<p>A math major (Ben Golub or neapol1s) would probably be better at answering this, but I’ll give it a shot: </p>
<p>Caltech’s math placement exam covers all of core math: </p>
<p>Math 1a: Single variable calculus
Math 1b: Linear algebra
Math 1c: Multi-variable calculus
Math 2a: Differential Equations
Math 2b: Probability and Statistics</p>
<p>For a more thorough look at differential equations, you would want to take ACM 95 abc (Complex analysis, ODE’s, and PDE’s, respectively). Math majors usually don’t take this class, but many other majors do (physics, geology, and most engineering majors). This class takes up a lot of time, and it is very rare that freshmen take it.</p>
<p>The undergraduate analysis class is Math 108abc (classical analysis), typically taken during a math major’s sophomore or junior year. I don’t know if any freshmen take it, but I do know there are some courses freshmen interested in math do take, like Math 3 (beginning number theory), 4 (introduction to mathematical chaos), 5 (introduction to abstract algebra), and 6(introduction to discrete mathematics). </p>
<p>Between the year-long courses, Math 5 and 6, more people take Math 5 as a freshman. However, you don’t need to pass out of anything to take either course. </p>
<p>If you’re interested in physics or an area of science or engineering that is not just pure math, you may be interested in the Physics 129 abc course. Ph 129 a covers analysis, 129 b covers group theory, and 129 c covers probability and statistics in physics (stuff like Monte Carlo methods, etc.)</p>
<p>I’m not quite sure what exactly you’re looking for in an analysis course, but ACM 95 might be your best bet as a combination of complex analysis and differential equations. For more information on any of the courses I mentioned, here are the course listings: <a href=“http://pr.caltech.edu/catalog/courses/courses.html[/url]”>http://pr.caltech.edu/catalog/courses/courses.html</a></p>