Results 1 to 2 of 2

Math Help - Vector Calculus Identity

  1. #1
    Member
    Joined
    Aug 2008
    Posts
    88

    Vector Calculus Identity

    Show that \nabla \times (\Phi \vec{A})=\Phi \nabla \times \vec{A}-\vec{A}\times \nabla \Phi, where \Phi represents a scalar field.
    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 davesface View Post
    Show that \nabla \times (\Phi \vec{A})=\Phi \nabla \times \vec{A}-\vec{A}\times \nabla \Phi, where \Phi represents a scalar field.
    Let \vec{A}=f\vec{i}+g\vec{j}+h\vec{k}

    Note I will just the subscript notation for partials e.g \displaystyle \frac{\partial f}{\partial x}=f_x

    \displaystyle \begin{vmatrix} \vec{i} & \vec{j} & \vec{k}  \\ \frac{\partial f}{\partial x} & \frac{\partial f}{\partial y} & \frac{\partial f}{\partial z} \\ \phi f & \phi g & \phi h \end{vmatrix} = \left( \phi_y h+\phi h_y - \phi_z g -\phi g_z \right)\vec{i}-\left( \phi_x h +\phi h_x-\phi_z f -\phi f_z\right)\vec{j}+\left( \phi_x g +\phi g_x -\phi_y f -\phi f_y\right)\vec{k}

    Now we need to break this up into what we want, so lets collect all of the \phi with no derivatives

    \phi \left[ (h_y-g_z)\vec{i}-(h_x-f_z)\vec{j}+(g_x-f_y)\vec{k}\right-\left[(g\phi_z-h\phi_y)\vec{i}-(f\phi_z-h\phi_x)\vec{j}+(f\phi_y-g\phi_x)\vec{k} \right]

    \phi \nabla \times \vec{A}-\vec{A}\times \nabla \phi
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Vector Calculus (Position Vector)
    Posted in the Calculus Forum
    Replies: 6
    Last Post: August 23rd 2011, 01:43 PM
  2. vector identity
    Posted in the Calculus Forum
    Replies: 2
    Last Post: September 27th 2010, 05:45 PM
  3. Qustion about vector identity
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: August 21st 2010, 02:07 AM
  4. vector calculus - vector feilds
    Posted in the Calculus Forum
    Replies: 3
    Last Post: February 25th 2010, 01:17 AM
  5. Vector Identity
    Posted in the Advanced Applied Math Forum
    Replies: 0
    Last Post: May 16th 2007, 07:00 AM

Search Tags


/mathhelpforum @mathhelpforum