Studying a vector calculus course in a different way

[mr.tea]

Hi,

I am currently in the first year of my undergraduate mathematics degree and I am taking a course in vector calculus.
The course content is: line integrals, conservative field, divergence, gradient, curl, the divergence theorem, green’s formula, Stokes' them., field theory.

I have seen that a lot of books are treating those subjects using differential forms. The course I am taking doesn't use differential form. When asked the lecturer about it, he also said that differential forms are the right way to deal with those subjects.

My question is, should I study this course using differential forms? Is the transition from one to another approach is that hard?

The course is using the book Adams and Essex(calculus, a complete course), but we can see that even the lecturer doesn't like the course book.

What should I do?

Thank you.

 

[micromass]

You're right, differential forms are the right way to study it. I don't think that learning differential forms is all that difficult to learn on the side while seeing the usual vector calculus in a course. The problem with differential forms is not that it is difficult. Differential forms are in fact quite easy to learn. It's that it's not obvious at all what they mean or how they should be interpreted.

I highly recommend Hubbard & Hubbard "Vector calculus, linear algebra and differential froms: a unified approach". This book has the best explanation and intuition I've seen in any book. Try to get one of the later editions which are better.
A quite different approach is the one of geometric algebra. This approach is even more superior to differential forms (it contains differential forms as a special case actually), but is little known. I suggest definitely learning this if you have the time. I suggest the two books of MacDonald. https://www.amazon.com/dp/1453854932/?tag=pfamazon01-20 (only II is relevant if you already know vector spaces) and https://www.amazon.com/dp/1480132454/?tag=pfamazon01-20 Only with geometric algebra/calculus do you get to know the real story!

 

[mr.tea]

You're right, differential forms are the right way to study it. I don't think that learning differential forms is all that difficult to learn on the side while seeing the usual vector calculus in a course. The problem with differential forms is not that it is difficult. Differential forms are in fact quite easy to learn. It's that it's not obvious at all what they mean or how they should be interpreted.

I highly recommend Hubbard & Hubbard "Vector calculus, linear algebra and differential froms: a unified approach". This book has the best explanation and intuition I've seen in any book. Try to get one of the later editions which are better.
A quite different approach is the one of geometric algebra. This approach is even more superior to differential forms (it contains differential forms as a special case actually), but is little known. I suggest definitely learning this if you have the time. I suggest the two books of MacDonald. https://www.amazon.com/dp/1453854932/?tag=pfamazon01-20 (only II is relevant if you already know vector spaces) and https://www.amazon.com/dp/1480132454/?tag=pfamazon01-20 Only with geometric algebra/calculus do you get to know the real story!

Thank you for your informative comment.
I am definitely going to get the Hubbard & Hubbard book. (by the way, I have already taken two courses in linear algebra).
Are there any good resources for studying the usual way? I really don't think the course book is good(even the lecturer seems like he doesn't like the book).

If differential forms are the right way, why do we learn it in the regular way?

I am definitely going to learn all this stuff in the summer, thoroughly and better then we are doing now, and probably I'll do it using your advice.

Thank you again!

 

[micromass]

Hubbard and Hubbard really do contain everything about the old-fashioned vector calculus too. But I would also recommend the following resource: http://mathinsight.org/thread/multivar Very good insights

If you already know multivariable calculus, you can probably start reading the chapter on differential forms in Hubbard right away. I don't think you need any other resource, but do ask if you feel like you miss something.

Why don't they teach differential forms right away? Good question. A lot of it is because it's not the usual way to do things. There are a lot of things in math which are taught wrong, and are only taught because it is standard. This is very sad, but that's the way it is.
Also, the very concept of differential forms is very hard to grasp. It's very easy to see what a vector field is. But what exactly is a differential form? That is hard. Hubbard has the best explanation of forms out of any books, but it's still not easy to grasp. It is probably thought that most non-math majors can't handle forms so soon in their education since they are not intuitive.

 

[mr.tea]

Hubbard and Hubbard really do contain everything about the old-fashioned vector calculus too. But I would also recommend the following resource: http://mathinsight.org/thread/multivar Very good insights

If you already know multivariable calculus, you can probably start reading the chapter on differential forms in Hubbard right away. I don't think you need any other resource, but do ask if you feel like you miss something.

Why don't they teach differential forms right away? Good question. A lot of it is because it's not the usual way to do things. There are a lot of things in math which are taught wrong, and are only taught because it is standard. This is very sad, but that's the way it is.
Also, the very concept of differential forms is very hard to grasp. It's very easy to see what a vector field is. But what exactly is a differential form? That is hard. Hubbard has the best explanation of forms out of any books, but it's still not easy to grasp. It is probably thought that most non-math majors can't handle forms so soon in their education since they are not intuitive.

So with Hubbard I can learn both the old-fashioned and differential forms way? cool! I really wanted something that could help me compare both ways.

Well, I am not going to be part of the herd...

What editions of the book do you recommend on? according to Amazon UK, I am in trouble if I need to buy the latest edition: http://www.amazon.com/dp/B008VRPQV2/?tag=pfamazon01-20
(I think I lost a few years in my life just seeing the price)

Are there any book stores in the US that ships to Europe?

Thank you!

 

[micromass]

So with Hubbard I can learn both the old-fashioned and differential forms way? cool! I really wanted something that could help me compare both ways.

Well, the differential forms approach contains the old-fashioned approach as a special case. It's very easy to grasp the old-fashioned approach when knowing differential forms.

What editions of the book do you recommend on? according to Amazon UK, I am in trouble if I need to buy the latest edition: http://www.amazon.com/dp/B008VRPQV2/?tag=pfamazon01-20

You can get the 5th edition from the publisher http://matrixeditions.com/5thUnifiedApproach.html

 

[PeroK]

So with Hubbard I can learn both the old-fashioned and differential forms way? cool! I really wanted something that could help me compare both ways.

Well, I am not going to be part of the herd...

What editions of the book do you recommend on? according to Amazon UK, I am in trouble if I need to buy the latest edition: http://www.amazon.com/dp/B008VRPQV2/?tag=pfamazon01-20
(I think I lost a few years in my life just seeing the price)

Are there any book stores in the US that ships to Europe?

Thank you!

Just to give you something to think about, I'd put another point of view:

Why do you learn matrices at school rather than starting with full-blown linear algebra?

Why do you learn a shaky version of calculus rather than the rigorous "epsilon-delta" version?

Why do you learn classical gravitation rather than General Relativity?

Once you become expert in something, you have a very different perspective of that topic than a beginner. And, your perspective may be difficult to grasp immediately by someone. You may only become expert step-by-step.

I wouldn't discourage you from looking beyond your course-work, but you need to be careful not to take too long to master the material because you are appproaching it from a more advanced, more general and potentially more time-consuming perspective.

 

[jtbell]

according to Amazon UK, I am in trouble if I need to buy the latest edition: http://www.amazon.com/John-H-Hubb...bra+and+differential+forms+a+unified+approach

The 4th edition is from 1899?

Amazon US sells the 5th edition (2015) from third-party vendors for US$120 or a bit less.

https://www.amazon.com/dp/0971576688/?tag=pfamazon01-20