TITLE: Why do we need tensors in modern physics?
QUESTION [24 upvotes]: I am wondering why do we really need the concept of tensor. I think it is like vectors, just as a notation of a set of related parameters. I could write the Navier–Stokes equations with scalars, or vectors, or tensors. If this is the case, we struggle to learn the algebra and calculus rules of tensors only to simplify the notation of a complex equation.
My question is:
Are there some examples to show the power of tensors instead of just simplifying the notation?
I know that when I use tensors of rank 3 or higher, it is often hard to use a matrix (equivalent to a rank-2 tensor). Tensors do make life easier in this kind of situation.
Here is a quote from the preface to Wilhelm Flügge's Tensor Analysis and Continuum Mechanics:

Many of the recent books on continuum mechanics are only "tensorized" to the extent that they use cartesian tensor notation as a convenient shorthand for writing equations. This is a rather harmless use of tensors. The general, noncartesian tensor is a much sharper thinking tool and, like other sharp tools, can be very beneficial and very dangerous, depending on how it is used. Much nonsense can be hidden behind a cloud of tensor symbols and much light can be shed upon a difficult subject. The more thoroughly the new generation of engineers learns to understand and to use tensors, the more useful they will be.

REPLY [1 votes]: For many applications, the use of tensor notation or non-tensorial notation is just a matter of convenience or personal preference. For instance, if you're an engineer calculating something about the motion of a car going fast around a curve, it may be helpful to work with a moment of inertia tensor in Cartesian 3-space, and write it in expressions using notation such as the Einstein summation convention. But this is all optional.
The area where it gets to be non-optional is general relativity. GR tells us that time and space simply don't work the way we think they do, and in particular it tells us that there is not even any such thing as a global frame of reference or a global set of space-time coordinates that is uniquely well suited to a particular observer in a particular state of motion.
Furthermore, GR tells us that parallel transport of vectors is not well defined unless we specify the path along which the vector is transported. As a concrete example, the Gravity Probe B satellite transported a gyroscope along a certain path through spacetime, and it measured how much that gyroscope's axis had wobbled at the end of that process, relative to a gyroscope axis such as the one defined by the earth's rotational axis (which amounts to the same thing as a wobble relative to the "fixed stars").
So in the context where GR effects are significant, you could tell me about some quantity you've defined that is notated like a "vector," but if it isn't  really a vector in the sense of being a rank-1 tensor, then you haven't really told me how you're defining this thing. Specifically, suppose we point this "vector" toward Arcturus, seal it in a box, and transport it along some path through spacetime, then open the box back up at the end. We want to know whether it will still be pointing toward Arcturus or not, and if not, then in what direction. If you can't guarantee me that it's an actual rank-1 tensor, then we don't know the answer to this question, and I would submit that you haven't really defined what this thing is at all.
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