# Covariant and contravariant tensors

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• dsaun777
In summary: J/m}, \text{rad/s}).##In summary, covariant and contravariant tensors are used in physics for different reasons. Mixed components are used to simplify calculations.
dsaun777
Is there a purpose of using covariant or contravariant tensors other than convenience or ease in a particular coordinate system? Is it possible to just use one and stick to one? Also what is the meaning of mixed components used in physics , is there a physical significance in choosing one over the other? For instance when breaking down a vector into its component form you can use covariant basis and contravariant components and vice versa but why? Sorry if this question seems redundant but I'm just puzzled.

[Moderator's note: Moved from a mathematical forum. This is about physics. A mathematical answer would probably be a different one and might increase confusion rather than clarify it.]

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Here is a mathematical answer: https://www.physicsforums.com/insights/what-is-a-tensor/
but as mentioned, the physical answer is a different one, because in physics tensors are tools, whereas in mathematics they are just certain objects. As the covariant parts are isomorphic to a contravariant version of it, there is little difference mathematically, whereas it is important to distinguish them in physics.

fresh_42 said:
Here is a mathematical answer: https://www.physicsforums.com/insights/what-is-a-tensor/
but as mentioned, the physical answer is a different one, because in physics tensors are tools, whereas in mathematics they are just certain objects. As the covariant parts are isomorphic to a contravariant version of it, there is little difference mathematically, whereas it is important to distinguish them in physics.
That is a nice brief generalization of what tensor can represent but it gives definitions based on undefined terms. Why even use contravariant and covariant tensors why not just stick to one, why are they mixed? There are no definitions of covariant and contravariant in your 'mathematical answer". I see that they are tools for physics but why have both. It seems like you show up to a job with two tape measures, one measuring meters and the other measuring feet. Both do the same operation. In terms of strictly theoretical physics can not we just disregard one?/

dsaun777 said:
There are no definitions of covariant and contravariant in your 'mathematical answer".
This is because it makes no mathematical sense. One part are ordinary vector spaces (contravariant), the other dual vector spaces (covariant). Physicists use it this way, the terms contravariant and covariant in mathematics are defined for functors and not for tensors. Furthermore a contravariant functor refers to the dual category, which is exactly the opposite of how physicists use it. So I avoided the terms as they belong mathematically in a different context, and physicists use them conversely than mathematicians do. This is a mess, and the reason why I said "mathematical answer". IIRC then large parts of the discussion beyond the article is about these contradictions.

Technically it makes a difference whether we consider a vector space ##V## or its dual space ##V^*##. They might be isomorphic, but one consists of directional arrows, the other one of functions which eat directional arrows. Hence physicists have to distinguish them carefully. E.g. a derivative can be both, the vector "slope at point" or the linear function "multiply by the slope factor", i.e.
$$f(x)=x^2 \Longrightarrow f'(x) = \{\,(x,x^2);(1,2x)\,\}$$
where we have a pair of point and attached direction, or
$$f(x)=x^2 \Longrightarrow f' \, : \,x \longmapsto 2x$$
where we have the function multiply by two.

It is still the same thing, the derivative of ##x^2##, but one is a vector at a point, the other the instruction how to apply the derivative. That's why co- and contravariant are distinguished in physics: do we mean the slope or how to apply the slope? Mathematically there is not much difference between ##2## and ##\text{ multiply by }2##.

dsaun777
dsaun777 said:
It seems like you show up to a job with two tape measures, one measuring meters and the other measuring feet.
One practical difference is that in applications contravariant components are typically measured in metres per something (e.g. m/s) but covariant components are typically measured in somethings per metre (e.g. J/m).

dsaun777
DrGreg said:
One practical difference is that in applications contravariant components are typically measured in metres per something (e.g. m/s) but covariant components are typically measured in somethings per metre (e.g. J/m).
Careful though, this depends on the dimensions of your coordinates. If coordinates have different dinensions (such as for polar coordinates) so will the components.

DrGreg
Orodruin said:
Careful though, this depends on the dimensions of your coordinates. If coordinates have different dinensions (such as for polar coordinates) so will the components.
That's a good point. The example I gave in post #5 was not a good one.

If you are using Cartesian coordinates with a Euclidean metric, it's difficult to notice the difference between contravariant and covariant (mathematically, there is a difference but the components look the same). But, for example, in 2D polar coordinates ##(r, \theta)## measured in ##(\text{metres}, \text{ radians})##, a typical contravariant vector's components might have units ##(\text{m/s}, \text{rad/s})##, and a typical covariant vector's components might have units ##(\text{J/m}, \text{J/rad})##.

DrGreg said:
One practical difference is that in applications contravariant components are typically measured in metres per something (e.g. m/s) but covariant components are typically measured in somethings per metre (e.g. J/m).
So the units of the covariant components usually contain mass or energy and the contravariant components contain velocities and their derivatives... Why?

dsaun777 said:
So the units of the covariant components usually contain mass or energy and the contravariant components contain velocities and their derivatives... Why?
Orodruin said:
Careful though, this depends on the dimensions of your coordinates. If coordinates have different dimensions (such as for polar coordinates) so will the components.
The covariant components are multilinear functions. What does a matrix entry for, say ##\varphi \, : \,\mathbb{R}\otimes \mathbb{R}\otimes \mathbb{R} \longrightarrow \mathbb{R}## has as units?

## 1. What is the difference between covariant and contravariant tensors?

Covariant and contravariant tensors are two types of tensors that differ in how they transform under a change of coordinates. Covariant tensors change their components in a specific way when the coordinates are changed, while contravariant tensors change their components in the opposite way. This is due to the fact that covariant tensors have indices in the lower position, while contravariant tensors have indices in the upper position.

## 2. How are covariant and contravariant tensors related?

Covariant and contravariant tensors are related through the metric tensor, which is a mathematical object that describes the relationship between the two types of tensors. The metric tensor allows us to convert a covariant tensor into a contravariant tensor and vice versa. This conversion is necessary in order to perform tensor operations and calculations.

## 3. What is the importance of covariant and contravariant tensors in physics?

Covariant and contravariant tensors play a crucial role in physics, particularly in the field of general relativity. They are used to describe the curvature of spacetime and the gravitational field. In this context, covariant and contravariant tensors are essential for formulating the equations of general relativity and making predictions about the behavior of matter and energy in the presence of gravity.

## 4. Can you give an example of a covariant and contravariant tensor?

One example of a covariant tensor is the stress-energy tensor, which is used to describe the energy and momentum distribution of matter and radiation in spacetime. An example of a contravariant tensor is the electromagnetic field tensor, which is used to describe the electric and magnetic fields in spacetime.

## 5. How are covariant and contravariant tensors used in machine learning and data analysis?

Covariant and contravariant tensors are used in machine learning and data analysis to represent and manipulate multidimensional data. They are particularly useful for analyzing data that has both spatial and temporal components, such as images or time series data. By using covariant and contravariant tensors, we can perform complex mathematical operations on this data and extract meaningful insights and patterns.

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