MATH200 December 2011
• Q1 (a) • Q1 (b) • Q1 (c) • Q2 (a) • Q2 (b) • Q3 (a) • Q3 (b) • Q4 • Q5 (a) • Q5 (b) • Q6 • Q7 • Q8 (a) • Q8 (b) i • Q8 (b) ii • Q8 (b) iii •
Question 02 (b)
Suppose z = f(x,y) has continuous second order partial derivatives and x = r cos(t), y = r sin(t). Express the following partial derivatives in terms of r,t and the partial derivatives of f.
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?
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You should use your answer from part (a) to solve this problem:
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To evaluate this derivative we merely need to recognize that we apply the first derivative operator to our answer from part (a):
We need to apply the chain rule again as in part (a):
where we have use the fact that x = r cos(t), y = r sin(t) to write the last equation above. From part (a), we found that
Using this, we continue to evaluate the partial derivatives above
finally giving us
Writing the result above in terms of r, t gives the final answer:
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MER QGH flag, MER QGQ flag, MER QGS flag, MER RT flag, MER Tag Partial derivative