Can mathematics help me understand and master chemistry?

[BioCore]

Hello Everyone,

I am finishing up my first year in university and over the year I noticed that I really enjoy doing Chemistry. This is different from my one year ago in grade 12 where I hated chemistry and just wanted to get over it. I have no idea why I suddenly start liking it, although it might have to do with the fact that I am able to understand it a bit better and have actually set aside the ideology that it (Chemistry) is just another course for me.

I have made my mind that I will study Biotechnology, and so I will need both Molecular Bio courses and chemistry courses.

What I noticed was that chemistry does have quite a bit of a connection to math especially Calculus. I am currently taking a university Calculus course but find that I did have some troubles with certain concepts - which caused me to have some trouble understanding chemistry principles and theories.

Well anyways, to the point. I decided that I would like to understand and master mathematics during the summer. Since I will have four months off I wanted to restudy most of my mathematics knowledge and try to do what I did not do when I was younger - that is to fully understand and try to master the theorems, and concepts in mathematics.

So I was wondering if you could all suggest on where to begin. If there were some good books to use for this. I will also be doing an introductory physics course over the summer so it will also compliment this nicely.

P.S.
I know some of you might say that this is not necessary and all, but I would really like to do this. I want to actually do this so that I can prove to myself that I am capable of mastering or semi-mastering mathematics.

BTW, can someone tell me how someone would know if they mastered a topic or not? Is it being able to fully understand and grasp a concept - in terms of mathematics being able to answer any form of a question. Or is it something else?

 

[JohnDuck]

Sounds like you're interested mostly in calculus of a single variable. If you want to understand it very formally and fundamentally, you'll want to look into books on introductory real analysis. "Principles of Mathematical Analysis" by Walter Rudin is regarded as particularly good. One word of warning, though--the calculus courses you've taken in the past focused on techniques for computing derivatives and integrals, probably with some applications in the physical sciences, and very little emphasis on proofs. The emphasis of analysis is entirely reversed. Proofs of theorems dominate the discussion. It will probably also be hard going if you're new to mathematical formalism.

If you just want to review techniques of computation (the stuff that's most likely to come up in your physics and chemistry courses), pick up your old calculus text and work through the problems. Basically any book at that level is interchangeable.

 

[BioCore]

Well I am not sure currently if my only interest is single variable calculus. What I wish to achieve is to study later on quantum chemistry a bit on my own. This is mostly so that I can start to get a feel of nanotechnology as well as people have told me that this is something that is helpful in current and especially future Biotechnology.

The type of quantum physics or chemistry that I have studied seems to be single variable, but is the more advanced stuff similar or different?

 

[JohnDuck]

I'm not really familiar with quantum mechanics, but I think it's safe to say that anything like that will at the very least be grounded in multivariate calculus, including ordinary and partial differential equations. Each of these is typically a semester long undergraduate course. I'm not going to say it's impossible to self-study all of these subjects over the course of a summer, but I think you would be hard pressed, especially if you are struggling with single-variable calculus. If you're really dead-set on studying quantum mechanics, figure out if its offered at your university, find out what the prerequisite courses are, and start taking those.

 

[BioCore]

Thanks for the tip, I will probably do that. I was just wondering since I plan on reviewing most of my past math skills, would you recommend the "...For the Practical Man" series by J.E. Thompson before reading "Principles of Mathematical Analysis" by Walter Rudin, or would just going over my past calculus texts be ok for this?

I ask because I have noticed that I might have a problem with some trigonometry mostly.

 

[BioCore]

Sorry to be bothersome with these questions and all. But I took your advice and I looked through some quantum mechanics courses, which also led me to some Biological Physics courses. Now at my school there is a specialist program in Biological Physics, and they tend to do a certain amount of math courses. As I don't really intend to do the specialist but would still like to study Biological Physics on my own I think I should be following somewhat of their curriculum and study some of the maths required in that field.

The maths that I noticed that were needed seemed to be calculus, Linear Algebra I, Calculus of several variables, Differential Equations I.

I was wondering if you could recommend some good books to use when self-studying these topics, or should I just follow the books that were used in these courses? OR, should I maybe stick with some Nelson educational books, because I tended to enjoy those very much over my high school years, and actually learned more from them by myself than with my teacher.

So if anyone could recommend what is best. What I think I need currently is to improve upon my Calculus skills. So I think what I am asking for is advice on how to do this? How did some of you improve or actually attain good skills and mastered calculus?

Thanks again.

 

[bravernix]

I took a biophysics course and I didn't like the text we used too much. I think it would have been better if we used Biological Physics by Philip Nelson. It seemed to be pretty comprehensive and had interesting problems in it.

The best way to improve on math skills for any topic (assuming you mean computational problems useful for the sciences) is to just do as many problems as you can. You mentioned you think your trigonometry understanding is weak which will definitely hinder your ability to do well in calculus. So, get a trig book and do tons of problems. Make sure your algebra skills are up to par as well. For learning calculus, you could go the real analysis route but I do not think this is quite what you are seeking. A book like Calculus by James Stewart seems to be a pretty standard one and is good for self-study I think (it covers single and multiple variables). I do not remember my linear algebra or ODE textbooks so maybe others can help. I think you will see that most difficulties with understanding in math and science can be remedied simply by doing many many problems (but read the text as well).

 

[BioCore]

I was looking through Biological Physics by Philip Nelson the other minute, and it does seem like a very good book to learn B. Physics from - my schools library has the updated first edition so it has a bit more problems.

Seeing as you recommended that I should also use the textbook Calculus by James Stewart - my T.A. recommended this one as well, I will have a look into it.

The calculus course that I took used a textbook that was difficult to understand and most of the problems did not reflect the type of test questions we were given. The book is Calculus for Biology and Medicine by Neuhauser, Claudia.

 

[LoneDragon]

Advanced topics that work with complex chemistry, biochemistry, and mathematics are:

1) Computational chemistry, where molecules are modeled by 3-D mathematics with models of the quantum physics models of atomic electron structures.

2) Combinatorial chemistry, where molecule interaction characteristics are modeled, and the combination of every product in a square matrix can show reactions that produce no products, reactions that produce products, reactions that are autocatalytic, chain reactions, that produce in a circle or hypercycle, a feedback loop that create a positive equilibrium, and reactions that inhibit other reactions. Permutational equation set mathematics are also involved as new products increase the size of the base matrix of chemical products in a complex chemical environment, like that found in biochemistry. The genetic and pharmeceutical industry use combinatorial chemistry for exploring for drugs and novel chemical products. Matrix mathematics of linear algebra types, are important in this field where the matrix contains the complex inter-action equations between chemical "specie" X and Y in the square matrix containing all products in a solution. Euler method, and runge kutta algorithms should be studied to simulate the concentration of these products when the interaction probabilities have been modeled for the matrix product calculations over time.

3) Nonlinear chemistry, where special reactions contain nonlinear terms not normally found in more basic linear reactions. These are of the Belousov-Zhabotinskii BZ type reactions. Chaos theory and nonlinear mathematics are important in this field of research, as they do not have asymptotic production modes.

4) In the inorganic realm of chemistry, one may study for interest, quasiperiodic crystals where there is no periodic crystal cell structure, but there are aperiodic atom placement symmetries like 10 fold symmetric crystals. Fourier transforms and related diffraction mathematics are involved in analysing these structures of inorganic chemistry. One may also study periodic crystals with the same methods of diffraction analysis.

Again, all of these methods of chemistry, related to mathematics, does require a base knowledge of steric or three dimensional electron structures, most accurately modeled by quantum physics, and parallel processing algorithms for partitioning such problems for computer analysis, which requires the math for the models of the chemistry. Also, methods to compute a model in detail, and then abstract the models, are mathematically /code important fields. Systems theory is also useful, aka cybernetics.

For biochemistry, hierarchical numerical data structures are also used in the computer code and math domain to "chunk" some processing for hueristic speed, when the minor models are characterized, and can be approximately combined to build larger models, like protien folding simulations on smaller computer systems.

Physical Chemistry, by George Woodbury, has a lot of math to start you off in mathematical chemistry, too, and covers some of these topics.