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1. Suppose the curves parameterized by r(t) = (1+2 sin t, e", 4t +2) and R(t) = (2t + 1, cos t - t,t +3 sin t+ 2) lie on a surface S. For the tangent plane to S at the point P(1, 1,2), (i) find a parametrization r(u, v), and (ii) find an equation (in x, y and z) for the plane. (Note r(0) = OP = R(0).) ,3t %3D %3D 2. Suppose a surface S is parameterized by r(u, v) = (2uv, u? - v², u? + v?), u + v? < 1. %3D a. Find an equation of the tangent plane to S at the point r (,4) = (4,- 1). b. Find the area of the surface in the given region. c. A sheet in the shape of the given surface has density proportional to the distance to the z- axis that. Find the mass of this sheet. 3. Let F(r, y, z) = (xy+ 3x, 4y- y + z, r2 + 10z + zy). Let S be the surface of the paraboloid z = 16 – r2 - y? for z > 0. Let D be the disk r? + y? = 16, z = 0. %3D %3D a. Evaluate f,F dS with the "upward" orientation. b. Evaluate [[F dS. with the "upward" orientation. You might want to think a bit before you start computing. (2u+3v, 2u-3v, uv), for (u, v) in region D in 4. Suppose a surface S is parameterized by r(u, v) = the uv-plane bounded by the curves v = u', v =8 and u = (0. For S oriented with the unit nor- mal vector pointing generally upward, evaluate JJs F dS, where F(r, y, z) = (r-y, z-y, (r+y)z). 5. Compute the line integral: fF dr, where F(r, y, z) = (r,-, y?- z2), and C consists of the three line segments that bound the portion of the plane z = 6- 3r with orientation of the curve compatible with the portion of the plane being oriented generally "upward". You may want to consider Stokes' Theorem here. y, in the first octant,

1. Suppose the curves parameterized by r(t) = (1+2 sin t, e", 4t +2) and R(t) = (2t + 1, cos t - t,t +3 sin t+ 2) lie on a surface S. For the tangent plane to S at the point P(1, 1,2), (i) find a parametrization r(u, v), and (ii) find an equation (in x, y and z) for the plane. (Note r(0) = OP = R(0).) ,3t %3D %3D 2. Suppose a surface S is parameterized by r(u, v) = (2uv, u? - v², u? + v?), u + v? < 1. %3D a. Find an equation of the tangent plane to S at the point r (,4) = (4,- 1). b. Find the area of the surface in the given region. c. A sheet in the shape of the given surface has density proportional to the distance to the z- axis that. Find the mass of this sheet. 3. Let F(r, y, z) = (xy+ 3x, 4y- y + z, r2 + 10z + zy). Let S be the surface of the paraboloid z = 16 – r2 - y? for z > 0. Let D be the disk r? + y? = 16, z = 0. %3D %3D a. Evaluate f,F dS with the "upward" orientation. b. Evaluate [[F dS. with the "upward" orientation. You might want to think a bit before you start computing. (2u+3v, 2u-3v, uv), for (u, v) in region D in 4. Suppose a surface S is parameterized by r(u, v) = the uv-plane bounded by the curves v = u', v =8 and u = (0. For S oriented with the unit nor- mal vector pointing generally upward, evaluate JJs F dS, where F(r, y, z) = (r-y, z-y, (r+y)z). 5. Compute the line integral: fF dr, where F(r, y, z) = (r,-, y?- z2), and C consists of the three line segments that bound the portion of the plane z = 6- 3r with orientation of the curve compatible with the portion of the plane being oriented generally "upward". You may want to consider Stokes' Theorem here. y, in the first octant,

Advanced Engineering Mathematics
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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1. Suppose the curves parameterized by r(t) = (1+2 sin t, e", 4t +2) and R(t) = (2t + 1, cos t - t,t +3 sin t+ 2) lie on a surface S. For the tangent plane to S at the point P(1, 1,2), (i) find a parametrization r(u, v), and (ii) find an equation (in x, y and z) for the plane. (Note r(0) = OP = R(0).) ,3t %3D %3D 2. Suppose a surface S is parameterized by r(u, v) = (2uv, u? - v², u? + v?), u + v? < 1. %3D a. Find an equation of the tangent plane to S at the point r (,4) = (4,- 1). b. Find the area of the surface in the given region. c. A sheet in the shape of the given surface has density proportional to the distance to the z- axis that. Find the mass of this sheet. 3. Let F(r, y, z) = (xy+ 3x, 4y- y + z, r2 + 10z + zy). Let S be the surface of the paraboloid z = 16 – r2 - y? for z > 0. Let D be the disk r? + y? = 16, z = 0. %3D %3D a. Evaluate f,F dS with the "upward" orientation. b. Evaluate [[F dS. with the "upward" orientation. You might want to think a bit before you start computing. (2u+3v, 2u-3v, uv), for (u, v) in region D in 4. Suppose a surface S is parameterized by r(u, v) = the uv-plane bounded by the curves v = u', v =8 and u = (0. For S oriented with the unit nor- mal vector pointing generally upward, evaluate JJs F dS, where F(r, y, z) = (r-y, z-y, (r+y)z). 5. Compute the line integral: fF dr, where F(r, y, z) = (r,-, y?- z2), and C consists of the three line segments that bound the portion of the plane z = 6- 3r with orientation of the curve compatible with the portion of the plane being oriented generally "upward". You may want to consider Stokes' Theorem here. y, in the first octant,
Transcribed Image Text:1. Suppose the curves parameterized by r(t) = (1+2 sin t, e", 4t +2) and R(t) = (2t + 1, cos t - t,t +3 sin t+ 2) lie on a surface S. For the tangent plane to S at the point P(1, 1,2), (i) find a parametrization r(u, v), and (ii) find an equation (in x, y and z) for the plane. (Note r(0) = OP = R(0).) ,3t %3D %3D 2. Suppose a surface S is parameterized by r(u, v) = (2uv, u? - v², u? + v?), u + v? < 1. %3D a. Find an equation of the tangent plane to S at the point r (,4) = (4,- 1). b. Find the area of the surface in the given region. c. A sheet in the shape of the given surface has density proportional to the distance to the z- axis that. Find the mass of this sheet. 3. Let F(r, y, z) = (xy+ 3x, 4y- y + z, r2 + 10z + zy). Let S be the surface of the paraboloid z = 16 – r2 - y? for z > 0. Let D be the disk r? + y? = 16, z = 0. %3D %3D a. Evaluate f,F dS with the "upward" orientation. b. Evaluate [[F dS. with the "upward" orientation. You might want to think a bit before you start computing. (2u+3v, 2u-3v, uv), for (u, v) in region D in 4. Suppose a surface S is parameterized by r(u, v) = the uv-plane bounded by the curves v = u', v =8 and u = (0. For S oriented with the unit nor- mal vector pointing generally upward, evaluate JJs F dS, where F(r, y, z) = (r-y, z-y, (r+y)z). 5. Compute the line integral: fF dr, where F(r, y, z) = (r,-, y?- z2), and C consists of the three line segments that bound the portion of the plane z = 6- 3r with orientation of the curve compatible with the portion of the plane being oriented generally "upward". You may want to consider Stokes' Theorem here. y, in the first octant,
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