1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Divergence of a vector field is a scalar field?

  1. Mar 10, 2010 #1
    Hello.

    How can I show the Divergence of a vector field is a scalar field(in [tex]E^{3}[/tex]) ?
    Should I show that Div is invariant under rotation?

    [tex]x^{i'}=a^{ij}x^{j},V^{'}_{i}(\stackrel{\rightarrow}{x})=a_{ij}v_{j}(\stackrel{\rightarrow}{x})[/tex]

    then

    [tex]\frac{\partial V^{'}_{i}(\stackrel{\rightarrow}{x})}{\partial x^{'i}}=\frac{\partial(a_{ij}V_{j} (\stackrel{\rightarrow}{x})) }{\partial(a^{ij} x^{j} ) } = \frac{\partial V_{i} (\stackrel{\rightarrow}{x})}{\partial x^{i}} [/tex]

    How can I prove the last equality?
     
  2. jcsd
  3. Mar 11, 2010 #2
    Start with the dot product of the del operator with the vector field; this is the expression for divergence.
     
  4. Mar 11, 2010 #3

    vela

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Education Advisor

    Use the chain rule to express the derivative with respect to x'i in terms of the derivatives with respect to the unprimed coordinates.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Divergence of a vector field is a scalar field?
  1. Scalar field (Replies: 4)

  2. Energy of Scalar Field (Replies: 3)

Loading...