Are 3D Matrixes of the form $\mathbb{R}^{u\times v\times w}$ defined and if so how?

Question

Are 3-dimensional matrices of the form $\mathbb{R}^{u\times v\times w}$ (where $u, v, w \epsilon \mathbb{N}$) defined?

If so:

• What do they represent?
• Would they be simply more powerful (e.g. represent all non-linear transformations)?
• Could they encode something outside of our current definition of spatial dimensions?
• What could be potential uses?

Answer 1

Those guys are called "tensors," and yes, they play an important role in physics. You can't represent anything nonlinear with them, but they do represent multi-linear maps, taking a number of vectors and returning a scalar.