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What is the most useful type of mathematics for physics?
Asked by: Don
This is the first time I am answering such a 'highly subjective' question. So, parts of my answer will probably range from 'educated advice' to 'wild speculation'.
Physics is probably the one area of science where many areas of mathematics have been directly applied. The reason is simple; nature seems to obey 'mathematical rules' rather than acting whimsically. In other words, it seems that natural laws can be expressed in terms of mathematics. Why this should be so, nobody knows.
If I were asked to single out one area of mathematics that is of absolutely maximum use in the study of physics, I would probably pick calculus. All of classical mechanics, thermodynamics, fluid dynamics, classical electromagnetism, statistical mechanics, and many other fields of physics make extensive (and sometimes exclusive) use of calculus.
Is this sufficient? Probably not for all areas of physics you might work in. The very next requirement would probably be differential equations, and can be thought of as part of calculus (although it is a vast area of study within itself). In addition, you may need probability theory and statistics, linear algebra, numerical methods and the like depending on the field you choose. If you are lacking in mathematical skills you can find an algebra tutor to get you up to speed. Some more recent theoretical work requires more mathematics than mere mortals such as me can hope to know.
The truth of the matter is, you can never know enough mathematics. To a physicist, mathematics is a toolbox. Before attacking a particular problem, you should have the necessary tools for the job. There are some tools (such as calculus) that should be in any physicist's toolbox, but as they specialize, they will add extra tools needed for the specific problems at hand.
The fact is, many different branches of mathematics are useful in a wide variety of physics applications so I will fall back on a personal view rather than seek a definitive, universal answer.
When I first formally studied Physics in high school, I started by examining kinematics with algebra. This generally worked because the curriculum stuck to simpler topics which algebra could handle, if the physics were approached intelligently (e.g. ignoring friction assuming g was constant over the altitude of a projectile, etc.). However when I learned calculus a whole new appreciation for kinematics (and physics in general) blossomed. I went from being able to do the problems (most of the time) to truly having a feel for what was going on. For this reason I have always felt that calculus is the keystone necessary for deep physics understanding.
With quantum mechanics and relativity, it also helps to have an appreciation of probability and statistics, but even here the mathematical techniques of these disciplines are not as necessary as calculus, in my opinion. You can get a pretty good sense of warped 4 D space time with out knowing what eigenvalues are, for example.
Mathematics is crucial to analysis in many fields of endeavor, so don't limit your studies based on this answer - keep learning and pursue many branches of math! It will train your mind as well as open the doors to success.
Well there are lots of very useful maths. I'll take you through the most important in terms of the developments in fundamental physics:
Classical Mechanics - Calculus
Each new development in physics often requires a new branch of mathematics. I would say that the older maths are the most widely used in physics now such as calculus - so are probably the most useful.
In my opinion, one has to view physics as a branch of applied mathematics. So, the question of what the most useful mathematics is, is a rather tricky one. In a way, it depends on what branch of physics you're interested in, as different maths is applied in different fields.
If you're interested in 'classical' physics (for eg. mechanics, thermodynamics and electrodynamics), then the field of calculus would suit you far better than subjects like algebra or statistics (This of course depends on how in-depth you go into the subject) On the other hand, algebra (and group theory) is very important in the quantum fields. I'll try to answer this question by basing it on my own experience.
To start of with, one should have a very good understanding of calculus, especially matters like vector calculus where topics like curl and divergence are covered (A good understanding of calculus of variations, where topics like maxima and minima of integrals are covered would also be very useful). Calculus lays the foundations for more advanced maths. Subjects like (partial) differential equations, and mathematical analysis all have their roots in calculus. These subjects are all essential in the 'classical' field. I've done courses in classical mechanics, and in reality, it's applied mathematics through and through.
On the other hand, in quantum mechanics you will deal more with algebraic techniques. For example matrix operations and transformations are very common. So, you might argue that algebra has more use. But before you think that quantum mechanics is predominantly a discrete field, I would like to make you aware that partial differential equations do creep in here (for eg. solving Schroedinger's equation for given energy values). However, a good understanding of modern physics can only be based on a good understanding of classical ideas. Of course, algebra can be extremely powerful in many fields, even beyond quantum mechanics (for eg. DSP and cryptography).
As a whole, I'd have to say that if you plan to pursue physics, you should do as many maths courses as you can. Many of the courses link up to each other very subtly. For example, eigenvalues and eigenvectors is an algebraic idea, but it is used widely in solving systems of differential equations; analysis (I like to think of this as the proof of mathematics) and calculus share very similar ideas (and in some sense are the same ideas-series etc).
So what do I think is the most useful type? I'd have to say a well rounded combination of all the above mentioned subjects. But if I have to choose one; I'd say differential equations. Many physical systems have a 'differential' touch to them. For example, spring systems or RLC circuits (electrodynamics) are actually differential eq's. Knowing how to solve differential equations (and how to set them up) is an invaluable tool in physics. Just imagine the difficulty you would have trying to do fluid mechanics without differential techniques.
For those interested in a good book on differential techniques (where topics like springs, circuits and population growth are covered), please refer to the book I've included as a reference.
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