Why do books on diff geometry suggest tensor calculus and differential forms are coord free, while others say tensors are coord dependent?

Question

There are many contradictions in literature on tensors and differential forms. Authors use the words coordinate-free and geometric. For example, the book Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers say differential forms are coordinate free while tensors are dependendent on coordinate. But when you look at the wikipedia article on tensor calculus it says that tensors are coordinate free representation. Another, mention would be Kip Thornes Modern Classical Physics where he explains that he develops physics in a coordinate free way using tensors. Other authors say, we develop differential geometry in a geometric way. Or we develop physics in a geometric way. Is geometry synonymous with coordinate free? This is all very confusing. There are many more examples in the literature but I dont see a definitive answer. The further I look the contradictions between authors. I am looking for an authoritative textbook that I can learn from. What do you think about Chris Isham's Modern Differential Geometry for Physicists? Also, is it better to use tensors vs differential forms in theoretical physics?

Answer 1

Yes, "in a geometric way" means coordinate-independent. Here's the story I tell my students: Although units (meters, grams, seconds, etc.) are used all the time in physics, we do not believe that the laws of physics change if the units are changed. Coordinates on 2-dimensional or 3-dimsnsional space are just higher dimensional versions of units. So, even though we do most of our calculations using coordinates, the initial assumptions and the final conclusions should not depend on the coordinates we used to show that the assumptions imply the conclusions.

The same is true with geometry. Geometric facts should not depend on coordinates. Recall that until Descartes came along, geometry was always done without the use of coordinates. Coordinates are extraordinarily useful, because the purely geometric arguments, such as what you learn in high school geometry, are painful to do and find. Descartes showed us how to do it all using algebra and, later, calculus far more easily.

By now I can identify at least 3 different ways to do calculations. One is to use local coordinates and write tensors as higher dimensional generalizations of vectors in $\mathbb{R}^n$. Another is to use differential forms and the method of moving frames as developed by Elie Cartan. The third is to avoid using coordinates completely and use abstract definitions for everything. I find that I use all three. Which one is easier to use depends on the specific calculation or proof.

But in the end you usually (but not 100% of the time) want to show that what you've derived is independent of coordinates.

Answer 2

I cannot suggest any particular textbook because mathematical physics is quite far from my area of expertise, although I will confess a fondness for Misner, Thorne and Wheeler.

But I will say that there does not exist any "authoritative textbook" in which all ambiguities of terminology and notation are erased and all terms and notations are used in manner that all physicists and mathematicians throughout the world will agree upon.

If you study a good book on mathematical physics then you'll learn that author's point of view on terminology and notation. But more importantly, you'll learn some math and physics. Then you'll have a solid foundation for further studies, and you'll be in a good position to navigate around the inevitable variations of terminology and notation that you will encounter in your further readings.

Answer 3

I have a book on tensors that teach both coordinate free (geometric) tensors and coordinate dependent. It is published by Springer and is calledIntroduction to Tensor Analysis and the Calculus of Moving Surfaces by Pavel Grinfeld. This could be why there is an argument because people think a mathematical tool is mutually exclusive to both systems.