Transcribed TextTranscribed Text

5.3. This problem works out a special case of part of the theorem in class that the composition of 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: X = 2 = 2 Z = =f+fxx+fyy + fax2 +2fory+fyxy2 + oll (x,y) | 12) 2 In these formulas I am using abbreviations like this: = &t = For the exercise, you should compose the functions and prove that Z = f + (fx8t + f,ht) + fxx82 + 2fxy8tht + fyrhh2 + fx8tt + fyhup + o(1t12). 2 as t 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 R, that = 1 dx (x) 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 that 0x1 1 dy1 (y) = ? 0y1 (x) 0x1 is not true, not even when the denominator is non-zero.

Solution PreviewSolution Preview

These solutions may offer step-by-step problem-solving explanations or good writing examples that include modern styles of formatting and construction of bibliographies out of text citations and references. Students may use these solutions for personal skill-building and practice. Unethical use is strictly forbidden.

    By purchasing this solution you'll be able to access the following files:

    for this solution

    PayPal, G Pay, ApplePay, Amazon Pay, and all major credit cards accepted.

    Find A Tutor

    View available Real Analysis Tutors

    Get College Homework Help.

    Are you sure you don't want to upload any files?

    Fast tutor response requires as much info as possible.

    Upload a file
    Continue without uploading

    We couldn't find that subject.
    Please select the best match from the list below.

    We'll send you an email right away. If it's not in your inbox, check your spam folder.

    • 1
    • 2
    • 3
    Live Chats