Let's use f(z) = 2 and evaluate along two curves. Consider breaking up the contour as showed in the figure below, A (0,0) C (1,1) s B (1,0) then C1=AB C2 = A C BC Now, to evaluate the integral S₁ (2) dz over the contour given by the curve C1. We can write this as, A[0,0], B[1,0], C[1,1] L₁ (2) dz -- S (2) dz + f (2) dz hence dz-dz + idy źdz=zdz+ydy + izdy — iydz For AB, we can take y = 0, hence dy = 0 and ze[0, 1]. For BC, z = 1, dz = 0 and ye[0, 1]. z=z+iy Task 2: a. Evaluate S₁ (2) dz by finding the integral over AB and BC. b. Also, evaluate S₂ (2) dz by finding the integral over AC. c. Hence identify, if contour integrals are path dependant or not?
Let's use f(z) = 2 and evaluate along two curves. Consider breaking up the contour as showed in the figure below, A (0,0) C (1,1) s B (1,0) then C1=AB C2 = A C BC Now, to evaluate the integral S₁ (2) dz over the contour given by the curve C1. We can write this as, A[0,0], B[1,0], C[1,1] L₁ (2) dz -- S (2) dz + f (2) dz hence dz-dz + idy źdz=zdz+ydy + izdy — iydz For AB, we can take y = 0, hence dy = 0 and ze[0, 1]. For BC, z = 1, dz = 0 and ye[0, 1]. z=z+iy Task 2: a. Evaluate S₁ (2) dz by finding the integral over AB and BC. b. Also, evaluate S₂ (2) dz by finding the integral over AC. c. Hence identify, if contour integrals are path dependant or not?
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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Solve in Google Colab and consider z= x+iy
![Let's use f(z) = 2 and evaluate along two curves. Consider breaking up the contour as showed in the figure below,
If
A (0,0)
then
C1 = AB
C2 = A → C
C (1,1)
Now, to evaluate the integral S₁ (2) dz over the contour given by the curve C1. We can write this as,
A[0,0], B[1,0], C[1,1]
hence
B (1,0)
BC
√₁₁ (2) dz - ₁B (2) dz + (2) dz
AB
dz-dz + idy
źdz = zdr+ydy + izdy — iydz
For AB, we can take y = 0, hence dy = 0 and ze[0, 1]. For BC, z = 1, dr = 0 and ye[0, 1].
z = x+iy
Task 2:
a. Evaluate S₁ (2) dz by finding the integral over AB and BC.
b. Also, evaluate S₂ (2) dz by finding the integral over AC.
c. Hence identify, if contour integrals are path dependant or not?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8fb69cb6-850d-4a3f-91d9-c6330f459c17%2F9c76f66e-c5d4-4fc3-8508-14488d29053a%2Fihzdy3_processed.png&w=3840&q=75)
Transcribed Image Text:Let's use f(z) = 2 and evaluate along two curves. Consider breaking up the contour as showed in the figure below,
If
A (0,0)
then
C1 = AB
C2 = A → C
C (1,1)
Now, to evaluate the integral S₁ (2) dz over the contour given by the curve C1. We can write this as,
A[0,0], B[1,0], C[1,1]
hence
B (1,0)
BC
√₁₁ (2) dz - ₁B (2) dz + (2) dz
AB
dz-dz + idy
źdz = zdr+ydy + izdy — iydz
For AB, we can take y = 0, hence dy = 0 and ze[0, 1]. For BC, z = 1, dr = 0 and ye[0, 1].
z = x+iy
Task 2:
a. Evaluate S₁ (2) dz by finding the integral over AB and BC.
b. Also, evaluate S₂ (2) dz by finding the integral over AC.
c. Hence identify, if contour integrals are path dependant or not?
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