I What is the magnitude of a tensor?

granzer

I know that a vector is a tool to help with quantities that have both a magnitude a direction. At a given point in space, a vector has a particular magnitude and direction and if we take any other direction at the same point we can get a projection of this vector in that direction.

Tensor is a tool to help with quantities that have a magnitude and 2 directions. But what are its magnitude and directions? Like if a tensor is defined at a point does it have a magnitude and 2 directions? For example, the stress tensor gives the stress at a point but as far as I have understood, at a given point as we change the plane (ie one of the direction) we get different stress value in a different direction. So at that point, the stress has different magnitude and direction as we keep changing the direction of the plane and in a given plane it's not like the projection of some 1 stress quantity.

When we say 'this' is a vector at 'this' point, we know what its magnitude and its direction are. Similarly, if we say 'this' is the tensor at 'this' point does it mean a particular magnitude and particular 2 directions? Can stress be defined by having a magnitude and 2 directions at a point like how a force is given by having a magnitude and 1 direction at a point?

EDIT: By defining a Coordinate system to space(consider a 3D space), we can define the vector by 3 coordinates and these 3 coordinates would give both magnitude and direction of the vector. Do the 9 coordinates of the tensor give a particular magnitude and 2 particular directions as well?

Orodruin

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Tensor is a tool to help with quantities that have a magnitude and 2 directions.
This is not really a good way of putting it as it is not necessarily possible to write it on the form $\vec v \otimes \vec w$. Instead, it is generally a linear combination of such objects. A better way of viewing a (rank 2) tensor is as a linear map from a vector space to itself, i.e., it maps a vector to another vector. In the case of the stress tensor, it is a linear transformation from directed surface normals to forces, both of which are vectors.

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fresh_42

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Real or complex scalars have an absolute value.
Vectors have a norm or length.
Matrices have a matrix or operator norm.
Bilinear mappings have a length, but this isn't a norm.

One could generalize the matrix norm to higher order tensors, but I doubt they will be of much use.

granzer

This is not really a good way of putting it as it is not necessarily possible to write it on the form $\vec v \otimes \vec w$. Instead, it is generally a linear combination of such objects. A better way of viewing a (rank 2) tensor is as a linear map from a vector space to itself, i.e., it maps a vector to another vector. In the case of the stress tensor, it is a linear transformation from directed surface normals to forces, both of which are vectors.
How is stress tensor written as a tensor product of 3 vectors? Which are the 2 vectors?

scottdave

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Orodruin

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How is stress tensor written as a tensor product of 3 vectors? Which are the 2 vectors?
Where did I say that it is a tensor product of 3 vectors? It is a linear combination of objects of the form $\vec v \otimes \vec w$. In general, such linear combinations cannot be written as a single tensor product.

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@granzer,
A matrix is a second order tensor (assuming it is defined in such a way that the tensor coordinate conversion rules apply) that you are familiar with. So any intuition you form regarding tensors should take that into account. Your statement in the original post regarding 2 directions does not apply to a matrix.

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fresh_42

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• granzer

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Getting back to your original question, the stress tensor (or any other 2nd order tensor) can be expressed in terms of 3 principal stress magnitudes in 3 principal perpendicular spatial directions. If you choose a plane perpendicular to any of these three directions, the normal stress on that plane is equal to the magnitude in that direction, and the shear stress on the plane is equal to zero. In terms of the principal stresses and directions, the stress tensor can be expressed mathematically in the following form:
$$\boldsymbol{\sigma}=\sigma_{11}(\mathbf{i_1}\otimes \mathbf{i_1})+\sigma_{22}(\mathbf{i_2}\otimes \mathbf{i_2})+\sigma_{33}(\mathbf{i_3}\otimes \mathbf{i_3})$$where the i's are unit vectors in the 3 principal directions. When you dot this tensor with a unit normal to a plane of arbitrary orientation, you obtain the traction vector (aka the stress vector) on that plane. This is known as the Cauchy stress relationship.

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granzer

Thank you all!!. All the material and explanations provided have been really helpful to get the right perspective to understanding the notion of tensors as an abstract object.
What is the difference between 2 vector spaces, V and W, when a transformation(linear map) is given as

T: V->W

is it just the difference between dimensionality of the 2 spaces?

If stress tensor is viewed as a transformation(linear map), are the area vector and traction vector in different vector spaces?

"What is the magnitude of a tensor?"

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