Let V = (2x)i + (3y)j + (z)k be a vector field. Let S be the upward-oriented surface given byz=-x²-5xy-5y² for 2

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Answered: Let V = (2x)i + (3y)j + (z)k be a…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Let F be the vector field (x², 2r, 2²). Let C be the boundary of Lv **---- the surface S given by 4x? + y? + 2² = 5, z > 1. Give C the counter-clockwise orientation when viewed from (above) the positive z-axis. Evaluate the integral // (7 × F) - dš by using a different surface bounded by C.
Let F = (3z + 3x²) ¿+ (6y + 4z + 4 sin(y²)) 3+ (3x + 4y + 6e²²) k (a) Find curl F. curl F - (b) What does your answer to part (a) tell you about SF. dr where C' is the circle (x − 15)² + (y – 10)² = 1 in the xy-plane, oriented clockwise? ScF. dr = 0 (c) If C is any closed curve, what can you say about ScF. dr? SoF. dr = 0 (d) Now let C be the half circle - (x − 15)² + (y − 10)² = 1 in the xy-plane with y > 10, traversed from (16, 10) to (14, 10). Find SF. dr by using your result from (c) and considering C plus the line segment connecting the endpoints of C. -0 Lc F. dr =
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6. Denote by V the vector ·(). Then we can define the operations grad (f) = f (gradient of f), div (F) = V F (divergence of F), and curl (F) = ▼ × F (curl of F). (a) Given F = (2x + 3y, 3x + z, sin x), calculate div(F) and curl(F). (b) Is the vector field F = (x + y,3y, z-x+1) conservative? What about F = (yz, xz, xy)? (c) Calculate V × (Vf) =curl(grad f) and V. (V × F) = div(curl F).
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Let V = (2x)i + (3y)j + (z)k be a vector field. Let S be the upward-oriented surface given by for 2 < x < 4 and -2 ≤ y ≤0. Find the value of the following surface integral to z=-x²-5xy-5y² four decimal places: [..v..nds= Submit Question -1.5 y -0.5 -5 -10 -15