This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Homework due 7th February Solutions to compulsory questions Problem ( 2.4 Exercise 19). A particle traveling along the path c ( t ) = (4 e t , 6 t 4 , cos t ) flies off on a tangent at time t = 0. Compute the position of the particle at time t = 1. Solution. The equation of the tangent at to c ( t ) time t = 0 is l ( t ) = c (0) + t c (0) = (4 , , 1) + t (4 , , 0) . Therefore, at time t = 1 the particle is at ( x,y,z ) = (4 , , 1) + (4 , , 0) = (8 , , 1) . Problem ( 2.5 Exercise 3c). Use the chain rule to compute h x where h ( x,y,z ) = f ( u ( x,y,x ) ,v ( x,y ) ,w ( x )) . Solution. The question is asking us to compute ( f g ) x where g ( x,y,z ) = ( u ( x,y,z ) ,v ( x,y ) ,w ( x )) . By the chain rule, ( f g ) x = f u u x + f v v x + f w dw dx . Note: It might be helpful to consider v ( x,y ) as a function v ( x,y,z ) which is constant in z , and w ( x ) as a function w ( x,y,z ) which is constant in y and z Then the question is asking us to compute ( f g ) x where g ( x,y,z ) = ( u ( x,y,z ) ,v ( x,y,z ) ,w ( x,y,z )) , which looks a little more familiar. Problem ( 2.5 Exercise 4). Verify the chain rule for h x , where h ( x,y ) = f ( u ( x,y ) ,v ( x,y )) and f ( u,v ) = u 2 + v 2 , u ( x,y ) = e x y , and v ( x,y ) = e xy . Solution. From the chain rule we expect h x = f u u x + f v v x = 2 u ( e x y ) + 2 v ( ye xy ) = 2 e 2 x 2 y + 2 ye 2 xy . We verify this by direct calculation: h x = x ( ( u ( x,y )) 2 + ( v ( x,y )) 2 ) = x ( e 2 x 2 y + e 2 xy ) = 2 e 2 x 2 y + 2 ye 2 xy . 1 Problem ( 2.5 Exercise 8). Let f : R 3 R be differentiable. Making the substitution x = cos sin , y = sin sin , and z = cos into f ( x,y,z ), compute f , f and f in terms of f x , f y and f z . Solution. We have f = f x x + f y y + f z z = cos sin f x + sin sin f y + cos f z , and f = f x x + f y y + f z z = sin sin f x + cos sin f y , and f = f x x + f y y + f z z = cos cos f x + sin cos f y sin f z ....
View
Full
Document
 Spring '08
 PARKINSON
 Math, Multivariable Calculus

Click to edit the document details