Taylor expansion metric tensor

Then ##g_{ij}## is the value of the metric at point ##p##.Now to your formula, which is just coordinate dependent, in general. In this chart, by the chain rule[tex]\partial_i g_{jk}=\frac{\partial g_{jk}}{\partial x^i}-\frac{\partial x^l}{\partial x^i}\frac{\partial g_{jk}}{\partial x^l}[/tex]But ##x^l=p^l=0##, so ##\partial_i g_{jk}=\frac{\partial g_{jk}}{\partial x^i}##.Now you can apply the formula from Wikipedia, with the multiindices being denoted by
  • #1
mertcan
344
6
hi, when I dug up something about metric tensors, I found a equation in my attached file. Could you provide me with how the derivation of this ensured? What is the logic of that expansion in terms of metric tensor? I really need your valuable responses. I really wonder it. Thanks in advance...
 

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  • #2
Actually I want to ask why do we have $$x^k x^l$$ instead of $$\partial x^k$$ $$\partial x^k$$ ? In taylor series, I know we always write the infinite smalls..

for instance in this link (http://mathworld.wolfram.com/TaylorSeries.html), they have (x-a) and x approaches to a. In short (x-a) becomes $$\partial x$$...
 
  • #3
I hope my question is clear and I really would like to ask Is there anyone who is capable of responding to my question?
 
  • #4
Guys, I do not know why you do not give answer. İs there a situation that bothers you in my question ? Uncertainity really makes me bad...
 
  • #5
mertcan said:
Guys, I do not know why you do not give answer. İs there a situation that bothers you in my question ? Uncertainity really makes me bad...
You are being too impatient. Sometimes it takes a while for someone to answer such a basic question.

Have a look at this Wikipedia page about Taylor series for several variables. The metric taylor series in your image just shows expansion around a point "p". I.e., x is like a displacement from the point p.
 
  • #6
strangerep said:
You are being too impatient. Sometimes it takes a while for someone to answer such a basic question.

Have a look at this Wikipedia page about Taylor series for several variables. The metric taylor series in your image just shows expansion around a point "p". I.e., x is like a displacement from the point p.
thank you strangerep, I consider that your answer is close to my first thought. You mean If we look at or make taylor expansion around so close points, "x" in the image becomes infinite small distance as I thought before.
 
  • #7
Given you accept the result in calculus of several variables (as given in the wikipedia page linked by strangerep), here is the explanation, for a smooth manifold of dimension ##d## with metric ##g##. Take a point ##p## in your manifold and take a chart ##(\mathcal{U},x)##, such that ##x(p)=0##, i.e. we set the coordinates of point ##p## in this chart, to be the zero vector in ##\mathbb{R}^d##.

Then the components of the metric in that chart, denoted by ##g_{ij}## are just mappings from ##x(\mathcal{U})## to ##\mathbb{R}##, i.e. they are functions of several variables. Note that ##x(\mathcal{U})## is a subset of ##\mathbb{R}^d##, which contains zero, so you can apply the formula from wikipedia, by setting ##a=0##. Just use contravariant indices for the coordinates, Einstein summation convention and note that to evaluate at ##p## means essentially to evaluate at ##0## in this chart.
 

1. What is a Taylor expansion of a metric tensor?

A Taylor expansion of a metric tensor is a mathematical approximation of the metric tensor at a given point in space. It involves expressing the metric tensor as a sum of terms, each of which is a product of the derivatives of the metric tensor with respect to the coordinates at that point.

2. Why is a Taylor expansion of a metric tensor useful?

A Taylor expansion of a metric tensor is useful because it allows us to approximate the metric tensor at a point without having to calculate the full tensor. This can be helpful in situations where the full tensor is difficult or impossible to calculate, or when we only need a rough estimate of the tensor at a specific point.

3. How is a Taylor expansion of a metric tensor related to the metric tensor's derivatives?

The Taylor expansion of a metric tensor involves taking the derivatives of the tensor at a given point. The coefficients in the expansion are related to these derivatives, allowing us to approximate the tensor at that point using only the derivatives.

4. Can a Taylor expansion of a metric tensor be used in higher dimensions?

Yes, a Taylor expansion of a metric tensor can be used in any number of dimensions. The expansion simply becomes more complex as the number of dimensions increases.

5. What are the limitations of using a Taylor expansion of a metric tensor?

One limitation of using a Taylor expansion of a metric tensor is that it is only a local approximation and may not accurately represent the full tensor over a large region. Additionally, the accuracy of the approximation decreases as the distance from the point of expansion increases. Another limitation is that the expansion may become increasingly complex in higher dimensions, making it difficult to use or interpret.

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