lec_week13

lec_week13 - MIT OpenCourseWare http://ocw.mit.edu 18.02...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 . 18.02 Lecture 30. – Tue, Nov 27, 2007 Handouts: practice exams 4A and 4B. Clarification from end of last lecture: we derived the diffusion equation from 2 inputs. u = concentration, F = flow, satisfy: 1) from physics: F = − k u , 2) from divergence theorem: ∂u/∂t = − div F . Combining, we get the diffusion equation: ∂u/∂t = − div F = + k div ( u ) = k 2 u . Line integrals in space. Force field F = P, Q, R , curve C in space, dr = dx, dy, dz Work = C F dr = C P dx + Q dy + R dz . ⇒ · Example: F = yz, xz, xy . C : x = t 3 , y = t 2 , z = t . 0 ≤ t ≤ 1. Then dx = 3 t 2 dt , dy = 2 tdt , dz = dt and substitute: 1 F dr = yzdx + xzdy + xydz = 6 t 5 dt = 1 · C C (In general, express ( x, y, z ) in terms of a single parameter: 1 degree of freedom) Same F , curve C = segments from (0 , , 0) to (1 , , 0) to (1 , 1 , 0) to (1 , 1 , 1). In the xy-plane, z = 0 = F = xy k ˆ , so F dr = 0, no work. For the last segment, x = y = 1, dx = dy = 0, so ⇒ · 1 F = z, z, 1 and dr = , , dz . We get 1 dz = 1 . Both give the same answer because F is conservative, in fact F = ( xyz ). Recall the fundamental theorem of calculus for line integrals: P 1 f dr = f ( P 1 ) − f ( P ) . · P Gradient fields. F = P, Q, R = f x , f y , f z ? Then f xy = f yx , f xz = f zx , f yz = f zy , so P y = Q x , P z = R x , Q z = R y . Theorem: F is a gradient field if and only if these equalities hold (assuming defined in whole space or simply connected region) Example: for which a, b is axy ˆ ı + ( x 2 + z 3 )ˆ j + ( byz 2 − 4 z 3 ) k ˆ a gradient field? P y = ax = 2 x = Q x so a = 2; P z = 0 = 0 = R x ; Q z = 3 z 2 = bz 2 = R y so b = 3. Systematic method to find a potential: (carried out on above example) f x = 2 xy , f y = x 2 + z 3 , f z = 3 yz 2 − 4 z 3 : f x = 2 xy f ( x, y, z ) = x 2 y + g ( y, z ). ⇒ f y = x 2 + g y = x 2 + z 3 g y = z 3 g ( y, z ) = yz 3 + h ( z ), and f = x 2 y + yz 3 + h ( z ). ⇒ ⇒ f z = 3 yz 2 + h ( z ) = 3 yz 2 − 4 z 3 ⇒ h ( z ) = − P 1 4 z 3 ⇒ h ( z ) = − z...
View Full Document

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.

Page1 / 5

lec_week13 - MIT OpenCourseWare http://ocw.mit.edu 18.02...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online