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Divergence of a vector field is a scalar field?

  1. Mar 10, 2010 #1

    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]\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


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    Use the chain rule to express the derivative with respect to x'i in terms of the derivatives with respect to the unprimed coordinates.
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