Results 1 to 2 of 2

Math Help - Divergence theorem proof

  1. #1
    Newbie
    Joined
    Feb 2010
    Posts
    4

    Divergence theorem proof

    How would I go about constructing this following proof?

    Let f = f (x, y, z) be a function. Show that for any other function v = v(x, y, z)

    \nabla \cdot (\nabla f \times \nabla v) = 0

    I understand you have to show that it is divergence free, but would you just write each vector term in terms of a derivative? :S
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
    Joined
    Feb 2008
    From
    Yuma, AZ, USA
    Posts
    3,764
    Thanks
    78
    Quote Originally Posted by chipette View Post
    How would I go about constructing this following proof?

    Let f = f (x, y, z) be a function. Show that for any other function v = v(x, y, z)

    \nabla \cdot (\nabla f \times \nabla v) = 0

    I understand you have to show that it is divergence free, but would you just write each vector term in terms of a derivative? :S
    Note this is only true if the scalar function are twice continously differentiable.

    Writing it out will work. Also if you know some other identities they could be helpful for example.



    \nabla \cdot (\vec{F} \times \vec{G})=\vec{G}\cdot( \nabla \times \vec{F})-F\cdot (\nabla \times \vec{G})

    Then for any scalar function with continous 2nd partial derviatives it is known that \nabla \times (\nabla f)=0

    Thus in the above identity if \vec{F}=\nabla f we get zero.

    Or you can also use the scalar triple product

    \nabla \cdot (\nabla f \times \nabla g)=\begin{vmatrix} \frac{\partial }{\partial x} & \frac{\partial }{\partial y} & \frac{\partial }{\partial z} \\ \frac{\partial f}{\partial x} & \frac{\partial f}{\partial y} & \frac{\partial f}{\partial z} \\ \frac{\partial g}{\partial x} & \frac{\partial g}{\partial y} & \frac{\partial g}{\partial z} \end{vmatrix}

    Just simplify from here
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Divergence Theorem
    Posted in the Calculus Forum
    Replies: 3
    Last Post: May 25th 2011, 08:40 AM
  2. Replies: 3
    Last Post: May 14th 2010, 11:04 PM
  3. Replies: 2
    Last Post: April 3rd 2010, 05:41 PM
  4. divergence theorem
    Posted in the Calculus Forum
    Replies: 0
    Last Post: February 26th 2010, 01:13 PM
  5. Divergence Theorem
    Posted in the Calculus Forum
    Replies: 1
    Last Post: April 26th 2009, 10:33 PM

Search Tags


/mathhelpforum @mathhelpforum