how to be an expert in programming

Of course, mathematics is desperately needed.

Most software developers know nearly nothing about it, but does it mean that you don't need to learn it. The lack of mathematical knowledge haunts the whole industry for years; developers fail to use proper unambiguous language to conduct simple ideas and requirements, to start with. I dare to say that the biggest gap between desirable and typical knowledge of engineers is the biggest in software development. Something should be done about it.

There is a historical problem that traditionally, most engineers learned calculus, which was mostly important for mechanical, electrical engineering, classical and big part or quantum physics, etc. By the way, calculus is not considered to be a serious part of mathematics by serious mathematicians. From the perspective of the level of strictness customary for modern mathematics, it is not strict enough, and from the perspective of the fundamental results, mathematicians often consider it as lacking a subject.

Now, computer science and practical development needs different fiend of mathematics. Not that it is more "advanced", but it some aspects it can be considered more fundamental, and in most places, practically simpler. However, there are many very difficult unsolved problems, very important practically. If is usually not feasible for an average but well educated software developer to delve into it, but knowledge of the basics is really important.

Now, please see my overview of some mathematical fields as I understand them, in my past answer:

Math needed in programming logics[^].

—SA

It depends what you want to do with it. If you want to go into something like 3D graphics or simulation you'll need to be able to deal with linear algebra and calculus. If you go into business-type software you can probably get away with just basic arithmetic, though some discrete mathematics is generally useful in software regardless.

The real connection between the two (in my opinion) is the ability to problem solve. If you struggle with simple word problems you probably won't be able to take requirements and turn them into working software, and you won't make it far. It's not a math skill per se, but the two are often presented together.

And personally, I'd actually say my Physics classes were more useful than math classes, because it taught me how to think and apply math I already knew to problems, which can easily be extended to using programming concepts to solve a programming problem.