Solutions2

# Solutions2 - 1 Mathematics 1c: Homework Set 2 Due: Monday,...

This preview shows pages 1–2. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 07/19/2010 for the course MA 1C taught by Professor Ramakrishnan during the Spring '08 term at Caltech.

### Page1 / 4

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online