{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

assignment7_soln

# assignment7_soln - AMath/Pmath 332 Assignment 7 Solutions...

This preview shows pages 1–3. Sign up to view the full content.

AMath/Pmath 332 Assignment 7 Solutions Section 4.2 E1. (a) Let C be the line segment with initial point i and terminal point 1. Evaluate i C z dz Solution: Parametrize the line by z ( t ) = i + (1 i ) t , 0 t 1. Then we have i C z dz = i 1 0 z ( t ) z ( t ) dt = i 1 0 ( i + (1 + i ) t )(1 i ) dt = (1 i ) p it + (1 + i ) 1 2 t 2 P v v v v 1 0 = (1 i ) p i + 1 2 (1 + i ) P = i (b) Let C be the portion of the unit circle with initial point i and terminal point 1. Evaluate i C z dz Solution: Parametrize C by z ( t ) = e it , π 2 t 0. Then i C z dz = i 0 π/ 2 z ( t ) z ( t ) dt = i 0 π/ 2 e it ( ie it ) dt = i i 0 π/ 2 dt = i ( π/ 2) = iπ/ 2 10. Let Γ be the perimeter of the square with vertices at the points z = 0, z = 1, z = 1 + i , and z = i traversed once in that order. Compute i Γ z 2 dz Solution: We parametrize each of the lines. We get z 1 ( t ) = t , 0 t 1, z 2 ( t ) = 1 + ti , 0 t 1, z 3 ( t ) = 1 + i t , 0 t 1, and z 4 ( t ) = i it , 0 t 1. Hence, i Γ z 2 dz = i 1 0 t 2 (1) dt + i 1 0 (1 ti ) 2 ( i ) dt + i 1 0 (1 t i ) 2 ( 1) dt + i 1 0 ( i + it ) 2 ( i ) dt = 2(1 + i )

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
14. (b) Let R be a positive real number. If C
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 4

assignment7_soln - AMath/Pmath 332 Assignment 7 Solutions...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online