# What is the magnitude of a tensor?

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?

• 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. – user856 Jun 1 '18 at 13:45
• @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. – GRANZER Jun 2 '18 at 7:28