Question about Jacobian change of variables

[mmmboh]

I'm not sure if this is a stupid question, but I'll go ahead anyway. I understand the math aspect of it, but one thing has me confused. If you have a uv plane, and then write x=x(u,v), y=y(u,v), why is it that no matter what the function transforming the uv plane to the xy plane is, we can assume the transformation will have infinitesimal areas in the shape of parallelograms? Couldn't there be times where the transformation has shapes that look nothing like parallelograms?

 

[lurflurf]

It will always be a parallelogram if a few conditionnsare met. The transformation must be continuously differentiable and locally invertable. We are talking about infinitesimal areas. Many shapes are infinitesimally parallelograms.

 

[mmmboh]

Why can't the infinitesimal area be a shape with say, only 1 pair of sides parallel and the other two pointing toward each other? so like a square but with the top shorter than the bottom.

 

[Hurkyl]

Why can't the infinitesimal area be a shape with say, only 1 pair of sides parallel and the other two pointing toward each other? so like a square but with the top shorter than the bottom.

(psst: "trapezoid")

The "shape" of the region is irrelevant; just a visual aid. The only relevant facts about it are what plane it lies in, and what its area is.


In the geometry of the tangent space at the point P (an "infinitessimal neighborhood", to a first-order approximation), ordinary geometric shapes can only appear as linear spaces: the point P itself, a directed line through P, an oriented plane through P, and so forth. We often "enlarge" these shapes in a drawing so as not to be infinitessimal -- e.g. draw the line as a tangent line.

Shapes also have magnitudes, in some sense. When we "enlarge" them, we might draw this by making an arrow with an appropriate length, or maybe a region of a plane with the right area. The Jacobian is the ratio between the magnitudes attached to the source and target planar shapes.

 

[mmmboh]

But when you do the determinant you are finding the area of a parallelogram, so doesn't the shape matter? which brings me back to my original question...I know you are right, I am just a little confused.