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jaskamiin
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There's a lot of questions that float around like "I'm an EE major, should I double/minor in math?", "What math classes should I take as a physics major?", etc. After I typed in the title, this already started to show. Maybe you will disagree with some philosophical points I make, but I think you'll find some good information in here nonetheless.
This is not an exhaustive list of information, but hopefully more like a guide for those considering math minor/double/electives.
I want to try to stay away from the whole "it depends..." thing. That being said... your mileage may vary. :-)
(This is geared towards Physics and engineering majors, of course [when I say "engineering", I'm including Computer Science, too], because of the assumptions I make, like your degree requiring some math anyways. Also, it could be some use for some high-school students who are curious about what they should study.)Part 1:
I'm a [STEM but non-math] major, should I double/minor in math?
In short, what it comes down to:
If you have the time, money, and dedication to stay in school for the extra time it would take, I think the answer to this should always be "absolutely". There is never going to be a time where you will complain about knowing too much math in your career; however, there will be plenty of times where you will kick yourself for not knowing enough. If you're asking yourself this question, chances are the real question is: "will it be helpful?". The general answer to that: There's hardly a time it would be unhelpful. University is about learning, too! So if you can do it, why not take advantage?
Basically, the only reasons you should not do a double/minor if you are not interested in the subject to continue studying it, or if you don't want to/can't spend the extra time or money.
Of course, maybe you're not totally sold for technical reasons. Each of the sections below has some reasoning as to why someone would do a minor/double major, and why someone wouldn't.
Let's be honest, most posts asking if you should pursue one of those have usually made up their minds, and are looking for reinforcement of the idea. So, I'd rather motivate you as to why you should, rather than if you should, because ultimately the why motivates the if. I'll put that in a spoiler box since it's pretty long, and espouses some of my biased opinion pretty strongly. It's not necessary to read, if you don't want to.
Okay, so I've thoroughly convinced you that you want to do something (right? right?). But what something should you do?
1.1. Minor in Mathematics:
Good news! If you're an engineering or physics major, you've probably almost completed a math minor already. A minor can be a really cool way to explore your interest in mathematics without having to spend the time and money that it takes to get a double major.
1.1.1. What does a math minor look like?
A typical minor ("type A") requires some combination of the basic calculus sequence (1-3) , linear algebra, differential equations, and one or two elective math courses.
A more involved minor ("type B") will extend that by adding some upper division courses like Analysis (real, complex, or both), or even a course in modern algebra.
Others are usually somewhere in between these, maybe requiring more hours like Type B, but allowing you to choose math electives to fill those up instead.
1.1.2. Some examples
Electives will be discussed in its own section, further down.
1.1.4. Why should I choose a minor? Pros/Cons? Other things to note?
Basically - choose a minor if you're just interested in math, or want to explore it a bit further! No one said you have to use every bit of information you learn from school after you're done. It's also a good choice if you don't feel like or can't spend the time and/or money on a double major. Either way, there's a 100% guarantee that the extra couple classes you take will help you understand something better from your primary education.
1.1.5. Will this help if I want to go to grad school (master's or PhD)?
In my opinion (and everyone will give you a different one on this), a minor won't do much more than your regular degree to prepare you for grad school. Of course it won't hurt you, either. This hasn't really been a huge roadblock for many, since many go into the same field as their primary degree, but it's something to keep in mind.
1.1.6. Will this help if I want to go into industry?
Employers will do one of two things: either look favorably on your minor, or neutrally. That is, it will never be a bad thing that you minored instead of double majoring. To many employers, it shows you have a broader field of interest and dedication to step outside of your required degree, and that's a very good thing.1.2. Double major in Mathematics:
For a lot of engineering and physics majors, a double major isn't going to be too much further either. But, it will require more than a minor, because ultimately you're grabbing a secondary specialization.
1.2.1. What does a math double look like?
In most cases, you will pretty much be taking the basic math degree requirements. For example, where I went to school, it states: "The second major contains at least 21 semester hours of upper division course work beyond the courses required for the students first major and general education requirements." So courses like the basic calculus sequence wouldn't count towards the math major, since you needed it for, say, an engineering major.
But, since the requirements and course availability vary by country and state (in the US), I can't put any concrete info here. So here's a general idea of the basic classes you may have to take:
Since most schools label the double major as basically the same requirements as the regular major, I'll instead direct you to a page from the University of Maryland with some cool advice.
1.2.3. What electives should I take?
Electives will be discussed in its own section, further down.
1.2.4. Why should I choose a double major? Pros/Cons? Other things to note?
If you're not on a strict deadline to graduate, and you're thinking about double majoring, do it. Overall it's not just helpful, but it's very fulfilling. You are qualifying yourself in another subject, and a double major will set you pretty far ahead of the crowd both on paper and mentally.
Just remember, as it's been said ad nauseum, it will take longer. There may be classes required for the major that you're not interested in (typically "pure" math courses). Also, if your school is like mine was, the math department may be small and offer necessary classes on an odd schedule, which can make the "extra time" thing really prevalent.
1.2.5. Will this help if I want to go to grad school (master's or PhD)?
Almost always. You don't need a double major to be admitted to grad school, though. Where the double major would really help is in mental preparation. A lot of grad-level physics and engineering courses are basically just math, and you'll start to see more rigor than in undergrad. Having a firm understanding of different mathematical tools can give you just the insight you need to do really well in these courses/your thesis.
1.2.6. Will this help if I want to go into industry?
Employers are likely to treat this in the same way as a minor (1.1.6), since they will care more about results and skill than your academic prowess. But, these things often go hand in hand, and a lot of employers like math majors for their problem solving and abstraction skills, as well as desire for exactness (unless you're a numerical analyst :-) ). Math majors are also often preferred for hot fields like data analysis, if you're into that.
Part 2:
I'm a [STEM but non-math] major, what math electives should I take?
Here's a general guide to what classes can be useful to any physics or engineering major, and how to choose others (since I will undoubtedly leave out something your school has). But first, a note...
"Pure Math" isn't useless...
It's pretty easy to get caught up in the idea that pure math is useless because it's not called "applied math". That's a very unfortunate way of thinking. Traditionally "pure" courses like Abstract Algebra or Graph Theory have very direct applications. A lot of modern physics is reliant on groups, and a huge portion of optimization problems are solved using graph theory.
Math courses are not there to strictly teach you the algorithm to solve a specific problem. Rather, your professors should be teaching you how to recognize when a problem can be solved using the abstract methods you are learning. So take courses outside of those with "direct application"; insight matters just as much.So what's a list of courses I should consider?
I'll split this up by area. I'll definitely forgo a few, but, for example, if you're studying Astronomy, the answer would be very similar to that of a Physics major. If your field isn't listed explicitly, use your best judgement -- if you're a mechanical engineering major, but you know you will be working a lot with programming, consider courses from the "CS" list as well.
Look in every list -- you may see something that would be helpful for you. There's always overlap.
I'm going to separate CS from "engineering", here.
I'm sure I forgot some stuff, but I'm on hour 3 of typing and my brain's shutting down. Hope this helps those who see it. Please add anything else in the comments.
This is not an exhaustive list of information, but hopefully more like a guide for those considering math minor/double/electives.
I want to try to stay away from the whole "it depends..." thing. That being said... your mileage may vary. :-)
(This is geared towards Physics and engineering majors, of course [when I say "engineering", I'm including Computer Science, too], because of the assumptions I make, like your degree requiring some math anyways. Also, it could be some use for some high-school students who are curious about what they should study.)Part 1:
I'm a [STEM but non-math] major, should I double/minor in math?
In short, what it comes down to:
If you have the time, money, and dedication to stay in school for the extra time it would take, I think the answer to this should always be "absolutely". There is never going to be a time where you will complain about knowing too much math in your career; however, there will be plenty of times where you will kick yourself for not knowing enough. If you're asking yourself this question, chances are the real question is: "will it be helpful?". The general answer to that: There's hardly a time it would be unhelpful. University is about learning, too! So if you can do it, why not take advantage?
Basically, the only reasons you should not do a double/minor if you are not interested in the subject to continue studying it, or if you don't want to/can't spend the extra time or money.
Of course, maybe you're not totally sold for technical reasons. Each of the sections below has some reasoning as to why someone would do a minor/double major, and why someone wouldn't.
Let's be honest, most posts asking if you should pursue one of those have usually made up their minds, and are looking for reinforcement of the idea. So, I'd rather motivate you as to why you should, rather than if you should, because ultimately the why motivates the if. I'll put that in a spoiler box since it's pretty long, and espouses some of my biased opinion pretty strongly. It's not necessary to read, if you don't want to.
It seems the majors that typically find their way to extra math education are Physics and Electrical Engineering. But I think that basing your decision off of what you're doing as a primary major is a mistake.
Mechanical engineers have made progress in PDEs, and Computer Scientists would be nowhere without a firm grasp of math. Fluid dynamics is hugely mathematical and aerospace engineers need some pretty advanced tools to study it. The idea is that everyone would benefit from taking the extra math, so your decision should not be based on if you are an EE or CE or chemistry major.
Instead, your decision should be based on, first of all, if you're interested in mathematics. If you aren't, turn back, because without interest, you won't have the desire to learn the subject more deeply. All of the mathematical tools a Physics major needs will (hopefully) be taught in the physics program. Taking a double major or a minor (or even just math electives) will teach you how those tools work on a more intimate level.
I like to use a hacker analogy here. The programmers considered the "best" and most well known are often those who have made some kind of creative development in the field. At the end of the day, we're all messing with 1's and 0's, but what makes them different from me? They understand the inner workings of a computer, the rules by which those 1's and 0's play. And with that understanding they are able to make things that a typical programmer would never think about. There's a big difference between knowing a programming language, and understanding how to program.
Mathematics is likewise. By taking the extra courses and spending time trying to really understand the rules by which the tools of physics and engineering play, you will open yourself up to a completely different way of thinking about problems in your field. It's this style of thinking that often leads to really unbelievable results.
That is why you should take the extra math education. If you agree with the why, I think the answer to "should I do a double major/minor
Mechanical engineers have made progress in PDEs, and Computer Scientists would be nowhere without a firm grasp of math. Fluid dynamics is hugely mathematical and aerospace engineers need some pretty advanced tools to study it. The idea is that everyone would benefit from taking the extra math, so your decision should not be based on if you are an EE or CE or chemistry major.
Instead, your decision should be based on, first of all, if you're interested in mathematics. If you aren't, turn back, because without interest, you won't have the desire to learn the subject more deeply. All of the mathematical tools a Physics major needs will (hopefully) be taught in the physics program. Taking a double major or a minor (or even just math electives) will teach you how those tools work on a more intimate level.
I like to use a hacker analogy here. The programmers considered the "best" and most well known are often those who have made some kind of creative development in the field. At the end of the day, we're all messing with 1's and 0's, but what makes them different from me? They understand the inner workings of a computer, the rules by which those 1's and 0's play. And with that understanding they are able to make things that a typical programmer would never think about. There's a big difference between knowing a programming language, and understanding how to program.
Mathematics is likewise. By taking the extra courses and spending time trying to really understand the rules by which the tools of physics and engineering play, you will open yourself up to a completely different way of thinking about problems in your field. It's this style of thinking that often leads to really unbelievable results.
That is why you should take the extra math education. If you agree with the why, I think the answer to "should I do a double major/minor
Okay, so I've thoroughly convinced you that you want to do something (right? right?). But what something should you do?
1.1. Minor in Mathematics:
Good news! If you're an engineering or physics major, you've probably almost completed a math minor already. A minor can be a really cool way to explore your interest in mathematics without having to spend the time and money that it takes to get a double major.
1.1.1. What does a math minor look like?
A typical minor ("type A") requires some combination of the basic calculus sequence (1-3) , linear algebra, differential equations, and one or two elective math courses.
A more involved minor ("type B") will extend that by adding some upper division courses like Analysis (real, complex, or both), or even a course in modern algebra.
Others are usually somewhere in between these, maybe requiring more hours like Type B, but allowing you to choose math electives to fill those up instead.
1.1.2. Some examples
- University of Florida ("Type A")
- https://www.as.uky.edu/math-minor-requirements ("Type A")
- Georgetown University (in between)
- UC Berkeley ("Type B")
Electives will be discussed in its own section, further down.
1.1.4. Why should I choose a minor? Pros/Cons? Other things to note?
Basically - choose a minor if you're just interested in math, or want to explore it a bit further! No one said you have to use every bit of information you learn from school after you're done. It's also a good choice if you don't feel like or can't spend the time and/or money on a double major. Either way, there's a 100% guarantee that the extra couple classes you take will help you understand something better from your primary education.
1.1.5. Will this help if I want to go to grad school (master's or PhD)?
In my opinion (and everyone will give you a different one on this), a minor won't do much more than your regular degree to prepare you for grad school. Of course it won't hurt you, either. This hasn't really been a huge roadblock for many, since many go into the same field as their primary degree, but it's something to keep in mind.
1.1.6. Will this help if I want to go into industry?
Employers will do one of two things: either look favorably on your minor, or neutrally. That is, it will never be a bad thing that you minored instead of double majoring. To many employers, it shows you have a broader field of interest and dedication to step outside of your required degree, and that's a very good thing.1.2. Double major in Mathematics:
For a lot of engineering and physics majors, a double major isn't going to be too much further either. But, it will require more than a minor, because ultimately you're grabbing a secondary specialization.
1.2.1. What does a math double look like?
In most cases, you will pretty much be taking the basic math degree requirements. For example, where I went to school, it states: "The second major contains at least 21 semester hours of upper division course work beyond the courses required for the students first major and general education requirements." So courses like the basic calculus sequence wouldn't count towards the math major, since you needed it for, say, an engineering major.
But, since the requirements and course availability vary by country and state (in the US), I can't put any concrete info here. So here's a general idea of the basic classes you may have to take:
- A course in proof methods
- At least one course in Real Analysis
- At least one course in Modern Algebra (Group Theory)
- Linear algebra
- Differential equations
- Discrete Mathematics (maybe a specific course like Graph theory or Combinatorics)
- Vector calculus
- Several electives
Since most schools label the double major as basically the same requirements as the regular major, I'll instead direct you to a page from the University of Maryland with some cool advice.
1.2.3. What electives should I take?
Electives will be discussed in its own section, further down.
1.2.4. Why should I choose a double major? Pros/Cons? Other things to note?
If you're not on a strict deadline to graduate, and you're thinking about double majoring, do it. Overall it's not just helpful, but it's very fulfilling. You are qualifying yourself in another subject, and a double major will set you pretty far ahead of the crowd both on paper and mentally.
Just remember, as it's been said ad nauseum, it will take longer. There may be classes required for the major that you're not interested in (typically "pure" math courses). Also, if your school is like mine was, the math department may be small and offer necessary classes on an odd schedule, which can make the "extra time" thing really prevalent.
1.2.5. Will this help if I want to go to grad school (master's or PhD)?
Almost always. You don't need a double major to be admitted to grad school, though. Where the double major would really help is in mental preparation. A lot of grad-level physics and engineering courses are basically just math, and you'll start to see more rigor than in undergrad. Having a firm understanding of different mathematical tools can give you just the insight you need to do really well in these courses/your thesis.
1.2.6. Will this help if I want to go into industry?
Employers are likely to treat this in the same way as a minor (1.1.6), since they will care more about results and skill than your academic prowess. But, these things often go hand in hand, and a lot of employers like math majors for their problem solving and abstraction skills, as well as desire for exactness (unless you're a numerical analyst :-) ). Math majors are also often preferred for hot fields like data analysis, if you're into that.
Part 2:
I'm a [STEM but non-math] major, what math electives should I take?
Here's a general guide to what classes can be useful to any physics or engineering major, and how to choose others (since I will undoubtedly leave out something your school has). But first, a note...
"Pure Math" isn't useless...
It's pretty easy to get caught up in the idea that pure math is useless because it's not called "applied math". That's a very unfortunate way of thinking. Traditionally "pure" courses like Abstract Algebra or Graph Theory have very direct applications. A lot of modern physics is reliant on groups, and a huge portion of optimization problems are solved using graph theory.
Math courses are not there to strictly teach you the algorithm to solve a specific problem. Rather, your professors should be teaching you how to recognize when a problem can be solved using the abstract methods you are learning. So take courses outside of those with "direct application"; insight matters just as much.So what's a list of courses I should consider?
I'll split this up by area. I'll definitely forgo a few, but, for example, if you're studying Astronomy, the answer would be very similar to that of a Physics major. If your field isn't listed explicitly, use your best judgement -- if you're a mechanical engineering major, but you know you will be working a lot with programming, consider courses from the "CS" list as well.
Look in every list -- you may see something that would be helpful for you. There's always overlap.
I'm going to separate CS from "engineering", here.
- General list: physics and engineering majors
- Partial Differential Equations. You will often learn some nuances of Fourier analysis that would be overlooked in physics courses (convergence is important!).
- Numerical analysis (numerical methods). Basically scientific computing with Matlab/Maple, etc. Calculations in the real world are never perfectly exact, and that's what this class shows you.
- Linear algebra/Matrix algebra. You will undoubtedly need at least one class in this, but if they offer another, take it.
- Complex Analysis. Goes by many names (functions of a complex variable, etc). Methods from this subject are often used to simplify otherwise hard problems (like simplifying difficult integrations).
- Functional Analysis. Provides a lot of insight into solving differential equations, for one. A 'pure' topic with a lot of applications. Especially useful for signal analysis.
- Dynamical Systems. Useful in predicting behavior of systems. Useful all around the board, even for astronomy/astrophysics, chemistry, biology, and more.
- Probability Theory and/or Statistical Methods. I can't list all the places you'll use this. It's a must-have.
- Topology. I hesitate to include this, but it's applied throughout all of engineering and physics, just non-explicitly. Manifolds are a study of topology (and relativity is pretty heavily based on this), and for engineers, topology is applied in sensor networks and robotics.
- Computer science
- Graph theory. A must have. Tons of algorithms (think: Dijkstra's) are graph problems and dozens of data structures are graphs themselves (any kind of tree). Optimizing these often requires some theoretical knowledge of how graphs work. Network theory is a cool "application" branch of this.
- Modern/Abstract algebra (group theory). Modern crypto is deeply drenched in group theory. Groups are found everywhere though, and find new uses every day. Carnegie mellon even offered a course called Group Theory and Its Applications in Robotics, Computer Vision/Graphics and Medical Image Analysis. Compression and encoding/decoding things in general are part of a subset called Coding theory. If there's a second semester of this, it will probably cover Field theory (different than physics), which has a lot of theorems that computers actually need to be able to work.
- Number Theory. Study of the integers. You'll learn a lot about primes, modern crypto standards, etc. Donald Knuth even said: "virtually every theorem in elementary number theory arises in a natural, motivated way in connection with the problem of making computers do high-speed numerical calculations".
- Combinatorics: Another must-have. It studies finite/discrete structures, so it's pretty naturally suited for CS. Again, Knuth even named one of the Art of Computer Programming by this: "Combinatorial algorithms". I'd say this drives a lot of the underlyings of the 'science' in computer science.
- Linear algebra/Matrix algebra. Matrices are everywhere in programming, and a lot of times you can use a matrix to represent data really efficiently (what is a database, after all? ;-) ).
- Numerical analysis (numerical methods). By far, this math class had the most CS majors in it when I took it. A lot of cool algorithms are found here. Google's PageRank comes from a numerical eigenvalue algorithm called the Power Method. If you're into graphics or high performance computing, this is a must-have.
- Probability Theory and/or Statistical methods. Like the above list, it's too useful to even be able to quantify it.
- Physics - other
- Modern/Abstract algebra (group theory). A ton of modern physics uses groups. You will learn about them somewhat in your degree, but an in depth class on them will really show you why they work the way they work. It was one of the most beautiful classes I took. Lie groups and representation theory are probably graduate-level, but you'll at least have a basis.
- Differential Geometry. If your school offers this, take it. It's key to understanding relativity, and tensor algebra is really useful, too.
- Calculus of Variations (variational calculus). Think - variational methods in QM for approximating the ground state, and Lagrangians and Hamiltonians can be seen as tools of this.
- Real analysis. It doesn't have direct applications in solving physical problems, but it's key to understanding the "math behind physics". Often called "advanced calculus".
- Engineering - other
- Optimization. I'm just going to link this, and let it describe why this is a good course. Any optimization course is good, and if your school offers them, be proud.
- Linear Programming. A part of optimization. Useful for scheduling problems, and used in "transportation, energy, telecommunications, and manufacturing" (from wiki). These are basically really good problem-solving classes.
- Optimization. I'm just going to link this, and let it describe why this is a good course. Any optimization course is good, and if your school offers them, be proud.
I'm sure I forgot some stuff, but I'm on hour 3 of typing and my brain's shutting down. Hope this helps those who see it. Please add anything else in the comments.