Directional Derivatives .... Notation .... D&K ....

In summary, the conversation is about the book "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk, specifically focusing on Chapter 2. The topic of discussion is D&K's notation for directional and partial derivatives, and the correct use of notation according to their schema. The conversation also mentions Proposition 2.3.2 and different possible ways of writing component functions.
  • #1
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I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ...

I am focused on Chapter 2: Differentiation ... ...

I need help with an aspect of D&K's notation for directional derivatives ... ...

D&K's definition of directional and partial derivatives reads as follows:
D&K - Start of Section 2.3 on Directional and Partial Derivatives ... .png

I am assuming that under D&K's definitions and notation one can write:##D_v f(a) = \begin{pmatrix} D_v f_1 (a) \\ D_v f_2 (a) \\ D_v f_3 (a) \\ ... \\ ... \\ ... \\ D_v f_p (a) \end{pmatrix}####= Df(a)v#### = \begin{pmatrix} D_1 f_1(a) & D_2 f_1(a) & ... & ... & D_n f_1(a) \\ D_1 f_2(a) & D_2 f_2(a) & ... & ... & D_n f_2(a) \\ ... & ... & ... & ... &... \\ ... & ... & ... & ... &... \\ ... & ... & ... & ... &... \\ D_1 f_p(a) & D_2 f_p(a) & ... & ... & D_n f_p(a) \end{pmatrix} \begin{pmatrix} v_1 \\ v_2 \\ ... \\ ... \\ ... \\ v_n \end{pmatrix}##

## = \begin{pmatrix} D_1 f_1(a) v_1 + D_2 f_1(a) v_2 + \ ... \ ... \ + D_n f_1(a) v_n \\ D_1 f_2(a) v_1 + D_2 f_2(a) v_2 + \ ... \ ... \ + D_n f_2(a) v_n \\ ... \ ... \ ... \ ... \ ... \\ ... \ ... \ ... \ ... \ ... \\ ... \ ... \ ... \ ... \ ... \\ D_1 f_p(a) v_1 + D_2 f_p(a) v_2 + \ ... \ ... \ + D_n f_p(a) v_n \end{pmatrix} ##Is the above a correct use of notation according to D&K's schema of notation ...

Peter
=========================================================================================

Proposition 2.3.2 may well be relevant to the above post ... so I am providing the same ... as follows:
D&K - 1 - Proposition 2.3.2 ... .... PART 1 ... .png

D&K - 2 - Proposition 2.3.2 ... .... PART 2 ... .png
 

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  • D&K - Start of Section 2.3 on Directional and Partial Derivatives ... .png
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  • D&K - 1 - Proposition 2.3.2 ... .... PART 1 ... .png
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  • D&K - 2 - Proposition 2.3.2 ... .... PART 2 ... .png
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  • #2
What you have written makes sense. You have broken things up more than the text did, into component functions ##D_jf_i(a)## for ##1\leq i\leq p##. The text has not stated a choice of notion for those component functions. Other possibilities for writing them would be ##D_jf(a)_i## and ##D_jf(a)^i##. The latter uses a superscript rather than subscript to align with how these things tend to be written in tensor calculus.
 
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  • #3
andrewkirk said:
What you have written makes sense. You have broken things up more than the text did, into component functions ##D_jf_i(a)## for ##1\leq i\leq p##. The text has not stated a choice of notion for those component functions. Other possibilities for writing them would be ##D_jf(a)_i## and ##D_jf(a)^i##. The latter uses a superscript rather than subscript to align with how these things tend to be written in tensor calculus.

Thanks Andrew ...

Appreciate your help ...

Peter
 

Related to Directional Derivatives .... Notation .... D&K ....

1. What are directional derivatives?

Directional derivatives are a type of derivative that measures the instantaneous rate of change of a function in a specific direction. It is used to determine the slope of a function along a particular direction, rather than just the traditional x and y axes.

2. What is the notation used for directional derivatives?

The notation used for directional derivatives is ∇f, where ∇ represents the gradient operator and f represents the function. It is often read as "the gradient of f".

3. What is the significance of the D&K in directional derivatives?

The D&K in directional derivatives refers to the mathematicians Augustin-Louis Cauchy and Johann Peter Gustav Lejeune Dirichlet, who first developed the concept of directional derivatives. Their contributions have greatly influenced the study of multivariable calculus and its applications.

4. How do directional derivatives relate to tangent vectors?

Directional derivatives are closely related to tangent vectors, as they both involve the concept of direction. While directional derivatives measure the rate of change of a function in a specific direction, tangent vectors represent the direction of a curve at a particular point on a surface.

5. In which fields of science are directional derivatives commonly used?

Directional derivatives are commonly used in fields such as physics, engineering, and economics. In physics, they are used to calculate the direction of maximum force or acceleration. In engineering, they are used in the design of machines and structures. In economics, they are used to determine the direction of maximum profit or loss.

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