Let C1 be the semicircle given
by z = 0,y ≥ 0,x2 + y2 = 1 and C2 the semicircle given by y = 0,z ≥
0,x2 +z2 = 1. Let C be the closed curve formed by C1 and C2. Let F
= hy + 2y2,2x + 4xy + 6z2,3x + eyi. a) Draw the curve C. Choose an
orientation of C and mark it clearly on the picture. b) Use
Stokes’s theorem to compute the line integral ZC F · dr.
Homework Answers
(23 pts) Let F(x, y, z) = ?x + y, x + y, x2 + y2?,...
(23 pts) Let F(x, y, z) = ?x + y, x + y, x2 + y2?, S be the top
hemisphere of the unit sphere oriented upward, and C the unit
circle in the xy-plane with positive orientation.
(a) Compute div(F) and curl(F).
(b) Is F conservative? Briefly explain.
(c) Use Stokes’ Theorem to compute ? F · dr by converting it to
a surface integral. (The integral is easy if C
you set it up correctly)
4. (23 pts)...
5. (a) [6] Let C be a simple closed curve given by the intersection between the cylinder 2y2 1 an...
5. (a) [6] Let C be a simple closed curve given by the intersection between the cylinder 2y2 1 and the surface:-10 + 0.4xy, and F = 《2xz-2y, 2y2+ 2x, x2 + y2 + z?) is a given vector field. Find the circulation F dr
5. (a) [6] Let C be a simple closed curve given by the intersection between the cylinder 2y2 1 and the surface:-10 + 0.4xy, and F = 《2xz-2y, 2y2+ 2x, x2 + y2 + z?)...
1. Let S be the part of the paraboloid z = 6 - x2 - y2...
1. Let S be the part of the paraboloid z = 6 - x2 - y2 that lies above the plane z = 2 with upwards orientation Use Stokes' Theorem to evaluate orem to evaluate F. dr where F(x, y, z) = <4y. 2z, -x>.
= and z= 8. Let A be the part of the cylinder x2 + y2 1...
= and z= 8. Let A be the part of the cylinder x2 + y2 1 between the planes z = 2, where n points away from the z-axis. Let C be the counterclockwise boundary of A. Let F(x, y, z) = (2xz + 2yz, –2xz, x2 + y²). Verify Stokes' Theorem: (a) Evaluate the line integral in Stokes' Theorem. (Hint: C has two separate parts.] (b) Evaluate the surface integral in Stokes' Theorem. Hint: curl (F) = (2x +...
Compute in two ways the flux integral ‹ S F~ · N dS ~ for F= <2y, y, z2> and S the closed surface formed by the paraboloid z = x2 + y2 and the disk x2 + y2 ≤ 4 at z = 4. Use divergence theorem...
Compute in two ways the flux integral ‹ S F~ · N dS ~ for F=
<2y, y, z2> and S the closed surface
formed by the paraboloid z = x2 + y2 and the
disk x2 + y2 ≤ 4 at z = 4. Use divergence
theorem to solve one way, and use SSs F * N ds to solve the other
way. (This is a Calculus 3 problem.)
* 36.3. Compute in two ways the fux integral ф...
Let R be the region shown above bounded by the curve C = C1[C2. C1 is a semicircle with center at...
Let R be the region shown above bounded by the curve C = C1[C2.
C1 is a semicircle with center
at the origin O and radius 9
5 . C2 is part of an ellipse with center at (4; 0), horizontal
semi-axis
a = 5 and vertical semi-axis b = 3.
Thanks a lot for your help:)
1. Let R be the region shown above bounded by the curve C - C1 UC2. C1 is a semicircle with centre at...
Begin with the paraboloid z = 22 + y2 for 0 < < 4, and slice...
Begin with the paraboloid z = 22 + y2 for 0 < < 4, and slice it with the plane y = 0. Let S be the surface that remains for y> 0 (excluding the planar surface in xz-plane) oriented downward (i.e. n3 <0). Let C be the Semicircle and the line segment in the plane z = 4 with counterclockwise orientation and F =< 2z+y, 2x + z, 2y + 2 > ZA C 4 S 2 = x2...
Problem 1.20. Let f(z, y)-(X2-y2)/(z2 + y2) 2 for x, y E (0, 1]. Prove that...
Problem 1.20. Let f(z, y)-(X2-y2)/(z2 + y2) 2 for x, y E (0, 1]. Prove that f(x, y) dx dy f f(x,y) dy)dr. Jo Jo JoJo
Problem 1 You are given the maximum and minimum of the function f(x, y, z) =...
Problem 1 You are given the maximum and minimum of the function f(x, y, z) = x2 - y2 on the surface x2 + 2y2 + 3z2 = 1 exist. Use Lagrange multiplier method to find them. Let us recall the extreme value theorem we discussed before the spring break: Extreme Value Theorem (For Functions Of Two Variables) If f(x,y) is continuous on a closed, bounded region D in the plane, then f attains a maximum value f(x,y) and a...
10. Stokes' Theorem and Surface Integrals of Vector Fields a. Stokes' Theorem: F dr- b. Let S be the surface of the paraboloid z 4-x2-y2 and C is the trace of S in the xy-plane. Draw a sk...
10. Stokes' Theorem and Surface Integrals of Vector Fields a. Stokes' Theorem: F dr- b. Let S be the surface of the paraboloid z 4-x2-y2 and C is the trace of S in the xy-plane. Draw a sketch of curve C in the xy-plane. Let F(x,y,z) = <2z, x, y?». Compute the curl (F) c. d. Find a parametrization of the surface S: G(u,v)- Compute N(u,v) e. Use Stokes' Theorem to computec F dr
10. Stokes' Theorem and Surface Integrals...
(23 pts) Let F(x, y, z) = ?x + y, x + y, x2 + y2?, S be the top
hemisphere of the unit sphere oriented upward, and C the unit
circle in the xy-plane with positive orientation.
(a) Compute div(F) and curl(F).
(b) Is F conservative? Briefly explain.
(c) Use Stokes’ Theorem to compute ? F · dr by converting it to
a surface integral. (The integral is easy if C
you set it up correctly)
4. (23 pts)...
5. (a) [6] Let C be a simple closed curve given by the intersection between the cylinder 2y2 1 and the surface:-10 + 0.4xy, and F = 《2xz-2y, 2y2+ 2x, x2 + y2 + z?) is a given vector field. Find the circulation F dr
5. (a) [6] Let C be a simple closed curve given by the intersection between the cylinder 2y2 1 and the surface:-10 + 0.4xy, and F = 《2xz-2y, 2y2+ 2x, x2 + y2 + z?)...
1. Let S be the part of the paraboloid z = 6 - x2 - y2 that lies above the plane z = 2 with upwards orientation Use Stokes' Theorem to evaluate orem to evaluate F. dr where F(x, y, z) = <4y. 2z, -x>.
= and z= 8. Let A be the part of the cylinder x2 + y2 1 between the planes z = 2, where n points away from the z-axis. Let C be the counterclockwise boundary of A. Let F(x, y, z) = (2xz + 2yz, –2xz, x2 + y²). Verify Stokes' Theorem: (a) Evaluate the line integral in Stokes' Theorem. (Hint: C has two separate parts.] (b) Evaluate the surface integral in Stokes' Theorem. Hint: curl (F) = (2x +...
Compute in two ways the flux integral ‹ S F~ · N dS ~ for F=
<2y, y, z2> and S the closed surface
formed by the paraboloid z = x2 + y2 and the
disk x2 + y2 ≤ 4 at z = 4. Use divergence
theorem to solve one way, and use SSs F * N ds to solve the other
way. (This is a Calculus 3 problem.)
* 36.3. Compute in two ways the fux integral ф...
Let R be the region shown above bounded by the curve C = C1[C2.
C1 is a semicircle with center
at the origin O and radius 9
5 . C2 is part of an ellipse with center at (4; 0), horizontal
semi-axis
a = 5 and vertical semi-axis b = 3.
Thanks a lot for your help:)
1. Let R be the region shown above bounded by the curve C - C1 UC2. C1 is a semicircle with centre at...
Begin with the paraboloid z = 22 + y2 for 0 < < 4, and slice it with the plane y = 0. Let S be the surface that remains for y> 0 (excluding the planar surface in xz-plane) oriented downward (i.e. n3 <0). Let C be the Semicircle and the line segment in the plane z = 4 with counterclockwise orientation and F =< 2z+y, 2x + z, 2y + 2 > ZA C 4 S 2 = x2...
Problem 1.20. Let f(z, y)-(X2-y2)/(z2 + y2) 2 for x, y E (0, 1]. Prove that f(x, y) dx dy f f(x,y) dy)dr. Jo Jo JoJo
Problem 1 You are given the maximum and minimum of the function f(x, y, z) = x2 - y2 on the surface x2 + 2y2 + 3z2 = 1 exist. Use Lagrange multiplier method to find them. Let us recall the extreme value theorem we discussed before the spring break: Extreme Value Theorem (For Functions Of Two Variables) If f(x,y) is continuous on a closed, bounded region D in the plane, then f attains a maximum value f(x,y) and a...
10. Stokes' Theorem and Surface Integrals of Vector Fields a. Stokes' Theorem: F dr- b. Let S be the surface of the paraboloid z 4-x2-y2 and C is the trace of S in the xy-plane. Draw a sketch of curve C in the xy-plane. Let F(x,y,z) = <2z, x, y?». Compute the curl (F) c. d. Find a parametrization of the surface S: G(u,v)- Compute N(u,v) e. Use Stokes' Theorem to computec F dr
10. Stokes' Theorem and Surface Integrals...
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