The trouble with differentials

There has recently been a bit of discussion about a "flaw in basic calculus" (links are below). No worries, there is no flaw in calculus! As usual, the confusion comes from over interpreting notation, in this case abuse of ``differentials". We write dy/dx to denote y'(x), the derivative of a function y(x). We also write d²y/dx2 to denote the second derivative of y.
The notation dy/dx is often overstretched:

• An example is when solving differential equations using the method of seperation of variables. Like dy/dx = x²/y³, then y³ dy = x² dx which gives y⁴/4 = x³/3 + C which gives y=(4x²/3 +C)^(1/4). As you can see when teaching Unit 20, the class objected a bit to the abuse of notation when using separation of variables.
  
• A second example is when trying to catch the notion of differential forms (as we have seen in 22a). One writes F = P dx + Q dy for example. Of course, the dx and dy have a precise meaning there as differential forms (linear maps from the tangent space at a point to the reals). The notion dxdy then is an anti-symmetric bi-linear map.
• One place, where the notation of differentials is often overreached is when looking at implicit differentiation or more generally at related rates. In the case of related rates, the variables x and y are actually functions of an other variable t. So, one looks at y(t)/x(t). Now, dy/dx actually can mean (dy(t)/dt)/(dx(t)/dt). In this framework, which actually only understood in a multi-variable set-up one has by the product rule (lets write x' for dx/dt d²y/dt² = d/dt ( dy/dx x') = d^2y/dx^2 x'² + dy/dx x'' This is the chain rule formula for the second derivative mentioned as formula (1) in the paper of Bartlett and Khurshudyan. The paper then goes on explaining the difficulty of pushing differentials too much.

In any way, the story illustrates and confirms that ``everybody hates related rates" The upshot is something we stress a lot in calculus courses. It is better to avoid ``differentials". It is a term which is abused (in that sense, the headline "a flaw in calculus" is true). Many texts abuse differentials. Actually, one can do very well without. Both in single variable calculus and multivariable calculus, one can avoid differentials by just looking at linearizations. The notion of differentials comes back in a precise way in more advanced courses (like 22a) when looking at differential forms.

• Slashdot:
• Blog
• Bartlett and Khurshudyan