Taylor expansion and parallel transport

In summary, The image shows a parallel transported vector with a covariant derivative of zero along the "pr" path. The green box demonstrates a Taylor expansion of the contravariant component of the Z vector at points "p" and "r", while the red box shows a Taylor expansion at points "u" and "r". The confusion arises from the use of "u" as a point, which stands for ##u = h_{ds}(q)##, the result of transporting from ##p## to ##q## by ##dt## and then from ##q## to ##u## by ##ds##. This is similar to the construction shown in section 6.3 of Differential Geometry Fionn Fitzmaurice page
  • #1
mertcan
340
6
hi, first of all in this image there is a fact that we have parallel transported vector, and covariant derivative is zero along the "pr"path as you can see at the top of the image. I consider that p, and r is a point and in the GREEN box we try to make a taylor expansion of the contravariant component of Z vector at the "p" and "r" points. But in the RED box we try to make a taylor expansion of the contravariant component of Z vector at the "u" and "r" points. Here is my confusion: Although there is not a "u" point, instead is just "p" and "r" points, How can we use such a point? What does "u"point stand for?

I quoted from Differential Geometry Fionn Fitzmaurice page 36
 

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  • #2
I have seen so many views, but I have not been replied for a long time... I am waiting your valuable return
 
  • #3
Hi.

The point ##u = h_{ds}(q)##, i.e. transporting from ##p## to ##q## by ##dt## and then from ##q## to ##u## by ##ds##. It's the same construction as section 6.3.
 

1. What is a Taylor expansion?

A Taylor expansion, also known as a Taylor series, is a mathematical representation of a function as an infinite sum of terms. It is used to approximate a function at a specific point by using the function's derivatives at that point.

2. How is a Taylor expansion useful in science?

A Taylor expansion is useful in science because it allows us to approximate complicated functions and make predictions about their behavior. It is also used in various fields of science, such as physics, engineering, and chemistry, to model and understand natural phenomena.

3. What is the relationship between Taylor expansion and parallel transport?

Taylor expansion and parallel transport are both mathematical concepts used in the study of differential geometry. Taylor expansion is used to approximate a function at a specific point, while parallel transport is used to move a vector along a curve while keeping it parallel to its original direction. In some cases, Taylor expansion can be used to calculate parallel transport along a curve.

4. What is the difference between Taylor expansion and Maclaurin expansion?

The main difference between Taylor expansion and Maclaurin expansion is the point at which the function is approximated. Taylor expansion is used to approximate a function at any given point, while Maclaurin expansion is used to approximate a function at the point x=0. Additionally, Maclaurin expansion is a special case of Taylor expansion, where all the derivatives of the function are evaluated at x=0.

5. Can Taylor expansion be used to approximate any function?

No, Taylor expansion can only be used to approximate functions that are infinitely differentiable at the point of approximation. If a function has a discontinuity or is not differentiable at the point, the Taylor series will not converge and cannot be used to approximate the function.

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