Chapter 1.2, Problem 73EQ

2.30 Let r denote a position vector r = x = xiêį (r² = x₂x₁) and A be an arbitrary constant vector. Use index notation to show that: (a) (c) ² (r) = n(n+1) rn-2. V. (rx A) = 0. (b) (d) V(r. A) = A. ▼x (rx A) = -2A.

Express the vector below in Cartesian coordinate: B = 2r sin cospar+r cose cose aer sind a ax= [2x/√(x² + y² + z²)] + {xz / [(x² + y²) * √(x². • √(x² + y² + z²)]} + {[y² * √(x² + y² + z²)] / (x² + y²)} ay = [xy² / √(x² + y² + z²)] + {[xyz²] / [(x² + y²) * √(x² + y² + z²)]} − {[xy * (x² + y² + z²)] / (x² + y²)} - ax= [2x² / √(x² + y² + z²)] + {xz / [(x² + y²) * √(x² + y² + z²)]} + {[y² * √(x² + y² + z²)] / (x² + y²)} a₂ = x²² / √(x² + y² + z²) az ay = [xy / √(x² + y² + z²)] + {[xyz²] / [(x² + y²) * √(x² + y² + z²)]} − {[xy * √(x² + y² + z²)] / (x² + y²)} a₂ = (xz)/√(x² + y² + z²) a₂ = (xz) / (x² + y² + z²) ax= [x² / √(x² + y² + z²)] + {xz / [(x² + y²) * √(x² + y² + z²)]} + {[y² * √(x² + y² + z²)] / (x + y)} ay = [2xy / √(x² + y² + z²)] + {[xyz²] / [(x² + y²) * √(x² + y² + z²)]} - {[xy * √(x² + y² + z²)] / (x² + y²)}

A net is dipped in a river. Determine the flow rate of water across the net if the velocity vector field for the river is given by v = (x-y, z + y + 3, z²) and the net is decribed by the equation y = √1-x²-2², y ≥ 0, and oriented in the positive y- direction. (Use symbolic notation and fractions where needed.)