Green's theorem with a scalar function

[Amaelle]

Homework Statement

Look at the image

Relevant Equations

Green theorem
circuitation

Greetings!
My question is: is it possible to use the green theorem to compute the circulation while in presence of a scalar function ? I know how to solve by parametrising each part but just in case we can go faster? thank you!

 

[Mark44]

I might b

Homework Statement:: Look at the image
Relevant Equations:: Green theorem
circuitation

Greetings!
My question is: is it possible to use the green theorem to compute the circulation while in presence of a scalar function ? I know how to solve by parametrising each part but just in case we can go faster? thank you!
View attachment 297061

I could be wrong, but I don't believe that Green's Theorem is a reasonable approach here. It seems much more straightforward to parameterize the boundary into three curves, and then evaluate the integral of each.

Going counterclockwise along $\gamma$, we have
1. $x = t, y = t, 0 \le t \le 2$
2. $x = 2\cos(t), y= 2\sin(t), \pi/4 \le t \le \pi/2$
3. $x = 0, y = t, 2 \ge t \ge 0$

BTW, your handwriting is not the easiest to read, particularly your shorthand abbreviations. Everything you wrote in longhand could be done using LaTeX, which would make it easier for us to understand.

For example:
Ex. 2: Compute $\int_\gamma xy ds$ where $\gamma$ is the parameterization of the boundary of
$D = \{(x, y) : x^2 + y^2 \le 2, x \ge 0, y \ge 0, y \ge x\}$

 

[Amaelle]

I might b

I could be wrong, but I don't believe that Green's Theorem is a reasonable approach here. It seems much more straightforward to parameterize the boundary into three curves, and then evaluate the integral of each.

Going counterclockwise along $\gamma$, we have
1. $x = t, y = t, 0 \le t \le 2$
2. $x = 2\cos(t), y= 2\sin(t), \pi/4 \le t \le \pi/2$
3. $x = 0, y = t, 2 \ge t \ge 0$

BTW, your handwriting is not the easiest to read, particularly your shorthand abbreviations. Everything you wrote in longhand could be done using LaTeX, which would make it easier for us to understand.

For example:
Ex. 2: Compute $\int_\gamma xy ds$ where $\gamma$ is the parameterization of the boundary of
$D = \{(x, y) : x^2 + y^2 \le 2, x \ge 0, y \ge 0, y \ge x\}$

Thank you so much!