Calculus: Vectors Line Integrals, etc. going forward to upper level math classes?

Question

With the exception of engineering, where would one likely see things related to Vector Calculus, line integrals, conservative fields, Curl & Divergence, Green & Stokes theorem, etc?

I am going to major in probability/statistics and am planning on reviewing the things from Calculus that will be most pertinent to the upper level math classes I am taking. This includes differential equations and that is probably the class I most want to know about with respect to what I mentioned.

I hope this question is clear enough.

Most of my other math classes will be related to stats.

I would appreciate as detailed an answer as you can give me and 10 points will go to the one with the most specific details.

Answer 1

All of those Calc III topics show up a lot in Physics, especially Electricity and Magnetism, but since you aren't going into that you will probably never see those topics again.



The only calc III that showed up in Differential Equations for me is the partial derivatives in Exact DEs and Partial DEs; you won't be seeing any vectors in this course. This course is 90% calc II since you are constantly integrating to find the solution to DEs.



Prob/Stats is a very different kind of math (though very important) - it's a different way of thinking and I found it extremely difficult. However it's probably the only branch in math that can get you a job such as an actuary. Double integrals is the only Calc III I remember in Probability, namely when doing continuous random variables. The calculus is the easiest part in probability. The tough part is interpreting situations and setting up the integrals correctly - that takes skill.



The only other math course I had that contained those topics like line integrals and green's theorem is Complex Variables, but you probably won't need that for probability.

Answer 2

I would say Honors and AP but the school you are describing sounds weird. Calculus and Trigonometry are usually what colleges want a student to have completed. Many take geometry early and don't consider it that difficult. However, I found it harder than Trig but I'm weird when it comes to math. The best suggestion I have is to discuss your concerns with the school's guidance counselor as they would have a better idea. The other option is just to have your son take as many math classes as possible, but I would definitely recommend Trig and Calc.

Answer 3

All of those things will indeed be covered in differential equations. I just took differential equations in the spring; I'll give you a list of the chapters from my book, and include the more important subtopics, so you can see what's covered:



Introduction

---Initial value problems

---Direction fields

---Euler's method

First order differential equations

---Separable equations

---Linear equations

---Exact equations

---Integrating factors

Mathematical models and numerical methods involving first order equations

---Compartmental analysis (also known as tank problems)

---Newtonian mechanics

---Improved Euler method

---Runge-Kutta and Taylor methods

Linear second-order equations

---Homogeneous linear equations

---Auxiliary equations with complex roots

---Nonhomogenous equations: the method of undetermined coefficients

---The superposition principle

---Variation of parameters

Introduction to systems and phase plane analysis

---Elimination method for systems with consistent coefficients

---Solving systems and higher order equations numerically

---Introduction to the phase plane

---Coupled mass spring systems

Theory of higher order linear differential equations

---Basic theory of linear differential equations

---Homogeneous linear equations with constant coefficients

Laplace transforms

---Definition and properties of the Laplace transform

---Inverse Laplace transform

---Solving initial value problems

---Transforms of discontinuous and periodic functions

---Impulses and the Dirac delta function

Series solutions of differential equations

---Cauchy Euler equations

Matrix methods for linear systems (my class skipped this chapter)

Partial differential equations

---Method of separation of variables

---Fourier series

---Fourier cosine and sine series

---The heat equation

---The wave equation

---Laplace's equation



I hope this is something like what you were looking for. Looking back, I think though that most of the topics you mentioned must have been covered in calculus 3 (I remember learning all those topics but apparently I didn't learn them in differential equations). So I'm not sure I had to type all that out...oh well.