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 diﬀerentials and evaluating φ for u = 1 , v = 2 , w = 1. 1...
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 Fall '07
 Harris
 Euclidean geometry, Line segment, dz surface

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