Insights Blog
-- Browse All Articles --
Physics Articles
Physics Tutorials
Physics Guides
Physics FAQ
Math Articles
Math Tutorials
Math Guides
Math FAQ
Education Articles
Education Guides
Bio/Chem Articles
Technology Guides
Computer Science Tutorials
Forums
Science and Math Textbooks
STEM Educators and Teaching
STEM Academic Advising
STEM Career Guidance
Trending
Featured Threads
Log in
Register
What's new
Search
Search
Search titles only
By:
Science and Math Textbooks
STEM Educators and Teaching
STEM Academic Advising
STEM Career Guidance
Menu
Log in
Register
Navigation
More options
Contact us
Close Menu
JavaScript is disabled. For a better experience, please enable JavaScript in your browser before proceeding.
You are using an out of date browser. It may not display this or other websites correctly.
You should upgrade or use an
alternative browser
.
Forums
Science Education and Careers
Science and Math Textbooks
Motivating definitions from differential geometry
Reply to thread
Message
[QUOTE="Adam Marsh, post: 6041419, member: 639749"] I'm not sure if this will be what you are looking for, but it certainly does the opposite of what most textbooks do: [URL]https://mathphysicsbook.com[/URL] The idea behind the book is to avoid historical motivations, long proofs and derivations, and (here's where I may lose you) tools for practical calculations. Instead the focus is on the ideas behind the definitions, with lots of geometrical viewpoints and detailed illustrations. For example, tensors originate from a tensor product defined by linearity under various operations, but then are often most profitably viewed as multilinear mappings or multi-dimensional arrays. Exterior forms originate from adding the rule that the product of a vector with itself vanishes, but then are often best viewed as multilinear mappings, anti-symmetric tensors, or anti-symmetric multi-dimensional arrays. Differential forms then smoothly assign an exterior form to each point on a manifold. But...as multilinear mappings, what do they operate on? Vector fields...and what are vectors? Little arrows? To me at least, going at least once through what a tangent vector field actually is (a mapping between real functions on the manifold) is really helpful in avoiding reliance on simplified pictures which may not actually be correct. Plenty of confusion also arises from hidden assumptions made in various texts. For example, there are at least two different ways to associate an exterior form with a multilinear mapping or an anti-symmetric array! And the inner product of those exterior forms is not the same as the inner product of the associated tensors. The good news is that once all this is surfaced and explicitly laid out, everything at least has a chance of becoming much clearer, and more traditional textbooks can be gone through without so many unanswered questions hanging over things. Or at least that's the idea. Hopefully the book ends up being helpful for you. [/QUOTE]
Insert quotes…
Post reply
Forums
Science Education and Careers
Science and Math Textbooks
Motivating definitions from differential geometry
Back
Top