I am quite new to tensors, with my knowledge based on Daniel Fleisch’s “Student’s guide to vectors and tensors” and Neuenschwander’s “Tensor calculus for physics”. I had the following questions: 1. What are the higher rank tensors with physical meaning attached to them? I know tensors up to rank-4: Rank-0 tensors: Scalars such as temperature etc. Rank-1 tensors: Vectors Rank-2 tensors: Inertia tensor, metric tensor, electromagnetic field tensor, etc. Rank-3 tensors: Trifocal tensor Rank-4 tensors: Riemann curvature tensor 2. (This question might not be well-formed) When are tensors necessary and when can we be satisfied with matrices (3x3, 3x3x3, 3x3x3x3)? I can understand why the tensors mentioned above – the vectors, inertia tensors, EM field tensor – should be tensors, namely to keep the tensor itself invariant upon coordinate transformation, but when would a physicist be satisfied with a matrix of numbers, without bothering about how they should transform if the coordinates are changed? I am guessing, for example, solving simultaneous equations using matrix method need not be tensors because presumably the numbers involved are without units? Would that mean that if the numbers involved were, say, pressure/unit length, we would have to make sure the matrix of numbers transforms correctly in a new coordinate system? The transformation matrix itself, , being without units, need not be tensors for that reason? Thanks a lot.