Vector Analysis Resources

[romistrub]

So here I am, trying to learn vector analysis (vector calculus, call it what you want) and I'm all but befuddled by the number of ways to present the topic. I'm looking for a resource (preferably free) that expands on:

Lame coefficients / scale factors (the same thing, right?)
Metric coefficients
Forms (maybe?)
Generalized Stokes' Theorem
Jacobian matrix

Basically, my problem is this: my professor introduced scale coefficients without explaining how to calculate them, and in trying to learn it by myself, I stumbled upon all sorts of confusing and/or revealing generalizations. Unfortunately, I don't understand how they fit together.

For example:
- I know how to calculate the Jacobian given a transformation
- I know how to derive div/grad/curl operators for orthogonal coordinate systems from reading the Appendix of Griffith's Electrodynamics. These forms depend on scale factors, which Griffiths, although including them, never explains how to find them or even what they really *are*.
- I understand, to a degree, where scale factors and metric coefficients come from, from reading "Geometrical Vectors" and trying to understand wikipedia.
- Wikipedia sucks.

Is there anything that just gives the briefest (and I mean briefest) summary of how to compute these factors given a general curvlinear coordinate system? I can handle a sentence or two explaining the steps, but I really just need some worked examples. I'm also short on time! :)

Thanks!

 

[jvicens]

I can understand your frustration with the overwhelming amount of information on vector analysis and the lack of clear explanations for certain concepts. However, I can assure you that with some practice and the right resources, you will be able to understand and apply these concepts effectively.

Firstly, let's clarify the terms you mentioned. Lame coefficients and scale factors are indeed the same thing. They refer to the factors that relate the coordinates in a curvilinear coordinate system to the coordinates in a Cartesian coordinate system. These coefficients are necessary for transforming vector operations (such as gradient, divergence, and curl) from one coordinate system to another.

Metric coefficients, on the other hand, refer to the coefficients that define the length and angle measurements in a curvilinear coordinate system. These coefficients are important for calculating line and surface integrals, as well as for understanding the geometry of a particular coordinate system.

Forms, or differential forms, are mathematical objects that are used to represent vector operations in a coordinate-independent way. They are useful in vector analysis because they allow us to generalize the concepts of gradient, divergence, and curl to any coordinate system, not just orthogonal ones.

Now, onto the resources you can use to expand your understanding of these concepts. One great resource is the book "Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach" by John Hubbard and Barbara Burke Hubbard. This book covers all the topics you mentioned and provides clear explanations and examples for each concept.

Another useful resource is "A Student's Guide to Vectors and Tensors" by Daniel Fleisch. This book focuses on explaining vector and tensor concepts in a clear and concise manner, with plenty of examples and exercises to practice.

For a more visual approach, you can check out various online video tutorials on vector analysis. The Khan Academy and MIT OpenCourseWare both have free tutorials on vector calculus that cover the concepts you mentioned.

Lastly, if you're short on time, I recommend focusing on understanding the steps for calculating scale factors and metric coefficients for a specific coordinate system, rather than trying to understand the general procedure for any curvilinear coordinate system. This will give you a foundation to build upon and make it easier to understand the generalizations you come across.

I hope these resources and tips help you in your learning journey. Remember to practice and ask for help when needed. Vector analysis can be challenging, but with perseverance and the right resources, you can master it. Best of luck!