Partial DerivativesDate: 03/24/2001 at 21:54:33 From: Neeraj Subject: Partial derivatives meaningless? Hi, We're currently studying partial derivatives. I am unable to determine what sense partial derivatives make in the following case: u is given as a function of two variables, say x and y. Also, y is itself a function of x. What does the partial derivative D_u/D_y mean? According to my reasoning, it is meaningless: To find u/y, we keep the other variables of u constant. Here, we have only one other variable, namely x. So, I keep x constant. But if x is constant, then y is constant (as y depends on x). But if y is constant then what sense does D_u/D_y make? After all, D_u/D_y determines the change in u when y is changed. Please help. Yours sincerely, Neeraj. Date: 03/24/2001 at 22:36:07 From: Doctor Peterson Subject: Re: Partial derivatives meaningless? Hi, Neeraj. In order to understand this, you have to focus on one thing at a time. You have been given a function u(x,y), and also a function y = f(x). These are separate things. When you take the partial derivatives of u, the function f is irrelevant; you are only considering how u depends on x and y, not how y may depend on x. As far as u is concerned, x and y could have any relation, or none. They are simply independent variables when we look at u. You might want to think of it this way. The equation z = u(x,y) defines a surface. The function y = f(x) defines a curve on the x-y plane, or a cylinder (in the generalized sense) in space, parallel to the z axis. Putting the two together, z = u(x, f(x)) and y = f(x) together determine a curve in space, where the cylinder cuts the surface. But even if we know that f exists, we can still think about u as if x and y were entirely independent, and that is what we do whenever we work with the function u alone. This does suggest some cautions when you work with such situations. Suppose z = u(x,y) = x^2 + y^2, and y = f(x) = 3x. Then, ignoring f, we can say that D_u/D_y = 2y; and we can say that when x = 2, this is 2f(2) = 12. But we can't substitute y = f(x) for y and say z = x^2 + (3x)^2 = 10x^2 and THEN ask for the partial derivative of z with respect to y. Once we have combined the functions, we no longer have any dependence on y; we have eliminated that variable. We can only talk about the partial derivative of the function u, not of the variable z in the whole context. As often happens in calculus, we must keep in mind what function and what variables are under consideration. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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