First approach to differential forms

[Math Amateur]

What do Physics Forums members regard as the best first introduction to differential forms ...?

 

[JonnyG]

What math background do you have?

 

[Math Amateur]

What math background do you have?

I am about at senior undergraduate level in algebra and analysis ...

Peter

 

[micromass]

I offer a very nontraditional approach (as seems to be usual with me). I think the best approach is first to understand geometric algebra (or Clifford algebra as mathematicians call it). In this way, you will appreciate the intuition behind the wedge product, cross product, etc. For this, I recommend Alan MacDonald's "Linear and geometric algebra". Just read part II, the rest you will likely you know already. This might be too easy, but there are much more mathematical approaches to this if you desire.

Understand infinitesimals is also very important, but as you're a senior in mathematics, you likely already have an appreciation for this. Anyway, a differential form is now nothing else but a representation of an "infinitesimal volume measurement". I first learned this from Lee's "introduction to smooth manifolds", which I still think is a very good place to learn this, certainly if you already intuitively know what a wedge is from geometric algebra.

 

[Math Amateur]

What do Physics Forums members regard as the best first introduction to differential forms ...?

I have a copy of Weintraub's book on differential forms (Differential Forms by Steven H. Weintraub) ... and was browsing it and wondering whether to use it on my first approach to differential forms ... then I found that Weintraub made the unconventional decision not to use the wedge ( $\wedge$ ) in his notation ... I was a bit put off by this decision as I wanted a conventional notational approach ... at least for my first approach ... but I have wondering whether I need worry about his notational approach ... maybe he just drops the wedge from the notation as it is superfluous as he says in Remark 1.1.8 on pages 9-10 ... as follows:

What do you think? Is this something that matters ... even for a first approach ... or do you need conventional notation for a first approach so you can recognise the theory in other books ... ?

Peter*** NOTE *** I note that Weintraub has a large number of very explicit and helpful looking computational examples ... which is a BIG plus in my opinion ...

 

[Math Amateur]

I offer a very nontraditional approach (as seems to be usual with me). I think the best approach is first to understand geometric algebra (or Clifford algebra as mathematicians call it). In this way, you will appreciate the intuition behind the wedge product, cross product, etc. For this, I recommend Alan MacDonald's "Linear and geometric algebra". Just read part II, the rest you will likely you know already. This might be too easy, but there are much more mathematical approaches to this if you desire.

Understand infinitesimals is also very important, but as you're a senior in mathematics, you likely already have an appreciation for this. Anyway, a differential form is now nothing else but a representation of an "infinitesimal volume measurement". I first learned this from Lee's "introduction to smooth manifolds", which I still think is a very good place to learn this, certainly if you already intuitively know what a wedge is from geometric algebra.

Thanks micromass ... appreciate the thought ... I will look up the text by MacDonald ...

Peter

 

[mathwonk]

I was always puzzled by them until I read the introductory article by Harley Flanders in this book:

https://www.amazon.com/dp/0883851296/?tag=pfamazon01-20

A more thorough and very geometric and friendly introduction was written by David Bachman and gone through here on the forum a few years back:

A geometric intro to diff forms

https://www.amazon.com/dp/0817683038/?tag=pfamazon01-20

 

[mpresic]

My second semester as a graduate student required a course in thermodynamics and statistical mechanics. The professor taught the subject using differential forms. One semester later I had to complete a course in Classical Electrodynamics, and I had the same professor. Believe it or not, he taught the course using differential forms. Funny.
Lucky.;'
I had the usual EM courses (out of Jackson) after I transferred to another school many years later. My employer also encouraged me to take EM courses in 1985-6 (out of Jackson) while working full-time in order to keep current.