Assignment 5

# Assignment 5 - dz dt at any point on the line. (c) Find the...

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Math 222 Assignment 5 (due Friday Oct. 26) (1) Let f ( x,y ) = 2 xy x 2 + y 2 ( x,y ) 6 = (0 , 0) , f (0 , 0) = 0. (a) ﬁnd f x (0 , 0) ,f y (0 , 0), ﬁnd both f x ( x,y ) and f y ( x,y ) at ( x,y ) 6 = (0 , 0) (b) Although both the partial derivatives f x (0 , 0) and f y (0 , 0) exist, show that f ( x,y ) is not continuous at (0 , 0). (c) show that at ( x,y ) 6 = (0 , 0) x 2 2 f ∂x 2 + 2 xy 2 f ∂x∂y + y 2 2 f ∂y 2 = 0 (2) Show that Laplace’s equation 2 V ∂x 2 + 2 V ∂y 2 = 0 is satisﬁed by each of the following functions: (a) V = x 3 - 3 xy 2 (b) V = e - y cos x (c) V = tan - 1 ( y x ) whereby ( x,y ) 6 = (0 , 0). (3) If w = f ( x,y,z ) = ( x 2 + y 2 + z 2 ) - 1 2 , ﬁnd (a) w at any point ( x,y,z ) 6 = (0 , 0 , 0) (b) the rate of increase of w at the point (2 , 1 , 2) in each of the following directions: (i) (1 , 2 , 2) (ii) (1 , 4 , 8) Leave the answers in terms of simple fractions. (4) For the surface z = xy deﬁned for x > 0 , y > 0, (a) ﬁnd the diﬀerential dz , and use your result to ﬁnd the tangent plane at ( x,y ) = (9 , 4) and the normal to the surface at this point. (b) Show that the straight line r ( t ) = ( kt, t k ,t ) , ( k > 0) lies on the surface. Now compute
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Unformatted text preview: dz dt at any point on the line. (c) Find the value of k which makes the straight line pass through the point deﬁned in the ﬁrst part of this question, and then show also that this line lies completely in the tangent plane at this point. (5) Let v = V ( x,y ) and x = r cos θ, y = r sin θ . (a) Find ∂V ∂r and ∂V ∂θ in terms of ∂V ∂x and ∂V ∂y . (b) Now show, for r 6 = 0 ± ∂V ∂x ¶ 2 + ± ∂V ∂y ¶ 2 = ± ∂V ∂r ¶ 2 + 1 r 2 ± ∂V ∂θ ¶ 2 (6) Let φ = Φ( x,y,z ) = x 2 + y 2 + z 4 and set x = uv + w, y = ( u + v + w ) 2 , z = w. (a) Find ∂φ ∂u , ∂φ ∂v , ∂φ ∂w . (b) Approximate φ if u = 1 . 01 , v = 2 . 01 , w = 1 . 02 by using diﬀeren-tials and evaluating φ for u = 1 , v = 2 , w = 1. 1...
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## This note was uploaded on 04/29/2008 for the course PHYS 230 taught by Professor Harris during the Fall '07 term at McGill.

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