Green's Theorem

Question 1

What is the relationship between Green’s Theorem and dq/dx - dp/dx(x,y)? Hint: Average value is computed here

Answer 1

[Lim a -> 0 (ʃʃ dq/dx - dp/dy dA) ] / (a^2

Question 2

What does Positive Orientation Mean?

Answer 2

Single Clockwise traversal around the border of the figure.

Question 3

For the vector function r(t), a <= t <= b, that traverses C around D. Then the region D is on what side of any given point along traversal C if you have positive orientation?

Answer 3

Left Side

Question 4

State Green’s Theorem

Answer 4

C is a positively oriented, piecewise-smooth, simple closed curve in the plane and let D be the region bounded by C. If P and Q have continuous partial derivatives on an open region that contains D, then
∫C Pdx + Qdy = ∫∫D (δQ/dx - δP/dy)dA

Question 5

What does ∮ represent?

Answer 5

This indicates the line integral is calculated using the positively oriented boundary curve of C.

Question 6

What does ∫∫δD Pdx + Qdy represent?

Answer 6

∫∫D (δQ/dx - δP/dy)dA Just another notation