Calculate the surface integral SSV × F-nd S With the help of Stokes' theorem, whenand S is the surface x² + y² + z² = 4, z ≥ 0, the positive side of which is the upper side.The value of the integral isF(x, y, z) =-(v + — + 2) · z¸ (² +v+x) · (²+1)3226.ey.cos(z)

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Answered: Calculate the surface integral Sf V x…

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 =
4- Teorema de stokes para calcular fr Fidő Fior con F(X₁Y. Z) = (y² + Z² ) î + (x² + Z²) Ĵ + (x² + y²jk alrededor de fo frontera (un trianguloj de la parte del plano x+y+z que queda en el primer octonte. La circulación es: Q14 b) 20 94 pr b) 2π cm do
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← A dietician is planning a snack package of fruit and nuts. Each ounce of fruit will supply zero units of protein, 3 units of carbohydrates, and 2 units of fat, and will contain 40 calories. Each ounce of nuts will supply 3 units of protein, 1 unit of carbohydrate, and 4 units of fat, and will contain 50 calories. Every package must provide at least 6 units of protein, at least 8 units of carbohydrates, and no more than 16 units of fat. Find the number of ounces of fruit and number of ounces of nuts that will meet the requirement with the least number of calories. What is the least number of calories? Let x be the ounces of fruit and y be the ounces of nuts. What is the objective function that must by minimized? z- 40 x+ 50 y The dietician should use ounce(s) of fruit and ounce(s) of nuts. These amounts will have a total of calories (Type whole numbers.)
Assume that the second partial derivatives of the surface X = x(0, 4) are given by Xo4 = 0, X4 - e, + 2pes X00 = 0e, – 2pe,, And has the unit normal vector N = sin0e, + cos o ez. Then the second fundamental form coefficients L, M, and N of the surface are given by L-O sin -2o cos . M-0,N- sin ở + 20 cos o L = 0 sin 0, M = 0, N = sin 0 The above answer The above a nswer L= 0 sin 0, M = 0, N - 20 cos o I. - O sin 0. M- 0. N- sin 0 + 20 cos o The a bove ans wer The above a ns wer
Which of the following is the equation of the tangent plane to the surface z=x2 - x+3y² +y at the point P(1, 1,4) ? (a) x+7y =6 (b) x+7y-z=4 (c) 3x+5y=8 (d) 3x+5y-z= 4 (e) x+7y+z=12
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Let z = g(r, y) = f(3 cos(ry), y + e") provided that f(3, 5) = 7, fi(3, 5) = 2, f2(3, 5) = 5. i) Find g, (0, 4). i) Find 92 (0, 4). iii) Find the equation of the tangent plane to the surface z = f(3 cos(ry), y + ev) at the point (0, 4). Türkçe: f(3, 5) = 7, fi(3,5) = 2, f2(3, 5) = 5 olmak üzere z = g(x, y) = f(3 cos(zy), y + e") olsun. i) 91 (0, 4) değerini bulunuz. in 92 (0, 4) değerini bulunuz. im) z = f(3 cos(ry), y + e*") yüzeyine (0, 4) noktasında teğet düzlemin denklemini bulunuz. O i) 20, ii) 5, ii 20x + 5y - z = 13 O i) 20, i) 5, ii) 20x - 5y - z = -1 O i) 20, i) 15, ii) 20x + 15y + z = 53 O i) 60, i) 5, ii) 60x - 5y - z = -47 O ) -40, i) 15, ii) -40x + 15y + z = -27

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Calculate the surface integral Sf V x Fnd S With the help of Stokes' theorem, when + x) · y y x) (z + 2x)= ((01 410) 2 (6 +212) 10-11-20) ((~² F(x, y, z) ,6.ey.cos(z) 3 and S is the surface x² + y² + z² = 4, z ≥ 0, the positive side of which is the upper side. The value of the integral is