Homework Help is Here – Start Your Trial Now!
Math
Calculus
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,

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
See similar textbooks
icon
Related questions
Question
100%

Question 4 please

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,
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 3 steps with 4 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Knowledge Booster
Basics of Inferential Statistics
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Precalculus
Precalculus
Calculus
ISBN:
9780135189405
Author:
Michael Sullivan
Publisher:
PEARSON
Calculus: Early Transcendental Functions
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning
SEE MORE TEXTBOOKS