# Electromagnetic Tensor: Vector Space Explained

• Gavroy
The vector space of type [2,0] tensors is called the cotangent bundle over the field of reals. It is the set of all linear extensions of the vector space of type [0,2] tensors.

#### Gavroy

Hey,

does anoyone of you know to which vector space the electromagnetic tensor belongs to?

Assuming the vacuum case, then the vector space of 2-forms on flat 4D space-time ?

okay, i don't think that it is the minkowski space.

but the vector space of 2-forms sounds good.

i saw somewhere the notation: [TEX] TM* \otimes TM* [/TEX]

sorry, this does not work: TM* (tensorproduct) TM*

what does this T stand for, does anyone know? this space, could have something to do with the 2-forms, but i am not really sure.

$$T_x M^*$$ is called the cotangent space of $$M$$ at the point $$x$$. It is the vector space of 1-forms. Higher forms are in the vector space made from tensor products of the cotangent space, so 2-forms are in $$T_x M^*\otimes T_x M^*$$.

When we write $$T M^*$$ we mean something different, but related. This is the cotangent bundle, which is the total space of the manifold $$M$$ together with the cotangent space at every point.

ah okay...thank you all

An electromagnetic field tensor, or Faraday tensor, F = Fuvdxvdxv is an element of the space of two forms over the field of reals, or a type [0,2] antisymmetric tensors. This is a subspace of all type [0,2] tensors, so any Faraday tensor with lower indeces is also a member of the space of type [0,2] tensors.

Sometimes the Faraday tensor is given with upper indeces. It is still antisymmetric but a member of the antisymmetric tensors over the field of reals, but with upper indeces, so is called a type [2,0] tensor.

Or it could be presented in mixed form, type [1,1]. A vector space doesn't need or involve a manifold in it's set of axioms but can, however, be identified with the tangent space of a point on a manifold, which fzero has discussed.

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## 1. What is an electromagnetic tensor?

The electromagnetic tensor is a mathematical object used to describe the electromagnetic field in a space. It contains information about the strength and direction of the electric and magnetic fields at each point in space.

## 2. What is a vector space?

A vector space is a mathematical concept used to describe a set of objects, called vectors, that can be added together and multiplied by numbers. These vectors have both magnitude and direction, and can be represented geometrically as arrows in space.

## 3. How is the electromagnetic tensor related to the electromagnetic field?

The electromagnetic tensor is a mathematical representation of the electromagnetic field. It contains information about the strength and direction of the electric and magnetic fields at each point in space, and can be used to calculate the behavior of the electromagnetic field.

## 4. What are the components of the electromagnetic tensor?

The electromagnetic tensor has 16 components, which can be organized into a 4x4 matrix. These components represent the electric and magnetic fields in different directions and at different points in space.

## 5. How is the electromagnetic tensor used in physics?

The electromagnetic tensor is an important tool in theoretical physics, particularly in the study of electromagnetism, special relativity, and quantum field theory. It is used to describe the behavior of electromagnetic fields, and is essential in understanding many phenomena such as light, electricity, and magnetism.