# A question about covariant representation of a vector

• guv
In summary: Thanks again.In summary, the conversation discusses the representation of vectors using covariant and contravariant bases, as well as the use of different notations for components of vectors. The author's conventions may be unconventional and it is recommended to refer to other resources for a better understanding of tensors.
guv

## Homework Statement

Hi I am reviewing the following document on tensor:
https://www.grc.nasa.gov/www/k-12/Numbers/Math/documents/Tensors_TM2002211716.pdf

## Homework Equations

In the middle of page 27, the author says:
Now, using the covariant representation, the expression $$\vec V=\vec V^*$$
then becomes

$$\vec V = V_j \vec e^{(j)}= V_j^* \vec e^{(j)*} = \vec V^*$$

## The Attempt at a Solution

How does this work? Earlier in the document, the Vector is always represented as
$$\vec V = V_j \vec e_{(j)} = V^j \vec e^{(j)}$$

I don't see how suddenly the coordinates or components of covariant basis combing with the contracovariant basis can represent the same vector? It's been made clear earlier,

$$V_j = V^i g_{ij}$$ and $$g_{ij}$$ is not 1 in general.

Thank you for clarification.

guv said:

## Homework Statement

Hi I am reviewing the following document on tensor:
https://www.grc.nasa.gov/www/k-12/Numbers/Math/documents/Tensors_TM2002211716.pdf

## Homework Equations

In the middle of page 27, the author says:
Now, using the covariant representation, the expression $$\vec V=\vec V^*$$
then becomes

$$\vec V = V_j \vec e^{(j)}= V_j^* \vec e^{(j)*} = \vec V^*$$
First of all, I should mention that this author's conventions are pretty unusual. You won't find them in math books.

K and K* denotes two arbitrary coordinate systems. There are two bases for ##\mathbb R^3## associated with each coordinate system. So there's a total of four bases. Any element of ##\mathbb R^3## can be expressed as a linear combination of the three elements of any of these four bases.

##\{\vec e^{(i)}\}## is the basis such that each ##\vec e^{(i)}## is in the direction of increasing ##i## coordinate. He calls this the contravariant basis associated with K. ##\{\vec e_{(i)}\}## is a basis such that each ##\vec e_{(i)}## is in a direction perpendicular to the plane in which the other two coordinates are constant. He calls this the covariant basis associated with K. The other two bases, ##\{\vec e^{(i)}{}^*\}## and ##\{\vec e_{(i)}{}^*\}## are defined similarly with respect to K*.

The reason why ##\vec V=V_j e^{(j)}=V_j{}^* e^{(j)}{}^*## is just that ##\{\vec e^{(i)}\}## and ##\{\vec e^{(i)}{}^*\}## are bases, and the ##V_j## and the ##V_j{}^*## are defined as the numbers such that these equalities hold. The reason why these things are equal to ##\vec V^*## is just that this symbol is a (pointless) notation for the linear combination ##V_j{}^* e^{(j)}{}^*##.

guv said:

## The Attempt at a Solution

How does this work? Earlier in the document, the Vector is always represented as
$$\vec V = V_j \vec e_{(j)} = V^j \vec e^{(j)}$$
It looks like he was just using a different notation for the components earlier. What he wrote as ##V^j## earlier is written as ##V_j## now.

I wouldn't recommend that you use this document to learn about tensors unless you have to. I like chapter 3 in Schutz's GR book. And the chapter on tensors in Sergei Treil's linear algebra book looks good too. Their conventions are more standard: They start with a vector space V and define V* as the vector space of linear functions from V into ##\mathbb R##. V* is called the dual space of V. Given an ordered basis ##(e_1,\dots,e_n)## for V, one can define an ordered basis ##(e^1,\dots,e^n)## for V* by ##e^i(e_j)=\delta^i_j## for all i,j. The latter ordered basis is said to be the dual of the former. Note that these two bases are bases for two different vector spaces.

Thanks for the clarification. I suspected that the author was a bit careless with that part of the discussion. The notations are actually okay for me coming from a physics background. I'll take a look at Schutz and Sergei Treil's books.

## 1. What is a covariant representation of a vector?

A covariant representation of a vector is a way of describing a vector in terms of its components in a specific coordinate system. In this representation, the vector's components are measured parallel to the coordinate axis, rather than perpendicular to it.

## 2. How is a covariant representation different from a contravariant representation?

In a contravariant representation, the vector's components are measured perpendicular to the coordinate axis. This means that the components in a contravariant representation will change when the coordinate system is rotated, while the components in a covariant representation will remain the same.

## 3. Why is covariant representation important in physics?

In physics, many equations and laws are written in terms of covariant representations, as they are independent of the choice of coordinate system. This makes it easier to compare and combine equations from different physical systems.

## 4. Can a vector have both a covariant and contravariant representation?

Yes, a vector can have both a covariant and contravariant representation. This is known as a mixed representation and is useful in certain calculations and transformations.

## 5. How is a covariant representation of a vector related to tensors?

Covariant representations are used to describe the components of tensors, which are mathematical objects that can represent quantities that have both magnitude and direction. Tensors have both covariant and contravariant components, which allows them to be transformed between different coordinate systems.

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