(c) What is fe (3 – ry) de + (-r'y)dy, where C is the line segment from (0, 0) to (2, 1). You'll need to parameterize the line segment first. (d) Repeat (b) and (c) for a different path defined by the parabola y (r) = 2r for 0 < * < 1. Did you get a different answer? Can you find a function f(x,y, 2) such that vf(r, y. z) = F(x, y, z)? (i.e., is there a potential function f(r. y, z) such that v/(2, y, z) = F(x, y. z)? )? %3D (e) Is fe F dr path independent? 3. GOAL: APPLY THE FUNDAMENTAL THEOREM OF LINE INTEGRALS: You are given the vector field F(r, y) = (r ln (y + 1) +1+ 3r*y")i + (5x*y* + -)j. 2 (y + 1) Answer the questions below. (a) Show that F(r, y) is a conservative field. (b) Find a potential function f(r, y) such that V f(r. y) = F(r, y). (c) Compute fe a (In (y + 1) +1+ 3²y®) dx+ ( 5x*y + 2y + dy, where C is any 2 (y + 1) smooth curve in region D from (-1,0) to (1, 2). 4. Find fe (e2sia" = In(r² + 1) – 4y) da + (12.ry + y"ebv) dy where C is the top half of the circle 1? + y = 4, traversed counterclockwise. - 4t. %3D 5. You argue with your friend that the path you have chosen to move a particle along requires more work than moving the same size particle under identical conditions along her path. You both start at the point (0,0, 0) and end at the point (1, 1, 1). You both apply the same force field defined by F(2, y, z) = (2ry – z) i+ (22 + 22)j+ (2y – 2xz) k. Your friend follows the black curve and you go through the red line segment, then up the blue and finally along the orange line segment. The parametric equation of the curve, call it C, that your friend is on is described by r(t) = ti+t*j+t°k, 0

Algebra for College Students
10th Edition
ISBN:9781285195780
Author:Jerome E. Kaufmann, Karen L. Schwitters
Publisher:Jerome E. Kaufmann, Karen L. Schwitters
Chapter13: Conic Sections
Section13.5: Systems Involving Nonlinear Equations
Problem 38PS
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Question 4 please

(c) What is fe (3 – ry) de + (-r'y)dy, where C is the line segment from (0, 0) to (2, 1).
You'll need to parameterize the line segment first.
(d) Repeat (b) and (c) for a different path defined by the parabola y (r) = 2r for 0 <
* < 1. Did you get a different answer? Can you find a function f(x,y, 2) such
that vf(r, y. z) = F(x, y, z)? (i.e., is there a potential function f(r. y, z) such that
v/(2, y, z) = F(x, y. z)? )?
%3D
(e) Is fe F dr path independent?
3. GOAL: APPLY THE FUNDAMENTAL THEOREM OF LINE INTEGRALS:
You are given the vector field F(r, y) = (r ln (y + 1) +1+ 3r*y")i + (5x*y* +
-)j.
2 (y + 1)
Answer the questions below.
(a) Show that F(r, y) is a conservative field.
(b) Find a potential function f(r, y) such that V f(r. y) = F(r, y).
(c) Compute fe a (In (y + 1) +1+ 3²y®) dx+ ( 5x*y + 2y +
dy, where C is any
2 (y + 1)
smooth curve in region D from (-1,0) to (1, 2).
4. Find fe (e2sia" = In(r² + 1) – 4y) da + (12.ry + y"ebv) dy where C is the top half of the circle
1? + y = 4, traversed counterclockwise.
- 4t.
%3D
5. You argue with your friend that the path you have chosen to move a particle along requires
more work than moving the same size particle under identical conditions along her path. You
both start at the point (0,0, 0) and end at the point (1, 1, 1). You both apply the same force
field defined by F(2, y, z) = (2ry – z) i+ (22 + 22)j+ (2y – 2xz) k. Your friend follows the
black curve and you go through the red line segment, then up the blue and finally along the
orange line segment. The parametric equation of the curve, call it C, that your friend is on
is described by r(t) = ti+t*j+t°k, 0<t<1. Your path is the union of the paths C1, which
is the line segment from (0,0,0) to (0, 1, 0), C2, the line segment from (0,1,0) to (0, 1, 1),
and C, the line segment from (0, 1, 1) to (1, 1, 1), that is the path L = C, UC,U C,?
Transcribed Image Text:(c) What is fe (3 – ry) de + (-r'y)dy, where C is the line segment from (0, 0) to (2, 1). You'll need to parameterize the line segment first. (d) Repeat (b) and (c) for a different path defined by the parabola y (r) = 2r for 0 < * < 1. Did you get a different answer? Can you find a function f(x,y, 2) such that vf(r, y. z) = F(x, y, z)? (i.e., is there a potential function f(r. y, z) such that v/(2, y, z) = F(x, y. z)? )? %3D (e) Is fe F dr path independent? 3. GOAL: APPLY THE FUNDAMENTAL THEOREM OF LINE INTEGRALS: You are given the vector field F(r, y) = (r ln (y + 1) +1+ 3r*y")i + (5x*y* + -)j. 2 (y + 1) Answer the questions below. (a) Show that F(r, y) is a conservative field. (b) Find a potential function f(r, y) such that V f(r. y) = F(r, y). (c) Compute fe a (In (y + 1) +1+ 3²y®) dx+ ( 5x*y + 2y + dy, where C is any 2 (y + 1) smooth curve in region D from (-1,0) to (1, 2). 4. Find fe (e2sia" = In(r² + 1) – 4y) da + (12.ry + y"ebv) dy where C is the top half of the circle 1? + y = 4, traversed counterclockwise. - 4t. %3D 5. You argue with your friend that the path you have chosen to move a particle along requires more work than moving the same size particle under identical conditions along her path. You both start at the point (0,0, 0) and end at the point (1, 1, 1). You both apply the same force field defined by F(2, y, z) = (2ry – z) i+ (22 + 22)j+ (2y – 2xz) k. Your friend follows the black curve and you go through the red line segment, then up the blue and finally along the orange line segment. The parametric equation of the curve, call it C, that your friend is on is described by r(t) = ti+t*j+t°k, 0<t<1. Your path is the union of the paths C1, which is the line segment from (0,0,0) to (0, 1, 0), C2, the line segment from (0,1,0) to (0, 1, 1), and C, the line segment from (0, 1, 1) to (1, 1, 1), that is the path L = C, UC,U C,?
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Author:
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