Solving Implicit Functions: F1(x), F2(x) near 0

In summary, the conversation is about a problem involving determining functions and computing partial derivatives. The problem statement involves an equation with two functions, F1(x) and F2(x), and the task is to determine if they implicitly determine functions near 0. The solution involves computing the determinant of a matrix and using a formula for the derivative given in class notes. There is some confusion about whether the derivative should be evaluated at x2 = 0 or kept in terms of phi1(x2), x2, phi2(x2). In the end, the conclusion is that if the calculations are correct, the derivative will be (0,0) at x2 = 0.
  • #1
I thought I had this but in doing the problem I realized I didn't (or maybe I didn't).
One of the problems was that in class and notes all examples were done in terms of f(x,y). Obviously in h/w, the problem is given as f(x1,x2,x3) - just to confuse me !

Problem statement

a- For the following equation, decide whether they implicitly determine functions near 0
b- If the equation implicitly determine a function, compute the partial derivatives at 0


F1(x) = ( x1[tex]^{2}[/tex] + x2[tex]^{2}[/tex] + x3[tex]^{2}[/tex] )[tex]^{3}[/tex] - x1 + x3 = 0;
F2(x) = cos (x1[tex]^{2}[/tex] + x2[tex]^{4}[/tex]) + exp(x3) - 2 = 0


For Part a-

I know I have to compute the determinant of a matrix. So for this example I played out with x1,x2,x3 and saw which determinant would not be 0. I came up with the following matrix.

B(x) = [partial_der_F1 (x1) partial_der_F1 (x3) ; partial_der_F2(x1) partial_der_F2(x3) ]. Evaluating this at 0, I get the following matrix

[-1 1; 0 1] and the determinant of this matrix is -1 != 0. So I can take the inverse of this matrix

Thus I can say that for each x2, the following function F(phi1(x2), x2, phi2(x2) ) = 0 where phi1 and phi2 are unique functions.

I think I have part a right. It is part b I am having problems with

For part b:

The formula for the derivative given in class notes is -B(x,g(x))[tex]^{-1}[/tex] A(x,g(x))

B = matrix calculated above
A = partial derivative of F1, F2 (2x1 matrix) with respect to x2.

Now my question is do I have to evaluate this at x2 = 0 or can I just leave it in terms of phi1(x2), x2, phi2(x2).

Whats confusing me is that if I compute A wrt x2 and evaluate at x2=0, I will get a 0 matrix.

Thanks in advance

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  • #2
I looked over the baby Rudin treatment of this theorem (pg. 224-228) and my conclusion is that the formula given for a derivative in your class notes holds when evaluated at the point. Thus [tex]\left(\phi_1^\prime (0),\phi_2^\prime (0)\right)=(0,0)[/tex] is my conclusion if your calculations are correct.

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