Which rules/theorems should you be able to prove?

[Dafe]

Hi,

I was thinking back at the days when I took calculus and how we had to prove things like the product rule etc.
Of course, most of that has magically disappeared from my mind.

I've decided to try and take a masters degree in applied mathematics, and have about a year to get my head back into the math-business.

What do you guys think are the most important rules/theorems one should be able to prove, say from calculus.
All of them? :)

Thanks

 

[HallsofIvy]

I am tempted to say "all of them"! Not from memory, of course. If I were you, I would take a calculus book and go through the theorems in it, first trying to prove each without looking at the proof in the book. If you are not sucessful, look at a few lines of the proof, then try to go on from there. Actually, that's the way you should always study mathematics.

 

[Dafe]

I was afraid you would say that :)
I just began going through my old calculus book, and will do as you say.

calculus, linear algebra, differential equations, physics, so many things in applied math.
No wonder you guys are crazy.

Thanks HallsofIvy

 

[travisjeffery]

Agreed; find a real good theoretical, rigorous textbook like Spivak Calculus or Hardy's Pure Mathematics so that you have the foundation for potentially being able to prove whatever you want.

 

[n!kofeyn]

travisjeffery has a good point. I recommend not picking up your old calculus book (which one was it?). Have you had an analysis course before? I would recommend going through either Calculus by Michael Spivak (as travisjeffery said) or Calculus by Tom Apostol. Apostol covers a lot of material, but Spivak may be more readable and has an answer book. I would then try to get some analysis in, try Mathematical Analysis by Tom Apostol. Analysis by Steven Lay is great introduction to doing proofs in calculus and basic analysis if you have never taken a proof class.

Since you are going into applied mathematics, I would also recommend refreshing or relearning linear algebra and differential equations. Also, partial differential equations would be a good thing to study as well.

 

[Dafe]

Hi, my calculus book(s) were the ones by Edwards and Penney.
I've heard the Spivak book mentioned on several occasions so I will have a look.

n!kofeyn: I bought Gilbert Strangs Linear Algebra book and am going through it while looking at his lectures.
The Linear Algebra course I took at uni was awful.
The mit ocw course on the other hand is wonderful! As soon as I complete it, I will go through the mit differential equations course.

Got a lot of learning/refreshing to do!

 

[n!kofeyn]

Hi, my calculus book(s) were the ones by Edwards and Penney.
I've heard the Spivak book mentioned on several occasions so I will have a look.

n!kofeyn: I bought Gilbert Strangs Linear Algebra book and am going through it while looking at his lectures.
The Linear Algebra course I took at uni was awful.
The mit ocw course on the other hand is wonderful! As soon as I complete it, I will go through the mit differential equations course.

Got a lot of learning/refreshing to do!

Cool. I'm really not for sure of the rigor of all applied programs, but at my school the applied students have to take a substantial amount of pure mathematics. So just be sure you get some practice in doing abstract proofs in analysis and linear algebra. Analysis seems to almost always be the kicker (it was for me!). Don't worry about proofs in differential equations, and like I said, if you have time, try to look into some PDEs.