# Vector field

• Apr 27th 2009, 06:59 AM
Apprentice123
Vector field
$\displaystyle F(x,y) = - \frac{y}{x^2+y^2}i + \frac{x}{x^2+y^2}j$

1) What domain of the field?

2) Prove that:
$\displaystyle \oint_{\alpha} - \frac{y}{x^2+y^2}dx + \frac{x}{x^2+y^2}dy = 2 \pi$

$\displaystyle \alpha$ is a circumference of center in origin and raio $\displaystyle r$

3) Check the green's theorem that:
$\displaystyle \oint_{\alpha} - \frac{y}{x^2+y^2}dx + \frac{x}{x^2+y^2}dy = 0$

My solution:

1) $\displaystyle IR- \{0,0\}$

2) $\displaystyle \int_0^{2 \pi} (sen^2t + cos^2t)dt = 2 \pi$

3) $\displaystyle \frac{ \partial (\frac{x}{x^2+y^2})}{ \partial x} - \frac{ \partial (\frac{y}{x^2+y^2})}{ \partial y} = 0$

$\displaystyle \int \int_S 0 dxdy = 0$

4) As the domain is $\displaystyle IR-\{0,0\}$ not is possible use the theorem of green

All this correct ?
• Apr 27th 2009, 11:44 AM
Apprentice123
• Apr 27th 2009, 11:54 AM
Jester
Quote:

Originally Posted by Apprentice123
$\displaystyle F(x,y) = - \frac{y}{x^2+y^2}i + \frac{x}{x^2+y^2}j$

1) What domain of the field?

2) Prove that:
$\displaystyle \oint_{\alpha} - \frac{y}{x^2+y^2}dx + \frac{x}{x^2+y^2}dy = 2 \pi$

$\displaystyle \alpha$ is a circumference of center in origin and raio $\displaystyle r$

3) Check the green's theorem that:
$\displaystyle \oint_{\alpha} - \frac{y}{x^2+y^2}dx + \frac{x}{x^2+y^2}dy = 0$

My solution:

1) $\displaystyle IR- \{0,0\}$

2) $\displaystyle \int_0^{2 \pi} (sen^2t + cos^2t)dt = 2 \pi$

3) $\displaystyle \frac{ \partial (\frac{x}{x^2+y^2})}{ \partial x} - \frac{ \partial (\frac{y}{x^2+y^2})}{ \partial y} = 0$

$\displaystyle \int \int_S 0 dxdy = 0$

4) As the domain is $\displaystyle IR-\{0,0\}$ not is possible use the theorem of green

All this correct ?

Very good! If $\displaystyle \int_c P\,dx + Q\, dy = \iint_R Qx-Py \,dA$ then to use Green's theorem requires that $\displaystyle P$ and $\displaystyle Q$are continuous and have continuous partial derivatives in $\displaystyle R$ which as you stated is not true. As classic problem is to show that for your vector field that

$\displaystyle \int_c P\,dx + Q\, dy = 2 \pi$

for any simple closed curve C enclosing the origin.
• Apr 27th 2009, 12:02 PM
Apprentice123
thank you