5.3. This problem works out a special case of part of the theorem in class that the composition
two Cr functions is also Cr. Let
Z = f(x,y) (x,y)=(8(t),h(t)),
where t, X, y, and Z are all scalar variables. Suppose that all three of the functions f(x,y),
g(t), and h(t) are C², , and that g(0) = h(0) = 0. Since the functions are C², we obtain these
power series at the origin:
Z = =f+fxx+fyy + fax2 +2fory+fyxy2 + oll
(x,y) | 12)
In these formulas I am using abbreviations like this:
For the exercise, you should compose the functions and prove that
Z = f + (fx8t + f,ht) + fxx82 + 2fxy8tht + fyrhh2 + fx8tt + fyhup + o(1t12).
0. Many terms are swept up into the total error o(1+12) at the end. Prove that these
terms are in fact o(1t12). (Note: You do not have to list every term separately in your reason-
ing. Instead, you can group the terms that disappear for the same reason.)
5.4. Recall one formulation of the inverse rule for a differentiable function f : R
when y = f (x) and the denominator on the right is not zero.
Let U², , V² C R 2 be open sets, let F : U² V² be a C¹ function with a C¹ inverse F-1:V²
U², , and let y = f(x) with X = (x1,x2) and y = (y1,y2). Find an example of all of that such
dy1 (y) = ? 0y1 (x)
is not true, not even when the denominator is non-zero.
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