- #1
Rasalhague
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The Wikipedia article Divergence, in the section Application to
The Wikipedia article Divergence, in the section Application to Cartesian coordinates, says of the del-dot formula for divergence, "Although expressed in terms of coordinates, the result is invariant under orthogonal transformations, as the physical interpretation suggests" ( http://en.wikipedia.org/wiki/Divergence ).
Suppose I take a vector field with Cartesan coordinates (x,y,-2z).
[tex]\text{div} (\vec{v})=\sum_{i=1}^{n}\frac{\partial v_i}{\partial x_i} = 1+1-2=0.[/tex]
But if I first apply the orthogonal transformation expressed by the matrix
[tex]A = \begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{bmatrix},[/tex]
then
[tex]\text{div} (A \vec{v}) = 1 + 1 + 2 = 4.[/tex]
So should the statement refer specifically to rotations (special/continuous orthogonal transformations)?
The Wikipedia article Divergence, in the section Application to Cartesian coordinates, says of the del-dot formula for divergence, "Although expressed in terms of coordinates, the result is invariant under orthogonal transformations, as the physical interpretation suggests" ( http://en.wikipedia.org/wiki/Divergence ).
Suppose I take a vector field with Cartesan coordinates (x,y,-2z).
[tex]\text{div} (\vec{v})=\sum_{i=1}^{n}\frac{\partial v_i}{\partial x_i} = 1+1-2=0.[/tex]
But if I first apply the orthogonal transformation expressed by the matrix
[tex]A = \begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{bmatrix},[/tex]
then
[tex]\text{div} (A \vec{v}) = 1 + 1 + 2 = 4.[/tex]
So should the statement refer specifically to rotations (special/continuous orthogonal transformations)?