Results 1 to 3 of 3

Thread: Scalar and vector fields

  1. #1
    Senior Member chella182's Avatar
    Joined
    Jan 2008
    Posts
    267

    Scalar and vector fields

    Show that $\displaystyle \nabla\times(f\mathbf{v})=\nabla f\times\mathbf{v}+f\nabla\times\mathbf{v}$

    Here $\displaystyle f(x,y,z)$ is a scalar field and $\displaystyle \mathbf{v}(x,y,z)$ is a vector field.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Nov 2009
    Posts
    283
    Thanks
    2

    Vector identities

    Rewrite your expression in component form:
    $\displaystyle
    \nabla \times (f \vec v) = \epsilon_{ijk} \partial_j (f \vec v_k)
    $
    where $\displaystyle
    \epsilon_{ijk}
    $ is the fully antisymmetric symbol which is 0 if any indices are the same, 1 if ijk is cyclic to xyz and -1 if ijk is cyclic to xzy.

    Concentrate on what the derivative is doing. This is just like d(uv) = (du)v + u(dv):
    $\displaystyle
    \partial_j (f \vec v_k) = (\partial_j f) \vec v_k + f (\partial_j \vec v_k)
    $

    So the proof goes:
    $\displaystyle
    \nabla \times (f \vec v) = \epsilon_{ijk} \partial_j (f \vec v_k)
    $

    $\displaystyle
    = \epsilon_{ijk} ( (\partial_j f) \vec v_k + f (\partial_j \vec v_k)
    )$
    $\displaystyle
    = \epsilon_{ijk} (\partial_j f) \vec v_k + \epsilon_{ijk} f (\partial_j \vec v_k)
    $

    $\displaystyle
    = \epsilon_{ijk} (\partial_j f) \vec v_k + f \epsilon_{ijk} (\partial_j \vec v_k)
    $

    $\displaystyle
    = (\nabla f) \times \vec v + f \nabla \times \vec v$
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member chella182's Avatar
    Joined
    Jan 2008
    Posts
    267
    Thanks.

    That looks nothing like the solutions I was given though plus, turns out I'd done it right in the first place I just thought I was writing BS, but apparently not
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Scalar by vector multiplication
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: Mar 10th 2011, 02:38 AM
  2. Replies: 0
    Last Post: Nov 5th 2010, 07:03 PM
  3. vector fields - graphing gradient fields
    Posted in the Calculus Forum
    Replies: 0
    Last Post: Mar 20th 2010, 05:53 PM
  4. vector & scalar projections
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Mar 7th 2010, 05:37 AM
  5. Scalar/Vector problems
    Posted in the Geometry Forum
    Replies: 2
    Last Post: Feb 21st 2010, 06:51 AM

Search Tags


/mathhelpforum @mathhelpforum