So, we had to do a thing in my Fundamentals of Engineering I class about how we got into STEM. I put down the parts about being singled-out by my Calc III TA as a good potential math major, that he cautioned my against engineering, and how, unlike most mechanical engineering majors who took things apart to see how they worked as kids, I've instead always been interested in "mapping out" the transfer functions of systems and learning how to "harmonize" with the systems, providing the proper inputs to get the system to do what I want -- to "dance like a marionette on a string". But, what I left out (principally due to time) is the thing that initially got me interested in Calc III and multi-dimensional math in the first place.
So, I had Calc I my junior year of high schhool,and I was the only junior in the first Calc I class of seniors at my high school. I was also the captain of the soccer team, and one of the things the guys on the soccer team would do for fun was to have "banana kick competitions": we would see how far we could line the ball up behind the bye line ("goal line") near the corner kick spot and still "banana kick" the ball into the goal; the person who scored a goal from furthest behind the bye line won.
So, I was concurrently enrolled in Calc I, which is 1D, and tried to come up with mathematical models of how the ball would curve in the air as it "banana'ed" into the goal mouth, but, try as I might, I couldn't even model the spin on the ball. Then, one of the first weeks into Calc III, we went over angular velocity, and all the pieces clicked into place: you take the ball's spin, find the plane through the center of the ball through which it's spinning, take a vector perpendicular to that plane, curl your RIGHT hand's fingers in the plane of motion in the direction of motion, and your thumb points in the direction of the angular velocity vector; the faster the ball is spinning, the longer the length of the angular velocity vector. This just solved so many problems in my mind, and really cleared up my muddled thinking!
I decided then and there that I wanted to devote my life to STEM, and seeing how many of the other "modeling" problems I was having difficulty conceptualizing they could teach about how to think properly. Maybe that wasn't the most intellighent career choice decision process in the history of the world, but, to my 18-year-old mind, it made sense.
So, that train of showing how to properly "think about" or model phenomena has never slowed down. Learning what manifolds are was a real water-shed moment for me: there are SO MANY physical phenomena and motion problems that are most accurately modeled as motion on a manifold, it's mind-boggling; it's still fascinating to me to model some random physical system as motion on a manifold, then implement control engineering on the system to get it to do what I want -- to "dance like a marionette on a string" -- that I do it for fun in my down-time, independent of whether I'm being paid to model or control that system in any capacity.
So, that's it: that's what got me into STEM. I'm happy as a clam to be working in robotics control engineering -- I literally despise geometric group theory and even PL manifold topology as being completely orthogonal to my research interests -- and I'm really looking forward to finishing my mechanical engineering degree and doing robotics control engineering for a living in real life.