In high school physics, vectors are introduced as quantities that have magnitude and direction. Watch this Vi Hart video Green Bean Matherole to find out more. We established that this means vectors are coordinate invariant; no matter what coordinate system we use to describe the arrow pointing in space, the magnitude and direction of the arrow remain the same. 

This may help us better appreciate the nature of vectors, but this more abstract view of vectors also allows us to look at a more general class of coordinate invariant quantities. Sound fun? Visit the Physics Classroom to learn more! Where do we find vectors in real life? How about football? Watch and learn at the Science of NFL Vectors.

To build up to this, we can step back and notice that scalars are also coordinate invariant. Scalars are quantities with magnitude, which remains the same no matter what coordinate system you use. Vectors have magnitude as well, and also another piece of information, direction. Note that vectors communicating more information does not make them “better” than scalars. It wouldn’t make sense to use a vector to describe mass, mass has no direction; we need a scalar. Likewise, a scalar is insufficient to communicate velocity because we need to know how fast and in what direction; we need a vector. Much like in the Goldilocks story, we need it to be just right! This NASA link provides a bit more information about Scalars and Vectors.

Perhaps, then, it’s not too hard to imagine that there may exist physical quantities for which we need something with more information than a vector. For example, let’s imagine a rubber block on a table. If I wanted you to move the block to a particular spot, I could give you a vector for its displacement and you could move it to precisely where I wanted. The vector had the right amount of information (magnitude and direction) for me to communicate displacement. Now suppose I want you to deform the cube into a particular shape. I want you to move the top face to the right while moving the bottom face to the left. In other words, shear the cube into a rhombus shape. If we used vectors and instructed you to apply forces, 10 N to the right and 10 N to the left, how would you deform the cube? Maybe you’d apply the forces to the top and bottom and shear, or maybe you’d apply the forces on the left and right and compress! To instruct how we want the cube deformed, we need to provide the magnitude and direction of the forces, and information on which faces they’re being applied, i.e. their distribution or orientation. We need one more piece of information, we need a second-order tensor (sometimes just called tensors). Just like vectors, these second-order tensors are coordinate invariant; no matter which coordinate system you use to write the second-order tensor, its magnitude, direction, and distribution remain the same. This video lesson: What's a Tensor? will give you a bit more information! How are tensors important? They are used in studying the physics of real-world fluids and structures.

With this extension to second-order tensors, we’ve arrived at the generalization that all coordinate invariant quantities are part of the tensor family. Scalars, vectors, and second-order tensors are all tensors that communicate differing amounts of information. Using the analogy from the video, they can be thought of as letters, words, and sentences, respectively. They have varying amounts of complexity, but all can be translated into different languages while retaining their meanings.

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