Note the DLR mole's last attempt to get to Mars was with the Bealge 2 lander, fingers crossed for this second attempt.
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Q2: " is it just a popular idea that you need to be a mathematician in order to program?" Is it a popular idea? I've certainly never encountered it. A mathematician is not someone who knows a little calculus. Modern mathematics (which, presumably, is studied by a current mathematician) is a wide-ranging and diverse field of which numerical calculus (I propose) is a rather tired and scantly addressed matter. Let's face it, most calculus that you'll probably ever meet would have been cutting-edge say, ooh, 150 years ago (Gauss' theorem of integrating over a surface... to pick an example from the top of my head)
Q3: "What are your experiences?" Programming since age 9, introduced to discrete calculus in programming in second year of Physics BSc (all in FORTRAN), 4 years writing numerical integrator-assisted tools for reusable launch vehicles, Physics MSc (no calculus), Physics PhD (trivial integrators written), postdoc (ditto), self-employed (no calculus).
But hey, why should one study something purely because you hear that it might improve your employment prospects? Do you find it satisfying and interesting? Aren't these reasons enough for studying it or is your schedule really crammed?