Lessons I wish I had learned before I started teaching differential equations [pdf] (1997)
&gt; 7. stay away from differentials<p>I always had issues with this. My undergraduate math classes were very formal and well thought, every object was defined, almost all theorems were proved in class.<p>In particular, we were taught differential equations theories, some general theories like systems of differential equations with constant coefficients, Cauchy&#x27;s theorem, and a bunch of recipes.<p>In a separate class, we were also taught differential forms (e.g. a.dx + b.dy being a linear combination of base elements dx and dy - if I remember well).<p>But then came the physics class, why we could go from dy&#x2F;dx = f(x) to dy = f(x).dx and integrate both sides remain a mystery. &quot;dx&quot; suddenly became an infinitely small things, when it used to be a notation for a linear form OR a notation for f&#x27;(x) === df&#x2F;dx.<p>Generally speaking, I had a lot of issues in physics because the maths used in that context were not formally defined and seemed to obey to some unwritten set of rules that you were supposed to find out by yourself during your final exam...