(e) Verify Green's theorem for a vector field F(x, y) = y²i+z²j and a triangle bounded by the lines a + y = 1 and -a + y = 1 and y = 0. (Hint: all the conditions in the Green theorem must be verified.)

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
icon
Related questions
Question
Please do (e)
Green's Theorem
(a) State the Green theorem in the plane.
(b) Express part (a) in vector notation.
(c) Give one example where the Green theorem fails, and explain how.
(d) Prove part (a) assuming that the domain D enclosed by the simple closed curve
C with positive orientation is of the form
D = {(x,y) | a ≤ y ≤ b, gi(y) ≤ x ≤92(y)},
where gi(y), 92(y) are continuous functions, and a, b are some real numbers.
(e) Verify Green's theorem for a vector field F(x, y) = y²i+z²j and a triangle
bounded by the lines a + y = 1 and -x + y = 1 and y = 0.
(Hint: all the conditions in the Green theorem must be verified.)
Transcribed Image Text:Green's Theorem (a) State the Green theorem in the plane. (b) Express part (a) in vector notation. (c) Give one example where the Green theorem fails, and explain how. (d) Prove part (a) assuming that the domain D enclosed by the simple closed curve C with positive orientation is of the form D = {(x,y) | a ≤ y ≤ b, gi(y) ≤ x ≤92(y)}, where gi(y), 92(y) are continuous functions, and a, b are some real numbers. (e) Verify Green's theorem for a vector field F(x, y) = y²i+z²j and a triangle bounded by the lines a + y = 1 and -x + y = 1 and y = 0. (Hint: all the conditions in the Green theorem must be verified.)
Expert Solution
Step 1

We are given vector field F(x,y)=y2i+x2j

For a triangle bounded by x + y = 1 , -x + y = 1 and y =0

steps

Step by step

Solved in 3 steps with 2 images

Blurred answer
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,