STOKE THEOREM
ANSWER ALL QUESTIONS QUESTION 1 (25 MARKS) Evaluate SI, F.ds where F = 3xî +221 + (1 - y2)k and S is the portion of z = 2 - 3y2 + x? that lies over the triangle in the xy-plane with vertices (0,0), (2.0) and (2,4) in the negative z-direction. -3 у 18 15 12 192 -2 0 2 2 Figure Q1
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16.1 Evaluate [[cz + 3y – x²dS where S is the portion z = 2 –...
16.1
Evaluate [[cz + 3y – x²dS where S is the portion z = 2 – 3y + x2 that lies over the triangle in the xy-plane with vertices (0,0), (2,0), (2, -4).
Q1. Evaluate the line integral f (x2 + y2)dx + 2xydy by two methods a) directly,...
Q1. Evaluate the line integral f (x2 + y2)dx + 2xydy by two methods a) directly, b) using Green's Theorem, where C consists of the arc of the parabola y = x2 from (0,0) to (2,4) and the line segments from (2,4) to (0,4) and from (0,4) to (0,0). [Answer: 0] Q2. Use Green's Theorem to evaluate the line integral $. F. dr or the work done by the force field F(x, y) = (3y - 4x)i +(4x - y)j...
1 Use Stokes' theorem to evaluate the integrals: F(x, y, z) dr a) where F(r, y,z)(3yz,e, 22) and C is the boundary of the triangle i the plane y2 with vertices b) where F(x, y,z (-2,2,5xz) an...
1 Use Stokes' theorem to evaluate the integrals: F(x, y, z) dr a) where F(r, y,z)(3yz,e, 22) and C is the boundary of the triangle i the plane y2 with vertices b) where F(x, y,z (-2,2,5xz) and C is in the plane 12- y and is the boundary of the region that lies above the square with vertices (3,5, 0), (3,7,0),(4,5,0), (4,7,0) c) where F(x, y,z(7ry, -z, 3ryz) and C is in the plane y d) where intersected with z...
7. (a) State Stoke's Theorem. (b) Use Stoke's theorem to evaluate curl(F)d where F(x, y, z)-< x2 sin(z), y2, xy >, and s is the part of the paraboloid z = 1-2-1/2 that lies above th...
7. (a) State Stoke's Theorem. (b) Use Stoke's theorem to evaluate curl(F)d where F(x, y, z)-< x2 sin(z), y2, xy >, and s is the part of the paraboloid z = 1-2-1/2 that lies above the xy-plane.
7. (a) State Stoke's Theorem. (b) Use Stoke's theorem to evaluate curl(F)d where F(x, y, z)-, and s is the part of the paraboloid z = 1-2-1/2 that lies above the xy-plane.
can you show me the work for 2,3,4,5, thank you 2. Evaluate ff curl F n...
can you show me the work for 2,3,4,5, thank you
2. Evaluate ff curl F n dS, where F = (a2yz, yz2, 23e#v), and S is the part of the sphere a2 + y2+225 that lies above the plane z 1, oriented upwards. - Solution: -4T 3. A metal sheet is bent into the shape of the parabaloid r = y2+ 2 where 0 (r, y, z) is 6(x, y, z) = z. Find the mass of the resulting metal...
QUESTION 5 Let the surface S be the portion of the cylinder x2 + y2 4 under z 3 and above the xy-plane Write the par...
QUESTION 5 Let the surface S be the portion of the cylinder x2 + y2 4 under z 3 and above the xy-plane Write the parametric representation r(z,0) for the cylinder x2 +y2 4 in term of z (a) and 0 (2 marks) Based on (a), find the magnitude of llr, x rell for the given cylinder (b) (6 marks) 1 1+ (e) Evaluate z dS for the given S (8 marks) Hence, use the divergence theorem to evaluate f,...
All of 10 questions, please. 1. Find and classify all the critical points of the function. f(x,y) - x2(y - 2) - y2 » 2. Evaluate the integral. 3. Determine the volume of the solid that is inside the...
All of 10 questions, please.
1. Find and classify all the critical points of the function. f(x,y) - x2(y - 2) - y2 » 2. Evaluate the integral. 3. Determine the volume of the solid that is inside the cylinder x2 + y2- 16 below z-2x2 + 2y2 and above the xy - plane. 4. Determine the surface area of the portion of 2x + 3y + 6z - 9 that is in the 1st octant. » 5. Evaluate JSxz...
1.1. Find the absolute and minimum values of f(x, y) = xy? on the set D=...
1.1. Find the absolute and minimum values of f(x, y) = xy? on the set D= {(x, y)\x² + y si 1.2. Find the extreme values of f(x,y) = x² + y2 + 4x-4y, using the Lagrange multipliers, with the constraint x² + y² 59 1.3. Evaluate the integral - Le*dxdy 1.4. Evaluate the integral L1.** sin(x+ + gydydx 1.5. Find the area of the surface x + y2 +22 - 4 that lies above the plane z = 1....
please provide explanations. (a) (7 points) Use the Green's Theorem to evaluate the line integral y dr+ry dy, wh...
please provide explanations.
(a) (7 points) Use the Green's Theorem to evaluate the line integral y dr+ry dy, where 2 C is the positively oriented triangle with vertices (0,0), (2,0) and (2,6) (b) (7 points) Let F(x, y) = (2xsin(y) + y2) i(x2 cos(y) +2ry)j. Find the scalar function f such that Vf F. equation of the tangent plane to the surface r(u, v) (u+v)i+3u2j+ (c) (7 points) Find an (u- v) k at the point (ro, yo, 20) (2,...
5 ve 6. Soru Dry - xdA over the triangle with vertices ( -1,0), (0,0) and...
5 ve 6. Soru
Dry - xdA over the triangle with vertices ( -1,0), (0,0) and (0,1) changing the variables by u = y - x and v = y + x. (DONOTEVALUATE INTEGRAL) 1 w 5. (15 points) Write the integral representing the area of the region al < x2 + y2 < band below the line y = x in polar coordinates.(DONOT EVALUATE INTEGRAL) ,y,z) as an iterated integral in cartesian coor- dinates. E is the region inside...
16.1
Evaluate [[cz + 3y – x²dS where S is the portion z = 2 – 3y + x2 that lies over the triangle in the xy-plane with vertices (0,0), (2,0), (2, -4).
Q1. Evaluate the line integral f (x2 + y2)dx + 2xydy by two methods a) directly, b) using Green's Theorem, where C consists of the arc of the parabola y = x2 from (0,0) to (2,4) and the line segments from (2,4) to (0,4) and from (0,4) to (0,0). [Answer: 0] Q2. Use Green's Theorem to evaluate the line integral $. F. dr or the work done by the force field F(x, y) = (3y - 4x)i +(4x - y)j...
1 Use Stokes' theorem to evaluate the integrals: F(x, y, z) dr a) where F(r, y,z)(3yz,e, 22) and C is the boundary of the triangle i the plane y2 with vertices b) where F(x, y,z (-2,2,5xz) and C is in the plane 12- y and is the boundary of the region that lies above the square with vertices (3,5, 0), (3,7,0),(4,5,0), (4,7,0) c) where F(x, y,z(7ry, -z, 3ryz) and C is in the plane y d) where intersected with z...
7. (a) State Stoke's Theorem. (b) Use Stoke's theorem to evaluate curl(F)d where F(x, y, z)-< x2 sin(z), y2, xy >, and s is the part of the paraboloid z = 1-2-1/2 that lies above the xy-plane.
7. (a) State Stoke's Theorem. (b) Use Stoke's theorem to evaluate curl(F)d where F(x, y, z)-, and s is the part of the paraboloid z = 1-2-1/2 that lies above the xy-plane.
can you show me the work for 2,3,4,5, thank you
2. Evaluate ff curl F n dS, where F = (a2yz, yz2, 23e#v), and S is the part of the sphere a2 + y2+225 that lies above the plane z 1, oriented upwards. - Solution: -4T 3. A metal sheet is bent into the shape of the parabaloid r = y2+ 2 where 0 (r, y, z) is 6(x, y, z) = z. Find the mass of the resulting metal...
QUESTION 5 Let the surface S be the portion of the cylinder x2 + y2 4 under z 3 and above the xy-plane Write the parametric representation r(z,0) for the cylinder x2 +y2 4 in term of z (a) and 0 (2 marks) Based on (a), find the magnitude of llr, x rell for the given cylinder (b) (6 marks) 1 1+ (e) Evaluate z dS for the given S (8 marks) Hence, use the divergence theorem to evaluate f,...
All of 10 questions, please.
1. Find and classify all the critical points of the function. f(x,y) - x2(y - 2) - y2 » 2. Evaluate the integral. 3. Determine the volume of the solid that is inside the cylinder x2 + y2- 16 below z-2x2 + 2y2 and above the xy - plane. 4. Determine the surface area of the portion of 2x + 3y + 6z - 9 that is in the 1st octant. » 5. Evaluate JSxz...
1.1. Find the absolute and minimum values of f(x, y) = xy? on the set D= {(x, y)\x² + y si 1.2. Find the extreme values of f(x,y) = x² + y2 + 4x-4y, using the Lagrange multipliers, with the constraint x² + y² 59 1.3. Evaluate the integral - Le*dxdy 1.4. Evaluate the integral L1.** sin(x+ + gydydx 1.5. Find the area of the surface x + y2 +22 - 4 that lies above the plane z = 1....
please provide explanations.
(a) (7 points) Use the Green's Theorem to evaluate the line integral y dr+ry dy, where 2 C is the positively oriented triangle with vertices (0,0), (2,0) and (2,6) (b) (7 points) Let F(x, y) = (2xsin(y) + y2) i(x2 cos(y) +2ry)j. Find the scalar function f such that Vf F. equation of the tangent plane to the surface r(u, v) (u+v)i+3u2j+ (c) (7 points) Find an (u- v) k at the point (ro, yo, 20) (2,...
5 ve 6. Soru
Dry - xdA over the triangle with vertices ( -1,0), (0,0) and (0,1) changing the variables by u = y - x and v = y + x. (DONOTEVALUATE INTEGRAL) 1 w 5. (15 points) Write the integral representing the area of the region al < x2 + y2 < band below the line y = x in polar coordinates.(DONOT EVALUATE INTEGRAL) ,y,z) as an iterated integral in cartesian coor- dinates. E is the region inside...
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