math Questions

Written Homework for July 3, 2024, Vector Calculus II First Name Last Name OSU ID Grading Some written homework problems are graded for correctness. To earn full credit for each problem you need to clearly and neatly show your work or provide a written explanation when appropriate. Partial credit will be awarded for logical, organized progress toward a solution when work is required and shown. Correct answers with no supporting work will not receive full credit. Problems 1. Let C be the line segment from point (0, 2, 1) to (3, 4, 4). Evaluate the line integral Z xy ds. C 2. Calculate the mass of a spring in the shape of a helix parameterized by ⃗r(t) = ⟨cos t, sin t, t⟩, 0 ≤ t ≤ 4π with a density function given by ρ(x, y, z) = x + 2z + 4. 1 3. Let R > 0. Let C be a circular path of radius R centered at the origin. Let D denote the solid disk of radius R centered at the origin. (C is the outer boundary of D.) (a) Calculate Z 2 2 e−x −y ds. C You can start by parameterizing C (or there might be a nice geometric argument.) (b) Calculate x 2 2 e−x −y dA. D (Start by choosing an appropriate coordinate system. Or there might be a way to reuse the result from (a) somehow…) (c) How are the computations from parts (a) and (b) related? 2 4. Find the area on the surface on the cylinder x2 + y 2 = 4 and between z = 0 and z = x2 + 2y 2 . 3 5. (Definition from OpenStax 6.1) A flow line (or streamline) of a vector field F is a curve ⃗r(t) r such that d⃗ = F (⃗r(t)). If F represents the velocity field of a moving particle, then the flow dt lines are paths taken by the particle. (a) Plot F (x, y) = ⟨−y, x⟩ and ⃗r(t) = ⟨2 cos(t), 2 sin(t)⟩. Prove that ⃗r(t) is a flow line of F . (b) Show that ⃗c(t) = ⟨e2t , ln|t|, 1t ⟩ for t ̸= 0 is a flow line of F (x, y, z) = ⟨2x, z, −z 2 ⟩. 4