Orthogonality of Absolute Derivative

Thread starter rbwang1225
Start date Jun 16, 2012
Tags

Absolute Derivative Orthogonality

In summary, the concept of orthogonality in relation to absolute derivative refers to the perpendicularity of two vectors, where one vector represents the direction of change of a function and the other vector represents the direction of change of its independent variable. This means that the absolute derivative is independent of the direction in which the function is changing. The orthogonality of absolute derivative is different from regular derivative in that regular derivative only considers the change in the function in one direction, while absolute derivative takes into account the change in all directions. This makes the absolute derivative a more comprehensive measure of change. The significance of orthogonality in the study of absolute derivative lies in its ability to provide a more accurate and complete understanding of the rate of change

Jun 16, 2012

rbwang1225

Hello!
Is any covariant vector orthogonal to absolute derivative of its contravariant counterpart?
I read a GR book, and it says the tangent vector of a curve is orthogonal to its absolute derivative, that is ##D\lambda^A/dst_A=0##, where ##t^A## is the unit tangent vector of some curve ##\gamma##.

Any comments would be appreciated.

Physics news on Phys.org

Jun 16, 2012

rbwang1225

Oh! I am very sorry. It's very straight forward. Since ##t^At_A=1, Dt^A/dst_A=0##

1. What is the concept of orthogonality in relation to absolute derivative?

The concept of orthogonality in relation to absolute derivative refers to the perpendicularity of two vectors, where one vector represents the direction of change of a function and the other vector represents the direction of change of its independent variable. This means that the absolute derivative is independent of the direction in which the function is changing.

2. How is the orthogonality of absolute derivative different from regular derivative?

The orthogonality of absolute derivative is different from regular derivative in that regular derivative only considers the change in the function in one direction, while absolute derivative takes into account the change in all directions. This makes the absolute derivative a more comprehensive measure of change.

3. What is the significance of orthogonality in the study of absolute derivative?

The significance of orthogonality in the study of absolute derivative lies in its ability to provide a more accurate and complete understanding of the rate of change of a function. By considering all directions of change, the absolute derivative can provide a more precise measurement of the function's behavior.

4. How is orthogonality of absolute derivative used in real-world applications?

The orthogonality of absolute derivative has many real-world applications, particularly in fields such as physics and engineering. It is used to study the motion of objects, the flow of fluids, and the behavior of systems, among other things. In these applications, the absolute derivative allows for more accurate predictions and models.

5. Can the concept of orthogonality of absolute derivative be extended to higher dimensions?

Yes, the concept of orthogonality of absolute derivative can be extended to higher dimensions. In higher dimensions, orthogonality refers to the perpendicularity of two vectors in a multi-dimensional space. The concept of absolute derivative can also be extended to higher dimensions, allowing for a more comprehensive understanding of the rate of change in multi-dimensional functions.