Use the Divergence Theorem to evaluate[/F.F. Ndsand find the outward flux of F through the surface of the solid S bounded by the graphs of the equations. Use a computer algebra system to verify your results.F(x, y, z) = (xy² + cos z)i + (x²y + sin z)j + e²kS: z = 1/² √√x² + y²₁ z = 8

Note the following result : Gauss divergence theorem : ∫∫SF⇀.N⇀dS =∫∫∫Ddiv(F⇀)dV ,where S is an…

Answered: Use the Divergence Theorem to evaluate…

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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Find the tangent plane to the elliptic paraboloid z = x² – 2y? at the point (1,1, 1). - O A. z = 2x – 4y – 3 B. z = 2x + 4y+3 C.z = 2x – 4y +3 %3D - D. No one correct. E. z = -2x – 4y +3
2. Consider the function f : R² → R² defined by f(x, y) = (xeª cos y — yeª sin y, eª y cos y + xeª sin y). It is given that the corresponding complex function ƒ : C → C is holomorphic on its domain. (b) Find f'(1 + in).
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Consider the function f : R² → R² defined by f(x, y) = (xeª cos y — yeª sin y, eª y cos y + xeª sin y). It is given that the corresponding complex function f : C → C is holomorphic on its domain. (a) State the Cauchy-Riemann equations. You do not have to show that they hold for f.
Find the flux of F-6zi- answer will involve π. 2yj + 2xk out of a sphere of radius 5 centered at the origin. Hint: Your
= -9zi+ (xe"z – 2xe"")j+ 12 k. Find f, F•JÃ, and let S be the portion of the plane 2x + 3z = 6 that lies in the first octant such that 0 < y < 4 (see figure to the right), oriented upward. Let F Find the flux of F through S. Show your work, and circle your final answer. X
(x, y, z), by in terms of Cartesian coordinates Sph = {(x, y, z) : (x — xo)² + (y — yo)² + (z − zo)² = a²}, where a, xo, yo and zo are constants, r(0, ) = (xo + a sin cos , yo + a sine sin d, %0 + a cos), giving the ranges of the parameters 0 and 6. = For the surface Sph, defined in part (a), show that a vector surface element is given by ds a sin 0 [r (xo, Yo, zo)] dedo, and justify the sign convention associated with this normal vector. Hence evaluate the flux f = = G # G.dS Sph of the vector field G = (1,0, z²) out of the surface.

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Use the Divergence Theorem to evaluate F.NO N ds and find the outward flux of F through the surface of the solid S bounded by the graphs of the equations. F(x, y, z) = (xy² + cos z)i + (x²y + sin z)j + e²k S: z = ²√√x² + y², z = 8