Taking the Partial Derivative of a FunctionDate: 09/06/2002 at 04:16:02 From: Robert Curl Subject: Partial derivative Dear Dr. Math, Given an equation, for example, x+y=0, is it true that if we perform the same operation on both sides we could have the same results on both sides? For example, if we add 6 to both sides, and on the left side we have (x+y)+6, and on the right side we have 0+6=6, we know that the results on the left and the right sides are equal, that is, (x+y)+6=0+6. But how about the operation of partial derivative? For the equation x+y=0, if we perform the operation of partial derivative on the left side and let F(x,y)=x+y, we have (partial F)/(partial x)=1 and (partial F)/(partial y)=1, but on the right, we know the result is zero, which is not equal to the result on the left. One must insert the solution, y=y(x)=-x, into the equation; that is, on the left side, F(x,y)=x+y, x=x, y=-x, then dF/dx=0, which is equal to the result obtained on the right. From the above, I think the left and right sides of an equation are not always symmetric. For the operations plus, minus, etc., we could always have same results on both sides after performing the same operation on both sides, but for the operation of derivative, the results on both sides are not always equal. Am I right? I need your help. Thank you very much. Sincerely, Robert Curl Date: 09/06/2002 at 08:46:01 From: Doctor Jerry Subject: Re: Partial derivative Hi Robert, The partial derivative with respect to a variable is an operation that is defined in terms of functions. That is, one takes the partial derivative of a function. If you apply the partial derivative operation to an equation, you are assuming that the equation defines one of the variables in terms of the others. Using x+y=0 as an example and assuming that we want to differentiate with respect to x, we think of y as the function of x that we would find if we could solve this equation for y in terms of x. Differentiating with respect to x then gives (y_x means the derivative of y with respect to x) 1 + y_x = 0 and so y_x = -1 This is correct becasue y = -x and we may differentiate the function f(x,y) = -x with respect to x and find f_x = -1. In this case we are using y as a symbol for f. - Doctor Jerry, The Math Forum http://mathforum.org/dr.math/ Date: 09/06/2002 at 08:47:31 From: Doctor Mitteldorf Subject: Re: Partial derivative Dear Robert, It is easy to get into trouble manipulating symbols according to preset rules, without thinking about what those symbols mean. In this case, the equation x+y=0 might mean that we are considering a function F(x,y) defined on the (x,y) plane to be the sum of the x and y coordinates. Perhaps we are interested in the locus of points on that plane where the function F has the value 0. In this case, it might be a meaningful operation to take the partial derivative of the function F with respect to x; but it wouldn't make sense to differentiate the right side, because the 0 on the right represents a value, not a function. Of course, we might have a different situation, and the same equation x+y=0 might mean something different: perhaps y is tied to x in such a way as to guarantee that x+y is always 0. Then the partial differentiation with respect to x would itself not be a meaningful operation, since y is inextricably tied to x, and x can never be varied while y is held constant. My perspective is that you start with an understanding of a situation in the world, and model that in relations among numbers. Ultimately it is our understanding of the world and of the way that our model represents it that determines what mathematical manipulations are permitted and appropriate. I am a scientist and statistician - not a pure mathematician. Mathematicians might have a different perspective; it is their discipline to define precisely the circumstances under which certain purely formal symbolic manipulations can be performed, with internal consistency in the results. I would be interested to see what a pure mathematician might say about the paradox which you have uncovered. - Doctor Mitteldorf, The Math Forum http://mathforum.org/dr.math/ |
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