<p>I have a slight concern about majoring in applied math or physics, which I am highly interested in.</p>
<p>My concern lies with geometry. I highly enjoy solving algebraic equations and the basic differential equations we have learned at school, but I hate geometrical problems.</p>
<p>Is geometry at all a big thing in applied math or physics programs? Will I need to do things such as coordinate proofs?</p>
<p>Thanks.</p>
<p>Hello, I am an applied math major.</p>
<p>You’re going to need some geometry because you’re required to take physics. Well, I should say trigonometry. However, nothing I’ve ever encountered was really all that hard geometrically that I needed to go to the library and refresh myself. “Algebraic geometry” will be very important. </p>
<p>I’m taking an upper-level differential equations class right now. In fact, I just took my midterm about an hour ago. I needed to know how to draw functions, basically. I needed to know how to make a vector field and show the directions of these vectors on the x-y plane to get a general picture of a phase portrait (slope fields pretty much). I needed to know how to take derivatives/partial derviatives, use linear algebra to compute eigen-values and eigen-vectors. I needed to know abou the stability of functions and what they look like in certain cases. I needed to know about bifurcations. I had to graph on each problem. </p>
<p>I’d say the most geometry I had (which I feel is just part of trigonometry) was mainly used in physics classes. Most of my math classes I haven’t done much geometrically as in finding angles and whatnot. I’d say knowing geometry is important, though. You’re going to have to do geometry at some point because to be honest you might end up in this line of work if you do an engineering job after graduation. There’s some classes that don’t use geometry like Operations Research where you just learn about all of these programming and math tools you can use if you work in business. Statistics doesn’t require any geometry I’m aware of.</p>
<p>Geometry is huge in essentially every area of physics, especially when you get to graduate classes as you delve deeper into the math. In Newtonian mechanics it’s mostly trigonometry, but once you get to electromagnetism there is a ton of vector calculus that is most naturally done in cylindrical or spherical coordinate systems. In many problems it’s very important to be able to use symmetries of the system to solve them efficiently. This become even more important in Lagrangian and Hamiltonian Mechanics as well as Quantum Mechanics.
For graduate courses and in current research, you see several notions in algebraic topology and differential geometry come up over and over again. For example, the Aharonov Bohm effect is observed when a solenoid is placed in between an electron and a double slit. As a result of the flux that pierces the solenoid, you notice a shift in the interference pattern, even though intuition says that the particle is in a region outside the solenoid with no magnetic field and does not experience any force. You can treat the flux enclosed by circling the solenoid as a sort of phase which is essentially like a winding number that appears in the electron wavefunction and causes the shift in the interference pattern.</p>
<p>Differential geometry and trigonometry are very important for physics, however in my career as a professional physicist, I have yet to need the kind of geometry that you spend a lot of time on in high school, that is formal geometric proofs.</p>