Results 1 to 6 of 6

Math Help - Try to solve that Question of Green theorm

  1. #1
    Newbie
    Joined
    Jun 2007
    Posts
    15

    Try to solve that Question of Green theorm

    "See Atach File"
    Attached Files Attached Files
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    9
    \oint_C (y^2dx + x^2 dy) = \iint_D \left( \frac{\partial x^2}{\partial x} - \frac{\partial y^2}{\partial y} \right) \ dA =\iint_D (2x - 2y) dA = 2\int_0^1 \int_0^{1-x} x - y \ dy \ dx

    You should be able to do it frum heir.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Forum Admin topsquark's Avatar
    Joined
    Jan 2006
    From
    Wellsville, NY
    Posts
    9,920
    Thanks
    332
    Awards
    1
    [quote]Find \oint_C y^2 \, dx + x^2 \, dy where the path is the boundary of the triangle given by x = 0, y = 0, and x + y = 1,[\quote]

    First we need to define our area of integration. This is a right triangle with verticies at (0, 0), (1, 0), and (0, 1).

    I will write
    \oint_C y^2 \, dx + x^2 \, dy = \oint_C \bold{F} \cdot d \bold{s}
    where \bold{F} = \left [ \begin{matrix} y^2 \\ x^2 \end{matrix} \right ]

    Green's theorem says that
    \oint_C \bold{F} \cdot d \bold{s} = \int_A \nabla \times \bold{F} \cdot d \bold{a}

    So we first find \nabla \times \bold{F}:
    \nabla \times \bold{F} = \left | \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ \partial_x & \partial_y & \partial_z \\ y^2 & x^2 & 0 \end{matrix} \right | = \hat{k} \left ( \frac{\partial (x^2)}{\partial x} - \frac{\partial (y^2)}{\partial y} \right ) = 2(x - y) \hat{k}

    Now, the area of this region is in the - \hat{k} direction so:
    \oint_C y^2 \, dx + x^2 \, dy = -2 \int_A (x - y)da = -2 \int_0^1 \int_0^{-x + 1}(x - y) \, dx \, dy = 0

    -Dan
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    9
    Quote Originally Posted by topsquark View Post

    Green's theorem says that
    \oint_C \bold{F} \cdot d \bold{s} = \int_A \nabla \times \bold{F} \cdot d \bold{a}
    I see you learned Green's theorem. But there is a problem.

    Actually Green's theorem says that:
    \iint_A (\nabla \times \bold{F})\cdot \bold{k} \ dA.

    You forgot the normal outward pointing vector \bold{k}.

    (Unless the notation d\bold{a} is so strange concept in Mathematical Physics. And it means \bold{k}\cdot dA.)
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Forum Admin topsquark's Avatar
    Joined
    Jan 2006
    From
    Wellsville, NY
    Posts
    9,920
    Thanks
    332
    Awards
    1
    Quote Originally Posted by ThePerfectHacker View Post
    (Unless the notation d\bold{a} is so strange concept in Mathematical Physics. And it means \bold{k}\cdot dA.)
    Actually, yes!

    -Dan
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    9
    Quote Originally Posted by topsquark View Post
    Actually, yes!
    Why am I not supprised?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 9
    Last Post: February 23rd 2011, 12:24 AM
  2. Replies: 2
    Last Post: February 20th 2011, 07:13 AM
  3. need help in residual theorm question
    Posted in the Calculus Forum
    Replies: 7
    Last Post: November 11th 2009, 05:44 AM
  4. Replies: 1
    Last Post: August 28th 2007, 10:44 AM
  5. Green's Theorm
    Posted in the Calculus Forum
    Replies: 1
    Last Post: April 27th 2006, 01:28 AM

Search Tags


/mathhelpforum @mathhelpforum