Fleisch D.A. A Student's Guide to Vectors and Tensors

Cambridge University Press, 2012. — 175 p. — ISBN: 978-0-521-19369-6. OCRThis book has one purpose: to help you understand vectors and tensors so that you can use them to solve problems. If you're like most students, you first encountered vectors when you took a course dealing with mechanics in high school or college. At that level, you almost certainly learned that vectors are mathematical representations of quantities that have both magnitude and direction, such as velocity and force. You may also have learned how to add vectors graphically and by using their components in the x-, y- and z-directions.
That's a fine place to start, but it turns out that such treatments only scratch the surface of the power of vectors. You can harness that power and make it work for you if you're willing to delve a bit deeper - to see vectors not just as objects with magnitude and direction, but rather as objects that behave in very predictable ways when viewed from different reference frames. That's because vectors are a subset of a larger class of objects called "tensors," which most students encounter much later in their academic careers, and which have been called "the facts of the Universe." It is no exaggeration to say that our understanding of the fundamental structure of the universe was changed forever when Albert Einstein succeeded in expressing his theory of gravity in terms of tensors.Vectors.
Vector operations.
Vector applications.
Covariant and contravariant vector components.
Higher-rank tensors.
Tensor applications.
Further reading.