# Variation of a tensor expression with indices

• spaghetti3451
In summary, the conversation discusses finding the expression ##\delta \bigg( \sqrt{- \eta_{\mu \nu} \frac{dx^{\mu}}{d \tau} \frac{dx^{\nu}}{d \tau}} \bigg)## and whether or not the proposed expression ##\delta \bigg( \sqrt{- \eta_{\mu \nu}} \bigg( \frac{dx^{\mu}}{d \tau} \bigg)^{-1/2} \bigg( \frac{dx^{\nu}}{d \tau} \bigg)^{1/2} \bigg)## is correct. The suggestion to use the chain
spaghetti3451
Say I want to find ##\delta \bigg( \sqrt{- \eta_{\mu \nu} \frac{dx^{\mu}}{d \tau} \frac{dx^{\nu}}{d \tau}} \bigg)##.

Is the following alright: ##\delta \bigg( \sqrt{- \eta_{\mu \nu}} \bigg( \frac{dx^{\mu}}{d \tau} \bigg)^{-1/2} \bigg( \frac{dx^{\nu}}{d \tau} \bigg)^{1/2} \bigg)##?

failexam said:
Say I want to find ##\delta \bigg( \sqrt{- \eta_{\mu \nu} \frac{dx^{\mu}}{d \tau} \frac{dx^{\nu}}{d \tau}} \bigg)##.

Is the following alright: ##\delta \bigg( \sqrt{- \eta_{\mu \nu}} \bigg( \frac{dx^{\mu}}{d \tau} \bigg)^{-1/2} \bigg( \frac{dx^{\nu}}{d \tau} \bigg)^{1/2} \bigg)##?

No. You have broken out terms of a sum and it is no longer clear what is being summed with what. I suggest you use the chain rule ##\delta f(x) = f'(x) \delta x##.

Thanks! I've got it.

## 1. What is a tensor expression with indices?

A tensor expression with indices is a mathematical expression that represents a tensor, which is a geometric object that describes the relationship between different coordinate systems.

## 2. What is the purpose of varying a tensor expression with indices?

The purpose of varying a tensor expression with indices is to calculate how the expression changes with respect to each of its indices, which can provide valuable information about the underlying geometric structure.

## 3. How is the variation of a tensor expression with indices calculated?

The variation of a tensor expression with indices is typically calculated using the Einstein summation convention, which involves summing over repeated indices and taking partial derivatives with respect to each index.

## 4. What are some applications of variation of a tensor expression with indices?

Variation of a tensor expression with indices has numerous applications in fields such as physics, engineering, and mathematics. It is commonly used in the study of general relativity, fluid mechanics, and elasticity, among others.

## 5. Are there any rules or properties that govern the variation of a tensor expression with indices?

Yes, there are several rules and properties that govern the variation of a tensor expression with indices, such as the product rule, chain rule, and symmetry properties. These rules can be used to simplify calculations and manipulate tensor expressions into different forms.

Replies
3
Views
1K
Replies
2
Views
988
Replies
12
Views
1K
Replies
5
Views
3K
Replies
2
Views
1K
Replies
12
Views
2K
Replies
22
Views
3K
Replies
12
Views
2K
Replies
7
Views
1K
Replies
28
Views
3K