jimgavagan said:
Thanks for the reply. I'm at a point where I think math would be great for me, and I'd really like to live and breathe it as you say, but I'm also at that point where I need to find out whether I have the potential to do so.
Here's an e-mail I just sent to a math professor I had, and I'm wondering if you guys could answer it too:
"Hello Dr. Brunner,
I'm a first-year undergrad student, and I'm interested in pursuing my interest in pure/theoretical mathematics. I'm not sure if the school offers a pure math major or concentration, as most of the classes look like in the end they're more tailored toward applied mathematics and combining that understanding with a degree from the sciences; however, I'm not really interested right now in pursuing a degree or career necessarily in pure math----only my raw interest.
In high school, I took your Calc II class at the university and received an A. However, as with most of the math I encountered in elementary and high school, I was left with an empty feeling that I really didn't understand the concepts and theories behind what I was doing---I felt I was really good at the "rote skill/memory" part of it, but I didn't feel I really understood what mathematics is. I'm therefore wondering if you are able to recommend any really good textbooks/books about getting a strong understanding of the theory behind math, with a focus on pure math. I ask about textbooks/books because I'd like to be self-taught in this respect and pursue this interest as a hobby from home, at least at first.
Thank you for your time.
Jim Gavagan"
I've said this in a few other threads, but calculus is simply a way to analyze arbitrary functions that can be non-linear instead of just linear.
In high school you would have learned about how to find the length, area, volume and so on of objects, but they would have all been objects that had straight lines. A straight lines length (in Euclidean Geometry) is just the length based on the pythagorean theorem.
Then you would have found the areas of things like triangles, rhombus' and volumes of things like cubes. The only non-linear thing you would have done would have focused on things with circles like a circle, cylinder, and maybe even something like the volume of a sphere.
Calculus is basically a framework for finding these things that are generalized to non-linear situations. For example you can prove the volume of a sphere is 4/3(pi)r^3 by using the volume formula of the equation x^2 + y^2 = r^2 and applying the appropriate formula.
Then in Calc III you look at systems with more than one variable and you find appropriate measures (like length/area/volume etc) in these situations. Then you look at operations involving vector calculus (dot product, cross product) to find results that again generalize to non-linear situations. In three dimensions a plane is a linear object (ie a surface), but in the non-linear circumstance you deal with things like spheres, paraboloids, and hyperboloids that are non-linear and you use the same framework to analyze all of these including the linear object (a plane).
Differential Geometry basically takes this approach in developing geometric properties of geometry that extends the "linear" geometry (ie Euclidean Geometry). In Euclidean Geometry straight lines are straight and the metric is given by the pythagorean theorem.
In differential geometry you deal with objects that have non-linear properties and you end up with metrics that are different from the pythagorean metric.
For the benefit of visualization Euclidean Geometry in two dimensions is an infinite plane, but a geometric object in differential geometry that is non-linear would be the surface of a sphere. In euclidean geometry you have a plane that goes on forever but with the sphere, the surface meets up with itself and therefore you have interesting properties that you don't have with the euclidean plane.
If you're interested though in the "theoretical" part of calculus you will have to take a course in Real Analysis. Mathematics at a foundational level is based on set theory, so if you want to get a feel for what theoretical maths involves get a book on set theory and after you take some calculus classes, get a book on topology and analysis.
And don't worry too much about not seeing the big picture straight away. Use your resources at your disposal and stay hungry and curious and you'll be fine.
