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

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MIT OpenCourseWare http://ocw.mit.edu 18.02 Multivariable Calculus Fall 200 7 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
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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 = 0 (constraint g = c ), or dz = So ∂z/∂x = 4 / 6 = 2 / 3 and dx 1 / 6 (taking the coefficients of dx and dy ). Or equivalently: if y is held constant then we substitute dy = 0 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 . Warning: notation can be dangerous! For example: f ( x, y ) = x + y , ∂f/∂x = 1. Change of variables x = u , y = u + v then f = 2 u + v , ∂f/∂u = 2. x = u but ∂f/∂x = ∂f/∂u !!
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