Hw06 Vector Analysis

Vector Analysis Z 1. Find the value of

→ − → − → − F · d ` along the circle x2 + y 2 = 2 from (1,1) to (1,-1) if F =

(2x − 3y)ˆi − (3x − 2y)ˆj. Answer: 6 Z Z Z → − → − ˆ 2. (∇ · F )dτ over the region x2 + y 2 + z 2 ≤ 25, where F = (x2 + y 2 + z 2 )(xˆi + yˆj + z k). Answer: 4π × 55 Z Z → − → − ˆ ˆ F ·n ˆ dσ over the part of the surface z = 4 − x2 − y 2 that is 3. If F = xi + y j, calculate above the (x, y) plane, by applying the divergence theorem to the volume bounded by surface and the piece that it cuts out of the (x, y) plane. Answer: 16π I → − − 4. V · d→ r around the boundary of the square with vertices (1,0), (0,1), (-1,0), (0,-1), if → − V = x2ˆi + 5xˆj. Answer: 10 → − → − → − → − 5. If a vecotr field A = Axˆi + Ayˆi satisfies V = ∇ × A for V = (zezy + x sin zx)ˆi + x cos xz ˆj − ˆ find Ax and Ay . z sin zxk, z 1 Answer: Ax = sin zx, Ay = − ezy + 2 ezy + cos zx y y

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