Topics in Calc 3 and Differential Eq.

[QuarkCharmer]

What topics should DEFINITELY be covered in multivariable calculus and differential equations?

My calc 3 course completely overlooked Lagrange Multipliers and Greens Theorem. I had to teach myself what a Jacobian was. I want to be sure that I am familiar with the subject but it's hard to find an accurate representation of what should be covered in an extensive course. It's specifically difficult to find DE topics, since there are frankly, so many.

 

[victor.raum]

I don't know much higher level calculus, but my impression from the vector calculus book that I did read was is that the following are pretty darn important for a vector analysis in 3D space (the order here is nothing especially sensible):

* Translation of area integrals into two iterated single-dimension integrals
* Translation of volume integrals into three iterated single-dimension integrals
* Path integrals
* Line integrals
* Surface integrals of scalar fields
* Surface integrals of vector fields
* Limits in multi-dimensional spaces
* Tangent planes
* Higher dimensional tangent spaces
* Basic Cross products
* Basic Dot Products
* The definition of elementary regions and elementary volumes
* Green's Theorem
* Stokes' Theorem
* The Divergence Theorem
* Gauss' Law (a simple but noteworthy application of the divergence theorem)
* Equality of mixed partials
* Gradient of a scalar field
* Curl of a vector field
* Divergence of vector a field
* The Laplacian Operator
* The dozen or so rules for dealing with divergences of curls, divergences of product functions, and all of the other sensible usages of chained vector operators
* The Transport Theorem
* Examples of how multi-dimensional calculus can be used to solve otherwise puzzling problems in single-dimensional calculus, like for example the Gaussian Integral $\int_{-\infty}^{\infty} e^{-x^2} \; dx$
* The derivatives of cross products and dot products
* The derivative matrix
* The Jacobian Determinant
* The multi-dimensional substitution rule
* The multi-dimensional chain rule
* The multi-dimensional product rule
* The multi-dimensional quotient rule
* Parametric curves
* Parametric surfaces

There was also a large section in that book of mine on finding the local minimums and maximums in multi-dimensional functions. I skipped it while reading, but I'm going into it soon. It does sure look like it has a lot of interesting goodies, and I hear there are applications to physics, so that's probably pretty important too.

If I missed out on any of those subjects and then later found them in other vector calculus texts then I would probably feel like I had been short changed (since they were all so interesting and useful). Then again, that could be bias speaking, as I've only read the one vector calculus text thus far.

 

[chiro]

Hey QuarkCharmer.

For DE's you should cover some kind of transform like Laplace, Fourier or something along those lines. You will also have to know how to change order of integration (this is in Multivariable calculus).

Also you will learn all the non-transform techniques like substitution tricks and stuff like that for very constrained classes of DE's.

Also need to know numerical schemes, Lipschitz continuity condition for DE's, and how to take a numerical scheme and put it into a computer to generate the resulting function and other DE's at various time steps. You will need to know when things will become unstable and blow-up, when answers are just plain wrong from the computer (don't make sense) and probably understand optimal conditions for using the right algorithm to give good accuracy and computation time.

So mainly the numerical schemes and the non-numerical schemes if you can find an analytic solution for DE's are definitely things to keep in the back of your mind.

 

[Jorriss]

For differential equations:
1) First order linear, higher order linear, variation of parameters and undetermined, separation of variables but don't spend nearly as much time on these as many courses do, they are truly easy.
2) Systems of linear diff. eq - eigenvalue problems
3) An introduction to integral transforms, probably spending the most time on the laplace transform.
4) Power series methods
5) Exact vs Inexact Differentials

I think that is most of what needs to be covered. There are other particular differential equations which are useful to know but should be, say, homework problems - such as the Euler diff eq.

Calc III depends on if it is multivariable calculus or multivariable AND vector calculus as some schools, mainly on quarter system, divide them.

 

[QuarkCharmer]

Thanks for the replies.

I'm pretty sure my calculus 3 course contains vector calculus, at least partially. We did everything up to the line integral and stopped. I self-learned the Jacobian, change of variables, the gradient and the curl. So I guess I just need to do Greens/Stokes and all that. We covered line integrals and whatnot. Also the laplacian, The Transport Theorem, and derivatives of the cross product, and triple products. I can manage that on my own I guess.

For DE, well, we did nothing involving Linear Algebra. We did cover Laplace transforms, no Fourier though. We did NOT do exact equations, or learn how to solve anything via computer. I don't know what the Lipschitz continuity condition is. Never touched a system of DE's...

Seems my calculus III course "almost" got it all covered. What is left for me to study seems simple enough, I know what all of those topics are from skimming my copy of Boas.

DE though, I am missing much. I thought that would be the case. It's now the end of the semester and I don't feel like I have learned anything. I can solve some second order differential equations 20 different ways though. I guess I should be doing that over the summer.