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?

  • $\begingroup$ No, a tensor does not have a magnitude and directions. Instead, one way to think about a (second-order) tensor is as a function that takes in a vector and gives you another vector. For example, you give the stress tensor a vector which is the normal of the plane, and it gives you another vector which is the stress across that plane. $\endgroup$ – user856 Jun 1 '18 at 13:45
  • $\begingroup$ @Rahul Tensor as linear mapping is fine, but I was wondering how to picture a tensor, just like how a vector can be given by an arrow. $\endgroup$ – GRANZER Jun 2 '18 at 7:28

This is a fantastic question that I have wrestled with myself. A second rank tensor is a linear map, but a tensor is also an object in itself. It can be rotated and scaled like other objects (it obeys certain laws under coordinate transformation).

Imagine an ellipsoid oriented arbitrarily in space. If you were to take the eigen values and vectors of your tensor, the eigen vectors correspond to a rotation that will align the coordinate system with the major axes of the ellipsoid, while the eigenvalues represent the major radii. The visualization breaks down if you have negative eigenvalues and perhaps other conditions as well.

Ellipsoid in space

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