505 words on Uni
I am not the biggest fan of Analysis but it has the bright side that it starts off fiddly with all the epsilons around and becomes clearer and more elegant later on. Somehow the good structures in there are just hidden in the simple cases. In the one dimensional case which you teach to freshers, for example, the very structure of what a derivate is and does just collapses and it becomes impossible to see what’s really going on because there are just numbers everyhwere.
I put the main blame for this on the fact that linear maps from a field to itself are nothing but multiplication by a number of the field. And this set of linear maps is the field itself. In a way I am tempted to put the blame for student not fully understanding derivatives on the isomorphism
which is hiding everywhere. You tell your freshers the definition of a derivative and you also tell them that it’s a linear approximation of the function at that point. But give them a week and they’ll have forgotten about all that linear nonsense and just think about a derivative as just another function that spits out numbers. However, these numbers are really to be thought of as linear maps, i.e. you get a linear map for each point of the function’s domain. That linear map just happens to be writable as a number. But people need to reach their second semester and learn about multi-dimensional differentiation to really see that.
This misconception really hurts when you look at the chain rule for taking derivatives. In the one dimensional case it is tempting (and working) to think of it in terms of ‘inner derivative times outer derivative’. However, keeping in mind the idea of linear maps, it may be conceptually better to think of the situation as follows: at each point of the domain you get a number which corresponds to the linear map that is the composition of the linear map which is the derivative of the ‘outer’ function at the result of the inner function of that point composed with the derivative of the ‘inner’ function at that point.
In formulas this means that conceptually you see better whats going on if you don’t write (f∘g)′(x) = f′(g(x))g′(x) but instead write Dx(f∘g) = Dg(x)f ∘ Dxg.
Of course it seems a bit crazy to use all those complicated words and notation for something that when doing the computations in a one dimensional exercise boils down to multiplying two numbers. It’s not that the simplified version is wrong. However, the ‘right’ idea, the idea that can be generalised to higher dimensional cases will remain hidden when using the simplified notation.
The ‘better’ notation also highlights that it’s really a good idea to not think about a derivative as a function which you plug points of the domain into. It’s more a collection of linear maps for each point of the domain. That could be another rant…
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