session 42 problem:non-independent variables
3.Now suppose w is as above and x^2y +y^2x = 1. Assuming x is the independent variable, find ∂w/∂x.
The answer is here:http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/2.-partial-derivatives/part-c-lagrange-multipliers-and-constrained-differentials/session-42-constrained-differentials/MIT18_02SC_pb_42_comb.pdf
I don't understand why we should set z = 0.
OCW Scholar - Multivariable Calculus
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Good question zane120000. We have a three variable equation, x, y, and z. We are told that x and y are related by a constraint equation. That means that any change in x produces a known change in y. The variable y is therefore totally dependent on x. This reasoning is reversible and x could be totally dependent on changes in y, but we are told in the question to assume that it is y that is dependent.
But that leaves a third variable hanging there, z. What do we do with that z? Well, we just want to know what the rate of change is for w when x changes. If we allow z to have a value that changes, this corrupts our calculation. We only want to know how w changes when x changes. Therefore we zero out any independent variables other than x.
Thank you for your response,Waynex! I got a deeper understanding with your help! Then I still have a question:When z have a different value from 0,will ∂w/∂x differ?Or will ∂w/∂x varies when a point moves vertically in cartesian coefficient?
That's a great point. And I don't know what the answer is supposed to be, but I did take a look at that while I was going through that pdf. If we did set z to a value other than 0, the x and y variables would show up in more places and the value does seem like it would change. Would that change be linear? If so, then the change is not as interesting as if might be if it was not linear.
I suspect that this is like calculating a single variable derivative at a specific point. For instance, if we calculate the derivative of 2x^3=y, we get 6x^2=dy. Perhaps we want to know the rate of change in y at the specific x location of 1, then we get 6(1^2)=dy. Similarly, we could ask what the partial of w with respect to partial x is when z is fixed to 5, and plug in 5 for z.