I have a simple question about differentials. I have been taught two ways to find the differential and my questions is in what situations do I use each one? simply speaking these are the 2 ways 1.) just take the partials of each component function and throw them in a matrix 2.) Let f be the function we want the differential for, where f:S→ℝm. You choose a curve [itex]\alpha[/itex](-ε,ε)→S such that [itex]\alpha[/itex](0)=p and [itex]\alpha[/itex]'(0)= v where p is in S and S is a surface, and v is in TpS. Then you compose f with [itex]\alpha[/itex] and take the derivative with respect to t. Thank you for your time.