Derivative of a Real-Valued Function of Several Variables: Junghenn Defn 9.1.3

In summary, the conversation discusses Chapter 9 of Hugo D. Junghenn's book "A Course in Real Analysis" and specifically focuses on Definition 9.1.3 which defines the derivative on $\mathbb{R}^n$. The question asks for clarification on how to demonstrate that the vector $f'(\mathbf{a})$ exists in $\mathbb{R}^n$, to which the response explains that the definition does not require proof of existence, but rather defines the vector as the derivative if it satisfies the given criteria.
  • #1
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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:
https://www.physicsforums.com/attachments/7865
View attachment 7866

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

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

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

Peter
 
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  • #2
You don't have to prove it. This is a definition (no proofs allowed). The definition is saying that if there's a vector $f'(\mathbf{a})$ satisfying the definition, then we call it the derivative. The definition is not actually claiming that the vector exists (it doesn't always, depending on $f$).
 
  • #3
Ackbach said:
You don't have to prove it. This is a definition (no proofs allowed). The definition is saying that if there's a vector $f'(\mathbf{a})$ satisfying the definition, then we call it the derivative. The definition is not actually claiming that the vector exists (it doesn't always, depending on $f$).
Thanks Ackbach ...

Appreciate the help...

Peter
 

Related to Derivative of a Real-Valued Function of Several Variables: Junghenn Defn 9.1.3

1. What is the definition of a derivative of a real-valued function of several variables?

The derivative of a real-valued function of several variables, as defined by Junghenn in Defn 9.1.3, is the rate of change of the function with respect to one of its independent variables. This is typically denoted by the symbol ∂f(x,y)/∂x or ∂f(x,y)/∂y.

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

The derivative of a real-valued function of several variables can be calculated using the same principles as a single-variable derivative. It involves taking the limit as the change in the independent variable approaches zero, and using the partial derivative notation to denote which variable is being held constant.

3. What is the significance of the derivative of a real-valued function of several variables?

The derivative of a real-valued function of several variables is an important concept in multivariable calculus, as it allows us to determine the slope or rate of change of a function in multiple dimensions. It is also used in optimization problems to find the maximum or minimum value of a function.

4. Can the derivative of a real-valued function of several variables be negative?

Yes, the derivative of a real-valued function of several variables can be negative. This indicates that the function is decreasing in the direction of the independent variable being considered. However, it is important to note that the derivative can also be negative if the function is decreasing in a different direction.

5. Are there any limitations on when the derivative of a real-valued function of several variables can be calculated?

There are some limitations on when the derivative of a real-valued function of several variables can be calculated. For example, if the function is not continuous or differentiable at a certain point, the derivative cannot be calculated at that point. Additionally, if the function is discontinuous or has sharp corners, the derivative may not exist at those points.

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