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.
b)
 ${\frac {\partial ^{2}z}{\partial t^{2}}}$

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?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint

You should use your answer from part (a) to solve this problem:
 ${\frac {\partial ^{2}z}{\partial t^{2}}}={\frac {\partial }{\partial t}}\left({\frac {\partial z}{\partial t}}\right).$

Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.
 If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work.
 If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies.
To evaluate this derivative we merely need to recognize that we apply the first derivative operator to our answer from part (a):
 ${\frac {\partial ^{2}z}{\partial t^{2}}}={\frac {\partial }{\partial t}}\left({\frac {\partial z}{\partial t}}\right).$
We need to apply the chain rule again as in part (a):
 ${\begin{aligned}{\frac {\partial ^{2}z}{\partial t^{2}}}={\frac {\partial }{\partial t}}\left({\frac {\partial z}{\partial t}}\right)&={\frac {\partial }{\partial x}}\left({\frac {\partial z}{\partial t}}\right){\frac {\partial x}{\partial t}}+{\frac {\partial }{\partial y}}\left({\frac {\partial z}{\partial t}}\right){\frac {\partial y}{\partial t}}\\&={\frac {\partial }{\partial x}}\left({\frac {\partial z}{\partial t}}\right)\cdot (r\sin(t))+{\frac {\partial }{\partial y}}\left({\frac {\partial z}{\partial t}}\right)\cdot r\cos(t)\\&=y{\frac {\partial }{\partial x}}\left({\frac {\partial z}{\partial t}}\right)+x{\frac {\partial }{\partial y}}\left({\frac {\partial z}{\partial t}}\right).\end{aligned}}$
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
 ${\begin{aligned}{\frac {\partial z}{\partial t}}&={\frac {\partial f}{\partial x}}\cdot (r\sin(t))+{\frac {\partial f}{\partial y}}\cdot r\cos(t)\\&=y{\frac {\partial f}{\partial x}}+x{\frac {\partial f}{\partial y}}.\end{aligned}}$
Using this, we continue to evaluate the partial derivatives above
 ${\begin{aligned}{\frac {\partial }{\partial x}}\left({\frac {\partial z}{\partial t}}\right)&=y{\frac {\partial ^{2}f}{\partial x^{2}}}+{\frac {\partial f}{\partial y}}+x{\frac {\partial ^{2}f}{\partial y\partial x}}\\{\frac {\partial }{\partial y}}\left({\frac {\partial z}{\partial t}}\right)&={\frac {\partial f}{\partial x}}y{\frac {\partial ^{2}f}{\partial y\partial x}}+x{\frac {\partial ^{2}f}{\partial y^{2}}}\end{aligned}}$
finally giving us
 ${\begin{aligned}{\frac {\partial ^{2}z}{\partial t^{2}}}&=y{\frac {\partial }{\partial x}}\left({\frac {\partial z}{\partial t}}\right)+x{\frac {\partial }{\partial y}}\left({\frac {\partial z}{\partial t}}\right)\\&=y\left(y{\frac {\partial ^{2}f}{\partial x^{2}}}+{\frac {\partial f}{\partial y}}+x{\frac {\partial ^{2}f}{\partial y\partial x}}\right)+x\left({\frac {\partial f}{\partial x}}y{\frac {\partial ^{2}f}{\partial y\partial x}}+x{\frac {\partial ^{2}f}{\partial y^{2}}}\right)\\&=y^{2}{\frac {\partial ^{2}f}{\partial x^{2}}}+x^{2}{\frac {\partial ^{2}f}{\partial y^{2}}}2xy{\frac {\partial ^{2}f}{\partial y\partial x}}y{\frac {\partial f}{\partial y}}x{\frac {\partial f}{\partial x}}\end{aligned}}$
Writing the result above in terms of r, t gives the final answer:
 ${\color {blue}{\frac {\partial ^{2}z}{\partial t^{2}}}=r^{2}\sin ^{2}(t){\frac {\partial ^{2}f}{\partial x^{2}}}+r^{2}\cos ^{2}(t){\frac {\partial ^{2}f}{\partial y^{2}}}2r^{2}\cos(t)\sin(t){\frac {\partial ^{2}f}{\partial y\partial x}}r\sin(t){\frac {\partial f}{\partial y}}r\cos(t){\frac {\partial f}{\partial x}}}$

Click here for similar questions
MER QGH flag, MER QGQ flag, MER QGS flag, MER RT flag, MER Tag Partial derivative