In Problems 1-10 use the divergence theorem to determinethe surface integral of the vector function F on the surface S.V.S.Scosesindi+sinesinoj+ cosok,where 00<2T, 0≤ ≤ TF = x²i + (x + y)j + (x+y+z)k

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Answered: In Problems 1-10 use the divergence…

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 = (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 =
7. Evaluate the line integral v dr where C is the curve formed by the intersection of the cylinder 2² + y² = 4 and the plane z+22-y=3, travelled counterclockwise as viewed from the positive z-axis, and v is the vector function v=zi + (x - y)j +yzk.
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1. It is sufficient to give the correct answer. a) Someone asks you to study a function f (r) for small input vectors r. What should be your first reaction? b) Give an example of a parameterized curve r(t) = (x(t), y(t)) such that t r(0) = (0, 1), r(1) = (1, 2), [r ( 1/2 )|= v2 c) Enter a continuous function f (x, y) that is not derivable at the point (0, 1) and has a local minimum with the value 2 at this point d) Is there any function f (x, y, z) such that all its level quantities are plane x + 3y + z = C and which are not lines? Enter one in this case e) Consider the function f(x, y) = y(y-x^2 ). In which areas of the planet are f positive, negative, resp 0? Draw a figure.
3. Let F = (2y² + z)i + 4.ryj + «k. (a) Show whether F is conservative. (b) Using the fundamental theorem of calculus for line integrals, prove that F. dr = -T along a curve C defined by a vector function r(t) = cos ti + sin tj + tk for 0

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In Problems 1-10 use the divergence theorem to determine the surface integral of the vector function F on the surface S. V. S S = cosesindi+sinesinoj + cosok, where 00<2T, 0≤ ≤ T F = x²i + (x + y)j + (x+y+z)k