Formalism: easy and useful vs. correct

A common problem in teaching is, I think, telling the [whole] truth vs. actually teaching something.  I have two examples of this problem that have been gnawing at my math conscience for a while:

Separable differential equations:

This is right out of the first day of an ODE course.  We look at a separable differential equation which, I am sure you recall, looks something like y' = xy, then write it using Leibniz notation (which is more precise, at the expense of being somewhat intimidating… a topic for another time!) as

\frac{dy}{dx} = xy.

We collect all the terms involving y on one side, and all the terms involving x on the other (the ability to do so may be taken as the definition of a separable equation):

\frac{dy}{y} = x~dx,

then integrate

\int \frac{dy}{y} = \int x~dx \Rightarrow \log{y} = \frac{x^2}{2}+C \Rightarrow y = Ae^{x^2}.

Now, dy/dx is not actually a fraction, and on the one hand, you hope a student calls you out on this.  But it is a useful formalism that, as far as I can tell, everyone uses.  The true derivation of the above solution should go:

\frac{1}{y}y' = x \Rightarrow \int \frac{1}{y} y'~dx = \int x~dx \Rightarrow \int \frac{1}{u}~du = \int x~dx,

then we proceed at usual.  I am using the variable u in the third equation to emphasize the change of variables u = y(x), so du = y'(x)~dx (which again is some formalism that I will duck for now).

In practice, I will not present the second derivation, since the point of the lecture is to learn how to solve separable equations, not to practice changing variables.  However, I will encourage students to call out the hand-waving in the first derivation.  In a perfect world, I would hope students could justify everything that happens in class, but the last thing I would want to do on the first day of classes is overwhelm students for my mathematical conscience.

Eigenvalues and the trace/determinant:

The other situation where I have often felt mathematical-moral-twinges is when eigenvalues make an appearance in a linear algebra course.  Two enormously useful identities for checking that you, at least, might have gotten the eigenvalues right is that the product of the eigenvalues is equal to the determinant of a matrix, and the sum of the eigenvalues is the trace.  The proof is not hard (namely, that a matrix is similar to its Jordan normal form, P^{-1}AP = J, where the diagonal of J is the [are the?] eigenvalues of A, and then recalling that the determinant of a product is a product of determinants, so det(A) = det(J), and that the trace of similar matrices is the same.)  However, Jordan normal form comes, necessarily, later than eigenvalues, and I have never had time to go back to present a complete proof.

I have given students this identity with a “cheat code” caveat in the past.  That is to say, I will not test them on it, and they cannot use it as justification for calculations, but I point out it would be foolish not to use it to double check calculations: if they get an eigenvalue calculation wrong on the final, I’m not mad, I’m just disappointed.

 

I am sure there are other examples of topics where this is dealt with, but these have bothered me the most!

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