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# tut5 - MATH 237 Tutorial 5 Wednesday June 7th 2006 The...

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MATH 237 Tutorial 5 Wednesday June 7th, 2006 The Chain Rule, Directional Derivative and the Gradient Vector 1) Atmospheric temperature depends on position and time. If we denote position by three coordinates x, y, z (in km) and time by t (in hours), then the temperature T is a function of four variables T ( x, y, z, t ). a) If a thermometer is attached to a weather balloon that moves through the atmo- sphere on a path given by x ( t ) , y ( t ) and z ( t ), what is the rate of change of T with respect to time recorded by the thermometer? b) Find the rate of change of T with time at t = 1 if T ( x, y, z, t ) = 100 5 + x 2 + y 2 (1 + sin[ πt/ 12]) - 20(1 + z 2 ) [ C ] and if the balloon moves along the curve x ( t ) = t , y ( t ) = 2 t , z ( t ) = t - t 4 4 . 2) Let u ( x, y, z ) = F 1 x - 1 y , 1 y - 1 z and assume that x 6 = 0 , y 6 = 0 and z 6 = 0 and F is differentiable. Show that x 2 ∂u ∂x + y 2 ∂u ∂y + z 2 ∂u ∂z = 0. 3) Let f ( x, y ) = H ( xy 2 ) where H is differentiable. Given that H 0 (3) = 2, calculate f (3 , 1). 4) Show that if u = u ( x, y ) and x = e s , y = e t then 2 u ∂s 2 + 2 u ∂t 2 = x 2 2 u ∂x 2 + y 2

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tut5 - MATH 237 Tutorial 5 Wednesday June 7th 2006 The...

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