Solutions2

Solutions2 - 1 Mathematics 1c Homework Set 2 Due Monday...

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Mathematics 1c: Homework Set 2 Due: Monday, April 12th by 10am. 1. (10 Points) Section 2.5, Exercise 8 Suppose that a function is given in terms of rectangular coordinates by u = f ( x,y,z ) . If x = ρ cos θ sin φ,y = ρ sin θ sin φ,z = ρ cos φ, express the partial derivatives ∂u/∂ρ,∂u/∂θ, and ∂u/∂φ in terms of ∂u/∂x,∂u/∂y , and ∂u/∂z . Solution. By the chain rule, ∂u ∂ρ = ∂u ∂x · ∂x ∂ρ + ∂u ∂y · ∂y ∂ρ + ∂u ∂z · ∂z ∂ρ = ∂u ∂x · cos θ sin φ + ∂u ∂y · sin θ sin φ + ∂u ∂z · cos φ ∂u ∂θ = ∂u ∂x · ( - ρ sin θ sin φ ) + ∂u ∂y · ρ cos θ sin φ + ∂u ∂z · 0 = - sin θ sin φρ ∂u ∂x + cos θ sin φρ ∂u ∂y ∂u ∂φ = ∂u ∂x · ρ cos θ cos φ + ∂u ∂y · ρ sin θ cos φ + ∂u ∂z · ( - ρ sin φ ) = ρ cos θ cos φ ∂u ∂x + ρ sin θ cos φ ∂u ∂y - ρ sin φ ∂u ∂z . 2. (10 Points) Section 2.5, Exercise 12 Suppose that the temperature at the point ( x,y,z ) in space is T ( x,y,z ) = x 2 + y 2 + z 2 . Let a particle follow the right circular helix σ ( t ) = (cos t, sin t,t ) and let T ( t ) be its temperature at time t . (a) What is T 0 ( t ) ? (b) Find an approximate value for the temperature at t = ( π/ 2) + 0 . 01 . Solution. (a) T ( t ) = T ( σ ( t )) = cos 2 t + sin 2 t + t 2 = 1 + t 2 . T 0 ( t ) = 2 t . (b) By the linear approximation, an approximate value is T ± π 2 ² + T 0 ± π 2 ² · ± π 2 + 0 . 01 - π 2 ² = 1 + ± π 2 ² 2 + 2 · π 2 · 0 . 01 3 . 4988 3. (10 Points) Section 2.6, Exercise 3(c) Compute the directional derivative of the function f ( x,y,z ) = xyz at the point ( x 0 ,y
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Solutions2 - 1 Mathematics 1c Homework Set 2 Due Monday...

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