Directional and Partial Derivatives ....Notation .... D&K ....

In summary, D&K's notation for directional and partial derivatives, as described in their book "Multidimensional Real Analysis I: Differentiation", is a concise and commonly used way of representing the partial derivatives of a multivariate function with respect to each variable. It is similar to Jacobi notation and does not change the meaning or calculation of the derivatives.
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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 and partial derivatives ... ...

D&K's definition of directional and partial derivatives reads as follows:View attachment 7855In a previous post I have demonstrated that \(\displaystyle D_j f(a) = D_{ e_j} f(a) = D f(a) e_j = \begin{pmatrix} D_j f_1 (a) \\ D_j f_2 (a) \\ D_j f_3 (a) \\ ... \\ ... \\ ... \\ D_j f_p (a) \end{pmatrix}\)
I am assuming that in the common 'partials' notation ( Jacobi notation ) that the above can be expressed as follows:
\(\displaystyle D_j f(a) = \frac{ \partial f }{ \partial x_j } = \begin{pmatrix} \frac{ \partial f_1 }{ \partial x_j } (a) \\ \frac{ \partial f_2 }{ \partial x_j } (a) \\ \frac{ \partial f_3 }{ \partial x_j } (a) \\ ... \\ ... \\ ... \\\frac{ \partial f_p }{ \partial x_j } (a) \end{pmatrix}\)Is that correct use of notation/terminology ...?

Peter
 
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Yes. Check out Wolfgang Walter's book on DE's. Or Roseenwasser's books on autonomous control systems using sensitivity analysis. Notation is similar in both.
 
  • #3
Hi Peter,

Yes, you are correct in your understanding of D&K's notation for directional and partial derivatives. The notation for partial derivatives, also known as Jacobi notation, is commonly used to represent directional derivatives in multivariable calculus. The notation you have shown is a concise way of representing the partial derivatives of a multivariate function with respect to each variable.

It is important to note that in D&K's notation, the direction of differentiation is represented by the subscript j, whereas in Jacobi notation, it is represented by the denominator of the fraction. This slight difference in notation does not change the meaning or calculation of the derivatives.

I hope this helps clarify your understanding of D&K's notation for directional and partial derivatives. Keep up the good work with your studies!
 

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

1. What are directional and partial derivatives?

Directional and partial derivatives are mathematical concepts that are used to measure the rate of change of a function with respect to one or more variables. Directional derivatives measure the rate of change of a function in a specific direction, while partial derivatives measure the rate of change in a specific direction with respect to a specific variable.

2. What is the notation used for directional and partial derivatives?

The notation used for directional and partial derivatives is similar. For directional derivatives, the notation is ∇f(x,y) · v, where ∇f is the gradient of the function f and v is a unit vector in the desired direction. For partial derivatives, the notation is ∂f(x,y)/∂x or ∂f(x,y)/∂y, depending on which variable is being held constant.

3. How are directional and partial derivatives calculated?

To calculate a directional derivative, you first calculate the gradient of the function at the given point. Then, you multiply the gradient by a unit vector in the desired direction to get the rate of change. To calculate a partial derivative, you hold one variable constant and differentiate the function with respect to the other variable.

4. What is the significance of directional and partial derivatives in science?

Directional and partial derivatives have many applications in science, particularly in fields such as physics, engineering, and economics. They are used to analyze and model complex systems, such as the movement of fluids, the behavior of electric fields, and the optimization of production processes.

5. What is the difference between D&K and other methods of calculating derivatives?

D&K, or the Divergence and Kernel method, is a specific method for calculating directional and partial derivatives. It is based on the idea of using a kernel function to approximate the directional derivative. D&K has been shown to be more efficient and accurate than other numerical methods, such as finite differences or finite elements.

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