What does the Jacobian mean in multivariable calculus?

[chaoseverlasting]

What does a jacobian mean? I know what it IS, as in, if given a set of multivariable functions, I can find out the jacobian, but what does it MEAN?

And why do we use it to change between coordinate systems (cartesian-> polar =|jacobian of polar|* function in polar coordinates)?

 

[quasar987]

The short answer is that when you take 2 vectors of R² and compute the determinant of the matrix whose lines or columns are the components of these vectors, you get the area of the parallelogram spanned by these 2 vectors.

In cartesian coordinates, the area of a little rectangle R is dxdy. It turns out that for a change of coordinates (x,y)<-->(u,v), then the area of the rectangle R in the new coordinates u,v is |J|dudv, where J is the jacobian determinant.

I recommend the calculus book by Stewart, where this is explained in many drawings with colours and excellent explanations.