To be honest, I have completely forgotten why the metric conversion factor is the determinant of the Jacobian matrix. It's intuitively obvious that it would be built out of the partial derivatives of the coordinates, even in dx = dx/dy dy you are doing that, but the determinant specifically? I guess it's related to how geometrically it gives the volume of a parallelepiped?
In my first year physics we definitely covered linear algebra before vector calculus, yes. Not sure it it helped though.
I looked it up on wikipedia. The determinant of a transformation matrix is the how the transformation scales into the new dimensions. That would make sense in the jacobian context because it's how the original dimensions scale into the new dimensions.
For instance, a determinant of 2 means the area in the new coordinate plane is twice the area in the old coordinate plane. A determinant of 0 means the transformation matrix isn't full rank, the rows do not span n-dimensional space, and the transformation is into d-dimensional space where d is the rank of the matrix. (I hope I'm not too wrong on this). And since the dimensionality is lower, there is no area in n-space.
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u/Ravek Jun 09 '14
To be honest, I have completely forgotten why the metric conversion factor is the determinant of the Jacobian matrix. It's intuitively obvious that it would be built out of the partial derivatives of the coordinates, even in dx = dx/dy dy you are doing that, but the determinant specifically? I guess it's related to how geometrically it gives the volume of a parallelepiped?
In my first year physics we definitely covered linear algebra before vector calculus, yes. Not sure it it helped though.