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What is metric

  1. Mar 10, 2015 #1
    When I study about the transformation of coordinates, especially while defining gradient, curl, divergence and other vector integral theorem in different co-ordinate system, a concept called metric is defined and it is said to used for transform these operators in different co-ordinates, it is given as from a rectangular co-ordinate to any system ,it is given as the differential element dxi=hi qi (dqi is the differential element in other co-ordinate system) and hi is scaling factor, but my question is what is this scaling factor, the book I referred just defined this transformation equation and did not derived it, please explain me how this equation came from and explain with diagrams if possible.
     
  2. jcsd
  3. Mar 11, 2015 #2

    Orodruin

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    The scale factor is (equivalently) the length of the vector
    $$
    \frac{\partial \vec r}{\partial q_i}, \quad \mbox{i.e.}\quad h_i = \left|\frac{\partial \vec r}{\partial q_i}\right|.
    $$
    These vectors can be used to form a normalised basis ##\vec e_i## according to
    $$
    \vec e_i = \frac{1}{h_i} \frac{\partial \vec r}{\partial q_i}.
    $$
    This is normally done in orthogonal coordinates in such a way that your basis becomes orthonormal.
    The metric is a more general concept and defines distances in a general manifold. It can be used to determine the length of vectors.
     
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