Jacobian determinant and orientation So in Jacobian determinant, it is often said that it gives information about whether Jacobian matrix changes orientation, but I cannot get what orientation exactly in this context.
 A: Consider a vector $v$.  $v$ (and all its scalar multiples) defines a one-dimensional subspace, the span of $v$.  $-v$ also spans the same subspace, but in the opposite direction.  So while it is the same subspace, in one sense, that is spanned by $v$ compared to $-v$, we say that such subspaces are in fact oppositely oriented from one another.  Again, you can think of this as the subspace having some directionality, not just being a collection of vectors.
In 2d, we can talk about a plane spanned by two vectors $u$ and $w$.  If $u$ is rotated to $w$ by a clockwise rotation, then the plane is clockwise oriented; if instead it is rotated counterclockwise, then it is counterclockwise oriented.
(You might realize that for any two vectors in a plane, you can rotate one vector to another using either a clockwise or a counterclockwise rotation.  I'm not saying that one or the other is impossible; rather, in assigning orientation to a plane, you choose which rotation you want to consider.  This can be better understood in terms of, say, the cross product.  The cross product assigns orientation also, and really what we're talking about is the difference between using $u \times w$ or $w \times u$.)
This pattern continues for higher dimensional spaces.  A linear map on an $N$-dimensional Euclidean space can either maintain the same orientation that the space has or reverse it.  So this would be like taking the clockwise oriented 2d plane and making it counterclockwise instead.  This is done in reflections, for instance.
