dfrac{\partial}{\partial x_1}\right|_{\vec{x}=\vec{a}}f(\vec{x}),\ldots ,\left. \dfrac{\partial}{\partial x_n}\right|_{\vec{x}=\vec{a}}f(\vec{x})\right)
\end{align*}
which is a linear map from ##\mathbb{R}^n## to ##\mathbb{R}## which is represented by a ##(1 \times n)-##matrix, i.e. a vector, if the coordinates ##x_1,\ldots ,x_n## are given, resp. the basis vectors ##e^1,\ldots ,e^n##. The mapping instruction is thus:
$$f\,'(\vec{a})(\vec{h}) = f'(\vec{a})\cdot \vec{h} = \sum_{i=1}^n \partial_i f(\vec{a}) \cdot h_i =\sum_{i=1}^n \left. \dfrac{\partial}{\partial x_i}\right|_{\vec{x}=\vec{a}}f(\vec{x}) \cdot h_i \in \mathbb{R}$$

I think it could really help you to read https://www.physicsforums.com/insights/the-pantheon-of-derivatives-i/ (part 1), in order to get a feeling of what is going on.

If you do so, please do the exercise and find out why my equation (1) is exactly the definition in 9.1 above and which letters translate to which here.

Last edited:
Math Amateur
Math Amateur said:
<Error filled quote deleted>

Thanks Mark44 ...

Just now reflecting on what you have written ...

Struggling a bit ...

If ##f'(\vec a)## is a scalar
That was a mistake on my part. I mistakenly thought that since f was a map from ##\mathbb R^n## to ##\mathbb R##, then f' would be, as well. The differential is a scalar, but the derivative is a vector. Apologies for any misunderstanding I caused.
Math Amateur said:
... that is ##f'(\vec a) \in \mathbb{R}## then how do we interpret the term ##f'(\vec a) \cdot \vec h## in equation (9.1) ... it looks as if to make sense of such an expression ##f'(\vec a)## has to be a vector in ##\mathbb{R}^n## ...

I am assuming that ## \vec h \in \mathbb{R}^n## and that ## \vec h = (h_1, h_2, \ ... \ ... \ , h_n) ##

Can you clarify ...
See above.
Math Amateur said:
Thanks again for your help ...

Peter

Math Amateur
fresh_42 said:
The author uses the following notations for the derivative of the in ##\vec{x}=\vec{a}## differentiable function ##f \, : \,\mathbb{R}^n\longrightarrow \mathbb{R}\,##:
\begin{align*}
f\,'(a) &= df_a = (\nabla f)(a) = \nabla f (a) \\
&= df_{\vec{a}}= (\nabla f)(\vec{a}) = \nabla f(\vec{a}) = f\,'(\vec{a}) = \left. \dfrac{d}{d\vec{x}}\right|_{\vec{x}=\vec{a}} f(\vec{x})\\
&= (\partial_1f(a),\ldots ,\partial_n f(a)) \\
& = (\partial_1f(\vec{a}),\ldots ,\partial_n f(\vec{a})) = \left(\left. \dfrac{\partial}{\partial x_1}\right|_{\vec{x}=\vec{a}}f(\vec{x}),\ldots ,\left. \dfrac{\partial}{\partial x_n}\right|_{\vec{x}=\vec{a}}f(\vec{x})\right)
\end{align*}
which is a linear map from ##\mathbb{R}^n## to ##\mathbb{R}## which is represented by a ##(1 \times n)-##matrix, i.e. a vector, if the coordinates ##x_1,\ldots ,x_n## are given, resp. the basis vectors ##e^1,\ldots ,e^n##. The mapping instruction is thus:
$$f\,'(\vec{a})(\vec{h}) = f'(\vec{a})\cdot \vec{h} = \sum_{i=1}^n \partial_i f(\vec{a}) \cdot h_i =\sum_{i=1}^n \left. \dfrac{\partial}{\partial x_i}\right|_{\vec{x}=\vec{a}}f(\vec{x}) \cdot h_i \in \mathbb{R}$$

I think it could really help you to read https://www.physicsforums.com/insights/the-pantheon-of-derivatives-i/ (part 1), in order to get a feeling of what is going on.

If you do so, please do the exercise and find out why my equation (1) is exactly the definition in 9.1 above and which letters translate to which here.
Thanks fresh_42 ...

You write:

" ... ... I think it could really help you to read https://www.physicsforums.com/insights/the-pantheon-of-derivatives-i/ (part 1), in order to get a feeling of what is going on.

If you do so, please do the exercise and find out why my equation (1) is exactly the definition in 9.1 above and which letters translate to which here. ... ... "Taking up your exercise ... ... we wish to show that equation (9.1) in Junghenn is equivalent to equation 1 in "Differentiation in a Nutshell" ... ..

Now ... Definition 9.1.3 in Junghenn (where we find (9.1) ) defines differentiation of a real valued function of several variables ...

... while equation 1 in "Differentiation in a Nutshell" defines the differentiation of a real-valued function of a single real variable ...

... but ... amending equation (1) for a real valued function of several variables ... we have

##f( x_0 + v) = f(x_0) + J(v) + r(v)## ... ... ... ... ... (1)

and

##\lim_{ v \rightarrow 0 } \frac{ r(v) }{ \| v \| } = 0## ... ... ... ... ... (1a)

where ##x_0, v## are vectors in ##\mathbb{R}^n## ... ...Equation (9.1) (with variables amended appropriately to match equation (1) ) effectively defines the derivative ##f'( x_0 )## as follows:

##\lim_{ v \rightarrow 0 } \frac{ f( x_0 + v ) - f ( x_0 ) - f' ( x_0 ) \cdot v }{ \| v \| } = 0## ... ... ... ... ... (9.1)Now ... (9.1) ##\Longrightarrow##

##\frac{ f( x_0 + v ) - f ( x_0 ) - f' ( x_0 ) \cdot v }{ \| v \| } = \frac{ \epsilon (v) }{ \| v \| }## ... ... ... ... ... (2)

where ##\epsilon (v)## is the "error" in using ##f' ( x_0 ) \cdot v## to estimate ##f( x_0 + v ) - f ( x_0 )## ... ...(2) ##\Longrightarrow##

##f( x_0 + v ) - f ( x_0 ) - f' ( x_0 ) \cdot v = \epsilon (v)##

... and ... we require ##\epsilon (v)## to converge faster to zero than linear which means:

##\lim_{ v \rightarrow 0 } \frac{ \epsilon (v) }{ \| v \| } = 0 ## ...... then ... rename ##\epsilon (v)## as ##r(v)## ... and we get (1) ... assuming, of course, that ##f' ( x_0 ) \cdot v = J(v)## ... ...

... so (9.1) is equivalent to (1) ...... BUT ... unsure if the above is rigorous and correct ... and, further ... VERY unsure how to justify that (9.1) implies that ##\epsilon (v)## converges faster to zero than linear ... ...Can you please confirm that the above is correct and/or point out errors and shortcomings ...

Can you please also explain how to justify that (9.1) implies that \epsilon (v) converges faster to zero than linear ... ...

Peter

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Yes, that's what I meant. More basically though, as ##x_0=a\; , \;h=v## and what you wrote ##J(v)=f\,'(a)\cdot h##. I took ##J## to stand for the Jacobian matrix evaluated at ##x_0=a## applied to direction ##v##. If we simply take ##r(v)=f(x_0+v)-f(x_0)-J(v)## form (1) and put it in (1a) then we get
$$\lim_{v\to 0} \dfrac{r(v)}{||v||} = \lim_{v \to 0} \dfrac{f(x_0+v)-f(x_0)-J(v)}{||v||} = \lim_{v \to 0} \dfrac{f(x_0+v)-f(x_0)-f\,'(a)(v)}{||v||} = \lim_{h \to 0} \dfrac{f(a+h)-f(a)-f\,'(a)(h)}{||h||} = 0$$
which is exactly (9.1). and your ##\varepsilon(v)=r(v)##. The faster than linear aspect comes from, although we divide by the norm of ##||v||=||h||## which is basically the linear component, we're still running towards zero. That's what is meant by faster than linear. This is usually written as ##r(v)=o(||v||)## or simply by ##f\,'(a)(h)=f(a+h)-f(a)+o(||h||)## instead of explicitly naming the error function ##\varepsilon\,,## resp. the remainder function ##r\,,## plus the limit, see https://en.wikipedia.org/wiki/Big_O_notation#Little-o_notation. In the Wiki article they say "##||v||## grows much faster than ##r(v)##" which is the same, just expressed the other way around: If ##||v||## overtakes ##r(v)##, then ##\dfrac{r(v)}{||v||}## still runs towards zero for ##v \to 0##.

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Math Amateur

## What is a derivative of a real-valued function of several variables?

A derivative of a real-valued function of several variables is a mathematical concept that measures the rate of change of a function in multiple dimensions. It represents the slope or tangent of a curve at a specific point.

## Why is the derivative of a real-valued function of several variables important?

The derivative of a real-valued function of several variables is important because it allows us to analyze how a function changes in multiple directions simultaneously. It is essential in fields such as physics, economics, and engineering for understanding and predicting changes in complex systems.

## How is the derivative of a real-valued function of several variables calculated?

The derivative of a real-valued function of several variables is calculated using partial derivatives. This involves taking the derivative of the function with respect to each independent variable and combining them to form a gradient vector.

## What is the relationship between partial derivatives and the derivative of a real-valued function of several variables?

Partial derivatives are the building blocks of the derivative of a real-valued function of several variables. They represent the rate of change of a function with respect to a single independent variable. The derivative of a function is the combination of all partial derivatives in multiple directions.

## What are some real-world applications of the derivative of a real-valued function of several variables?

The derivative of a real-valued function of several variables has numerous applications in fields such as physics, economics, and engineering. It is used to optimize processes, predict changes in complex systems, and solve real-world problems involving multiple variables.

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