# Why the concept of tensor was invented

#### chandran

why the concept of tensor was invented. I always see that tensors are provided in matrix format. example inertia tensor is there in a 3x3 matrix.
why?

#### pmb_phy

chandran said:
why the concept of tensor was invented. I always see that tensors are provided in matrix format. example inertia tensor is there in a 3x3 matrix.
why?
There are quantities used in physics for which the concept of a number or a vector is insufficient. A more general notion of a geometrical object was required. The number (scalar) and vector were defined as tensors of a lower "rank" and then tensors of higher rank were defined. Two such tensos come to mind. The tidal force tenso which are found here

http://www.geocities.com/physics_world/mech/inertia_tensor.htm
http://www.geocities.com/physics_world/mech/tidal_force_tensor.htm

To completely defined the stress on and inside a body a tensor is needed.

Pete

#### pervect

Staff Emeritus
Tensor notation is more general than matrix notation.

If you can scrape by with tensors of rank <=2, you can probably use matrix notation. However, there are important tensors with higher ranks (such as the rank 4 Riemann tensor in General Relativity). At this point matrix notation is not sufficient, and one needs the full power of tensor notation.

There really isn't that much additional difficulty in learning tensor notation as opposed to matrix notation, either - it seems to be standard to teach engineers matrix notation, and scientists tensor notation, however.

#### HallsofIvy

The nice thing about tensors (of which vectors and scalars are special cases) is that they change "homogeneously" when you change coordinate systems. Exactly what that means is a bit complicated. If you make a "linear" change- just rotate a coordinate system- you can think of the change in any tensor expressed in that coordinate system as just "multiply by the rotation matrix".

The crucial part is that if a tensor is represented by "all 0's" in one coordinate system then it is represented by "all 0's" in any coordinates system- even strange ones with curved axes.

That has a very nice property: if we have an equation that says A= B, where A and B are tensors, in some coordinate system, then A- B= 0 in that coordianate system and so A- B= 0 or, again A= B in every coordinate system- as long as we express everything in terms of tensors, the equations are true or false independent of the coordinate system.

That's especially important in physics where "coordinate systems" are things we impose on reality! A "law of physics" has to be true regardless of whatever coordinate system we choose. We can be sure of that if we write everything in terms of tensors.

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