Using Math in Physics: Real Analysis I, II & Complex Variables

[Nusc]

Well I wasted 15 minutes trying to write up a thread and unfortunately an error popped up so I had to close the window so I'll be brief.

Real Analysis I
Real Analysis II
Complex Variables

When does the material from these courses come up in upper division physics courses?

Is a course in statistics recommended for QM?

Mathematical physics courses won't suffice. I want the real deal.


Damn that was short.

 

[Kalimaa23]

What is covered in the real analysis courses?
You'll definitely need a course in complex analysis.

Statistics could be useful in QM, but it depends on how the course is taught. If it includes measure theory I'd say take it, else drop it.

 

[Nusc]

Real Analysis I

Numbers, sets, and functions: induction; supremum, infimum, and completeness; basic set theory; bejective and inverse functions; countable and uncountable sets.

Sequences: convergence, Cauchy sequence, subsequence, Bolzano-Weierstress theorem, limsup, and liminf.

Limits and continuity: basic theorems, intermediate value theorem, extreme value theorem, inverse function theorem, uniform continuity.

Derivative: basic theorems, mean value theorem, Taylor's theorem, trigonometric functions, exponential functions, l'Hopital's rule.

Riemann integral: basic definition and theorems, fundamental theorem of calculus.

Real Analysis II

Series: convergence tests, absolute convergence, conditional convergence, rearrangements, Cauchy product.

Sequences and series of functions: pointwise and uniform convergence, Weierstress M-test, power series.

Euclidean spaces: Basica topology, connectedness, compactness; metric spaces.

Functions of several variables: limits and continuity.

Derivative: linear transformations, differentiability, inverse function theorem, implicit function theorem.



These all sound like content from Calculus I, II, & III but of course I have no idea what is actually taught in the class.

Can you explain why a course in complex analysis is important and what physics courses would it apply in?

I hope that clarifies things...

 

[cronxeh]

This has been covered before in this same section. Basically Real Analysis is a rigorous overview of Calculus and introduces more advanced ideas and introduces the student to proofs and analytical methods. How is this stuff useful for a Physicist? I'm not sure but I do believe that math can not hurt a Physics major, and as far as I know this course will be a prereq for partial diff eq of mathematical physics or something similar sounding - which would include something of this nature:

Partial Differential Equations of
Mathematical Physics
First and second order partial differential
equations and systems of equations. Initial
and boundary value problems. Fundamental
solutions and Green’s functions. Theory of
characteristics. Eigenvalue problems.
Rayleigh-Ritz and Ritz-Galerkin methods.
Approximate and asymptotic methods.
Nonlinear equations. Applications.

As well as

Calculus of Variations:
Extension of elementary theory of maxima
and minima. Euler equations, conditions of
Weierstrass, Legendre, and Jacobi; Mayer
fields; Hamilton-Jacobi equations; transversality;
conjugate and focal points.
Applications to geodesics, minimal surfaces,
isoperimetric problems, Hamilton’s principle,
Fermat’s principle, brachistochrones.

And

Topological Methods in Analysis
Aspects of topological methods and applications
to existence theorems in analysis.
Use of fixed-point theorems and topological
degree to study properties of solutions
to ordinary and partial differential equations.
No previous courses in topology are
required

You will note that Real Analysis 2 is listed as a prerequisite for all 3 of those courses (well at least in my University). Whether you'd like to get to know that material is your call.