9 12 Greens Theorem Topics to review 1

9. 12 Green’s Theorem Topics to review 1) Double integral (9. 10) 2) Double integral in polar coordinate (9. 11)

Double integral z y a b x R

Double integral b R a

Example 1: Evaluate (9. 10) 2 1 R

Positive Direction We say the positive direction around a simple closed curve C is that the direction a person must walk on C, in order to keep the region R bounded by C to the left. R Positive Direction counterclockwise R Negative Direction clockwise

THEOREM 9. 13 Green’s Theorem in the plane R NOTE: 1)Dirction 2)Relation double 3)conditions

THEOREM 9. 13 Green’s Theorem in the plane How to convert Line integral Double integral George Green (1793 -1841) an English mathematician and physicist. One of the most important theorem WHY? ?

Example 1: using Green’s Theorem (1, 1) R + or -

Example 1: using Green’s Theorem R + or -

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Monday : 1)HW 9. 8+9. 9 2)Quiz 9. 7 + 9. 8 + 9. 9

Groups Write an equivalent statement to statement 1 ( as much as you can )

Example 4: use Green’s Let C be the closed curve consisting of the four straight line segments R

Green’s Theorem: (Region with Holes) R NOTE: 1) 2) Positive direction

Green’s Theorem: (Region with Holes) Proof d 1 R 1 d 5 R 2 d 3 d 6 d 4 d 2 E 1 closed curve bounding R 1 = d 1 U d 5 U d 3 U d 6 E 2 closed curve bounding R 2 = d 2 U (-d 6) U d 4 U (-d 5)

OFFICE HOURS SAT 12: 20 1: 10 SUN MON 12: 20 1: 10 TUE WED 12: 20 1: 10

Example 5: Where C = C 1 U C 2 is the boundary of the blue region R R

FACT: replace a complicated closed path with a path that is simpler R Proof

Example 6: use Green’s (example 4) Let C be the closed curve consisting of the four straight line segments METHOD 1 lengthy METHOD 2 Can we use Green’s METHOD 3 Simpler path NO R

Example 6: use Green’s (example 4) 1 2 3 4

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