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Last time we saw things about gradients and directional derivatives. Before that we studied how to look for minima and maxima of functions of several variables. And today we are going to look again at min/max problems but in a different setting, namely, one for variables that are not independent. And so what we will see is you may have heard of Lagrange multipliers. And this is the one point in the term when I can shine with my French accent and say Lagrange's name properly. OK. What are Lagrange multipliers about? Well, the goal is to minimize or maximize a function of several variables. Let's say, for example, f of x, y, z, but where these variables are no longer independent. They are not independent. That means that there is a relation between them. The relation is maybe some equation of the form g of x, y, z equals some constant. You take the relation between x, y, z, you call that g and that gives you the constraint. And your goal is to minimize f only of tho...

PROFESSOR: One of the things you will be thinking about in the homework is the definition of gauge-invariant observables. So an operator O is a gauge-invariant observable-- gauge environment observable-- if, see, under a gauged transformation, O can change. The operator can change under a gauged transformation. You'd say, oh, but isn't it supposed to be gauge-invariant? But, well, it's not quite the operator itself that is gauge-invariant. It is its expectation value. So you should have that psi prime of O prime psi prime is equal to psi O psi. So the operator may change. And that's OK. The wave function will change as well. So the question is that if you put the new wave function and the new operator and you find the expectation value, do you get the same thing? That's what you really can ask. You cannot ask more than that. And you will see very funny things when you do this. You will see that this operator, P, the momentum operator that you like, tha...