Hey Draco93.
If you are interested in applied mathematics (both for now and possibly the future), I would suggest Numerical Analysis and PDE's.
Complex Variables + Numerical Analysis
Complex Variables + PDEs
Numerical Analysis + PDEs
Intermediate Mechanics + Something
Summer school at my university is 2 quarters long. I'm planning on taking 2 math courses in the first quarter and 2 in the second quarter. Each course (e.g. Complex Variables 1) is 1 quarter long.
Background info: I'm a sophomore/junior applied mathematics major (after summer school, I will be a junior).
Upper division Math courses that I have taken:
Linear Algebra 1
Probability and Statistics 1
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If you were me, what upper division courses would you take in summer?
Math: Complex Variables 1 (Proof based)
Math: Complex Variables 2 (Proof based)
description (both): Complex numbers, functions, differentiability, series extensions of elementary functions, complex integration, calculus of residues, conformal maps, mapping functions, applications.
Math: Numerical Analysis 1
description: Numerical methods for the solution of nonlinear equations (Newton method), for integration (quadrature formulas and composite integration), and for the initial value problem for ordinary differential equations (Euler and Kutta methods).
Math: Numerical Analysis 2
description: Numerical methods for the solution of systems of linear equations (direct and iteractive methods), and the finite difference methods for boundary value problems for (ordinary and partial) differential equations.
Math: Partial Differential Equations 1
description: Wave, heat, and potential equations.
Math: Partial Differential Equations 2
description: Fourier series; generalized functions; and numerical methods.
Physics: Intermediate Mechanics 1
Physics: Intermediate Mechanics 2
description: Newtonian mechanics in 3D. Conservation laws. Simple harmonic oscillator (undamped, damped, driven and nonlinearities). Central force motion (orbits, Kepler’s laws). Collisions (elastic, inelastic, scattering, cross sections). Special relativity.
I plan on taking courses consecutively, so if I am taking PDE 1 in summer, then I will take PDE 2 as well in summer. So basically pick 4 courses out of the 8. I'm not sure if I should take Complex Variables right now.