# Tensors with both covariant and contravariant components

• noahcharris
In summary, the conversation discusses the concept of tensors in the context of general relativity (GR). The idea of fully co/contravariant tensors is mentioned, as well as the confusion about how a single tensor can have both covariant and contravariant indices. The conversation goes on to explain that a type (1,1) tensor is a linear map from the tangent vector space to itself and must have the correct transformation properties. Tensors are then defined as multilinear maps on a manifold, and a tensor field is a map that assigns a tensor to each point on the manifold. The possibility of a tensor having a mixture of contravariant and covariant indices is also discussed.
noahcharris
Hey all, I'm just starting into GR and learning about tensors. The idea of fully co/contravariant tensors makes sense to me, but I don't understand how a single tensor could have both covariant AND contravariant indices/components, since each component is represented by a number in each index. And yet there must be a mixture of sorts because of all the upper/lowercase indices in GR. Any illumination would be much appreciated, thanks!

Are you familiar with and understand how a type (1,1) tensor is a linear map from the tangent vector space to itself?

Orodruin said:
Are you familiar with and understand how a type (1,1) tensor is a linear map from the tangent vector space to itself?

I was not aware of that. Could you explain further or perhaps point me towards the right resources? I don't understand what it means for a tensor to have a tangent vector space. -_-

The tensor does not have a tangent vector space, the underlying manifold does. Tangent vectors are what you would also call vectors or contravariant vectors. A linear map from the tangent vector space at a point of the manifold to itself is a type (1,1) tensor - it must be for the vectors to have the correct transformation properties.

Orodruin said:
The tensor does not have a tangent vector space, the underlying manifold does. Tangent vectors are what you would also call vectors or contravariant vectors. A linear map from the tangent vector space at a point of the manifold to itself is a type (1,1) tensor - it must be for the vectors to have the correct transformation properties.

What does it mean for a tensor to have an underlying manifold? And I would think that a linear mapping from a point to a point in vector space would simply be the kronecker delta, but maybe that is a type (1,1) among others?

If you are studying GR you should be familiar with the concept of a manifold and vectors on the manifold, that is what we are dealing with is it not?

Yes, the Kronecker delta is a prime example of a type (1,1) tensor - it has one covariant and one contravariant index, but far from the only one.

Tensors are multilinear maps ##T:V^*\times\cdots\times V^*\times V\times\cdots\times V\to\mathbb R##, where V is a vector space and V* its dual space. In GR, and in differential geometry in general, V is the tangent space ##T_pM## of a smooth manifold M, at a point p in M. A tensor field is a map that takes each point p (in some subset of M) to a tensor at p.

I think that if you had a tensor where some indices were for distances or volumes (dx) and other indices were for rates or densities ( ∂y/∂x ), then the tensor would be a mixture of contravariant (dx) and covariant ( ∂y/∂x ). I am not well versed on this subject, so I don't have any real-world examples.

## 1. What are tensors with both covariant and contravariant components?

Tensors with both covariant and contravariant components are mathematical objects used in physics and engineering to represent quantities that have both direction and magnitude. They are represented as multi-dimensional arrays of numbers and are used to describe the relationships between different physical quantities in a coordinate-independent manner.

## 2. What is the difference between covariant and contravariant components in tensors?

Covariant components of a tensor are defined as the components that transform in the same way as the coordinate system changes, while contravariant components transform in the opposite way. This means that covariant components change when the coordinate system changes, while contravariant components remain the same.

## 3. How are tensors with both covariant and contravariant components used in physics?

Tensors with both covariant and contravariant components are used to describe the physical properties of objects and phenomena in a coordinate-independent way. They are used in various fields of physics, such as relativity, electromagnetism, and fluid mechanics, to model and solve complex problems.

## 4. Can tensors with both covariant and contravariant components be visualized?

Yes, tensors with both covariant and contravariant components can be visualized in different ways, depending on their dimensionality. For example, a 2D tensor can be represented as a matrix, while a 3D tensor can be represented as a cube. However, visualizing higher-dimensional tensors can be challenging and requires advanced mathematical techniques.

## 5. What are some real-life applications of tensors with both covariant and contravariant components?

Tensors with both covariant and contravariant components have a wide range of applications in engineering and physics. They are used in computer graphics to model 3D objects, in machine learning algorithms for image and speech recognition, and in fluid dynamics to model the flow of fluids. They are also used in physics to describe the curvature of space-time in general relativity and in electromagnetism to describe the electric and magnetic fields.

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