# Geometrical Interpretation of Tensors (intuition)

How would one describe to a non-mathematician (an undergraduate physicist actually- so please do use mathematics-i am not even sure if they can be described without mathematics!) what do tensors represent geometrically?

• Depends what you mean by tensors. Are we talking in the sense of "stress tensors" and general relativity, or in the sense of QM (i.e. the tensor product of Hilbert spaces) – Omnomnomnom Apr 22 '16 at 12:00
• Wow, i didn't think about that as i am yet to encounter them. I would say, all three kinds of tensors. – TheQuantumMan Apr 22 '16 at 12:19
• Have you taken linear algebra yet? – Omnomnomnom Apr 22 '16 at 15:47
• Yes. Also, i know a bit of multivariable calculus, differential equations and calculus of variations and complex analysis(i don't know if the last 3 are relevant). – TheQuantumMan Apr 22 '16 at 16:16
• ...@QuanticMan, linear algebra that includes vector-duality? – janmarqz Apr 22 '16 at 16:27

Here's the quick way to describe what's going on:

In linear algebra, you learned primarily about linear transformations. In particular, $T:V \to W$ is a function that takes a single vector, and produces another vector in a linear way. In particular $T$ satisfies $T(ax + by) = aT(x) + bT(y)$. It turns out that linear transformations can naturally be thought of as matrices, so that's what you end up working with most of the time.

A tensor, by contrast, is a map $T:V_1 \times V_2 \times \cdots \times V_k \to W$. That is, it takes several vectors as its input and produces an output-vector in a multi-linear way. That is, $T$ will satisfy $$T(v_1,v_2,\dots v_{k-1},ax + by,v_{k+1},\dots,v_n) =\\ aT(v_1,v_2,\dots v_{k-1}, x,v_{k+1},\dots,v_n) + bT(v_1,v_2,\dots v_{k-1}, y,v_{k+1},\dots,v_n)$$ Rather than thinking of tensors as matrices, we think of them as multidimensional arrays.

Another example of a Tensor that can be thought of as a matrix is a "bilinear form" $T:V_1 \times V_2 \to \Bbb R$. In particular, given a matrix $A$, the function $$T(v_1,v_2) = v_1^T A v_2$$ is exactly this kind of tensor. One commonly used tensor that works this way is the stress tensor. The question it answers is "what is the component of pressure on the plane perpendicular to $v_1$ in the direction of $v_2$?" Two others are the dot-product and, more generally, metric tensors.