lec_week6

lec_week6 - MIT OpenCourseWare http://ocw.mit.edu 18.02...

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Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu 18.02 Multivariable Calculus Fall 2007 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms . 1 2 18.02 Lecture 14. Thu, Oct 11, 2007 Handouts: PS5 solutions, PS6, practice exams 2A and 2B. Non-independent variables. Often we have to deal with non-independent variables, e.g. f ( P, V, T ) where PV = nRT . Question: if g ( x, y, z ) = c then can think of z = z ( x, y ). What are z/x , z/y ? Example: x 2 + yz + z 3 = 8 at (2 , 3 , 1). Take differential: 2 x dx + z dy + ( y + 3 z 2 ) dz = 0, i.e. 4 6 1 6 dy . 4 dx + dy + 6 dz = (constraint g = c ), or dz = So z/x = 4 / 6 = 2 / 3 and dx 1 / 6 (taking the coecients of dx and dy ). Or equivalently: if y is held constant then we substitute dy = to get dz = 4 / 6 dx , so z/x = 4 / 6 = 2 / 3. In general: g ( x, y, z ) = c g x dx + g y dy + g z dz = 0. If y held fixed, get g x dx + g z dz = 0, i.e. dz = g x /g z dx , and z/x = g x /g z ....
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This note was uploaded on 05/04/2011 for the course MATH 18.02 taught by Professor Auroux during the Spring '08 term at MIT.

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lec_week6 - MIT OpenCourseWare http://ocw.mit.edu 18.02...

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