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lec_week5

# lec_week5 - MIT OpenCourseWare http/ocw.mit.edu 18.02...

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18.02 Lecture 11. Tue, Oct 2, 2007 Diﬀerentials. Recall in single variable calculus: y = f ( x ) dy = f ( x ) dx . Example: y = sin 1 ( x ) x = sin y , 2 dx = cos y dy , so dy/dx = 1 / cos y = 1 / 1 x . Total diﬀerential: f = f ( x, y, z ) df = f x dx + f y dy + f z dz . This is a new type of object, with its own rules for manipulating it ( df is not the same as Δ f ! The textbook has it wrong.) It encodes how variations of f are related to variations of x, y, z . We can use it in two ways: 1. as a placeholder for approximation formulas: Δ f f x Δ x + f y Δ y + f z Δ z . 2. divide by dt to get the chain rule : if x = x ( t ), y = y ( t ), z = z ( t ), then f becomes a function of t and df = f x dx + f y dy + f z dz dt dt dt dt Example: w = x 2 y + z , dw = 2 xy dx + x 2 dy + dz . If x = t , y = e t , z = sin t then the chain rule gives dw/dt = (2 te t ) 1 + ( t 2 ) e t + cos t , same as what we obtain by substitution into formula for w and one-variable diﬀerentiation. Can justify the chain rule in 2 ways: 1. dx = x ( t ) dt , dy = y ( t ) dt , dz = z ( t ) dt , so substituting we get dw = f x dx + f y dy + f z dz = f x x ( t ) dt + f y y ( t ) dt + f z z ( t ) dt , hence dw/dt . 2. (more rigorous): Δ w f x Δ x + f y Δ y + f z Δ z , divide both sides by Δ t and take limit as Δ t 0. Applications of chain rule: Product and quotient formulas for derivatives: f = uv , u = u ( t ), v = v ( t ), then d ( uv ) /dt = f u u + f v v = vu + uv . Similarly with g = u/v , d ( u/v
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lec_week5 - MIT OpenCourseWare http/ocw.mit.edu 18.02...

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