2. Consider the vector field F(r, y) = [x, y). (a) Show that F is path independent by showing that F is a gradient, ie. find a function f(z, y) such that F = Vf. (b) Compute algebraically the line integrals over the three paths A, B,C as shown in the figure below, from (0,0) to (1, 1). Check that the value of the integrals are all equal. To be more precise, A is a line segment, B is part of the graph f(x) = 1, and C consists of two line segments meeting at a right angle. (1, 1) A C B (c) Use your answer to part (a) and the FTC for line integrals to evaluate F - dr, F - dř, F- di
2. Consider the vector field F(r, y) = [x, y). (a) Show that F is path independent by showing that F is a gradient, ie. find a function f(z, y) such that F = Vf. (b) Compute algebraically the line integrals over the three paths A, B,C as shown in the figure below, from (0,0) to (1, 1). Check that the value of the integrals are all equal. To be more precise, A is a line segment, B is part of the graph f(x) = 1, and C consists of two line segments meeting at a right angle. (1, 1) A C B (c) Use your answer to part (a) and the FTC for line integrals to evaluate F - dr, F - dř, F- di
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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Question
![2. Consider the vector field F(r, y) = [x, y].
(a) Show that F is path independent by showing that F is a gradient, i.e. find a
function f(z, y) such that F = Vf.
(b) Compute algebraically the line integrals over the three paths A, B,C as shown
in the figure below, from (0,0) to (1, 1). Check that the value of the integrals
are all equal. To be more precise, A is a line segment, B is part of the graph
f(x) = r², and C consists of two line segments meeting at a right angle.
(1, 1)
A
C
B
(c) Use your answer to part (a) and the FTC for line integrals to evaluate
F - dr,
F- dř, [F- d7](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F690bc708-737a-4036-8bde-cd8ee17ec8dd%2F7cbce6c8-4571-48f1-9ac3-fbc407257bf2%2Fz9nwrf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:2. Consider the vector field F(r, y) = [x, y].
(a) Show that F is path independent by showing that F is a gradient, i.e. find a
function f(z, y) such that F = Vf.
(b) Compute algebraically the line integrals over the three paths A, B,C as shown
in the figure below, from (0,0) to (1, 1). Check that the value of the integrals
are all equal. To be more precise, A is a line segment, B is part of the graph
f(x) = r², and C consists of two line segments meeting at a right angle.
(1, 1)
A
C
B
(c) Use your answer to part (a) and the FTC for line integrals to evaluate
F - dr,
F- dř, [F- d7
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