Understanding the definition of derivative

In summary, when differentiating a vector field along the direction of another vector field, we need to define either a further structure affine connection or a Lie derivative through flow. However, this can be a confusing concept, as differentiation is always a local procedure in one direction and requires a specific direction to be chosen. The definition of differentiating a vector field along another vector field may be ill-defined due to issues with coordinate dependence and the fact that the ordinary product XY is not necessarily a left or right invariant vector field.
  • #1
lriuui0x0
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As far as I understand, when we want to differentiate a vector field along the direction of another vector field, we need to define either further structure affine connection, or Lie derivative through flow. However, I don't understand why they are needed. If we want to differentiate ##Y## in the direction of ##X##, since each ##X_p## is itself a derivation operator, we can already do differentiation?

Consider the following definition for differentiate ##Y## along ##X##. Write ##Y## in components ##Y = (y^1, \dots, y^n)##, each ##y^i## is a scalar function and differentiate it along ##X## we get ##Xy^i##. Do it for all components we get ##XY##.

I believe there's problem with this definition, which might relate to coordinate dependence. Can somebody explain to me why is this an ill definition? With many different concepts in differential geometry, it gets confusing on what is coordinate dependent and what is not.
 
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  • #2
lriuui0x0 said:
As far as I understand, when we want to differentiate a vector field along the direction of another vector field, we need to define either further structure affine connection, or Lie derivative through flow. However, I don't understand why they are needed.
Differentiation is always a local procedure in one direction. You have to choose this direction, e.g. along a flow.
If we want to differentiate ##Y## in the direction of ##X##, since each ##X_p## is itself a derivation operator, we can already do differentiation?
This is too sloppy. In direction of an entire vector field? You have to specify one specific direction, e.g. ##X_p##. The ordinary product ##XY## is not necessarily a left invariant (or right invariant, depends on your convention) vector field again. What you can do is walk along ##X## and then along ##Y## but you will get different starting points.
Consider the following definition for differentiate ##Y## along ##X##. Write ##Y## in components ##Y = (y^1, \dots, y^n)##, each ##y^i## is a scalar function ...
No, the ##y^i## are scalars, the coefficients for the basis vectors, which are one forms. What you have is ##Y=\sum_k y^k\dfrac{\partial}{\partial x_k}## if the ##\{\,x_k\,\}## are your standard variables.
... and differentiate it along ##X## we get ##Xy^i##. Do it for all components we get ##XY##.
No, you don't. It isn't even defined. You can talk about the "matrix entries" of ##XY## but that doesn't give them meaning.
I believe there's problem with this definition, which might relate to coordinate dependence. Can somebody explain to me why is this an ill definition? With many different concepts in differential geometry, it gets confusing on what is coordinate dependent and what is not.
Yes, there is a problem. See above.
Here is an example (under section "B")
https://www.physicsforums.com/insights/pantheon-derivatives-part-iv/
 

Related to Understanding the definition of derivative

1. What is the definition of derivative?

The derivative of a function is the rate of change of the function at a specific point. It represents the slope of the tangent line to the function at that point.

2. How is the derivative calculated?

The derivative is calculated using the limit of the difference quotient, which is the change in the function's output divided by the change in the function's input as the change in input approaches zero.

3. What is the geometric interpretation of the derivative?

The derivative can be interpreted geometrically as the slope of the tangent line to the function at a specific point. It represents the instantaneous rate of change of the function at that point.

4. What is the difference between average rate of change and instantaneous rate of change?

The average rate of change is the overall rate of change of a function over a specific interval, while the instantaneous rate of change is the rate of change at a specific point on the function. The derivative represents the instantaneous rate of change.

5. Why is the concept of derivative important?

The concept of derivative is important because it allows us to analyze the behavior of functions at specific points and understand their rates of change. It is also essential in many areas of mathematics and science, such as physics, economics, and engineering.

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