Question

I'm confused about the nature of coordinate transformations (Jacobian) and how exactly they work, just want to have a conversation with someone on the question below.

Answer 1

Does this statement make sense/is it valid, really?

Transformations between rectangular, cylindrical, and spherical coordinate systems *do not* change the shape of the object, but do conveniently change the axes along which you look at the object and the way those axes relate to one another.

Other transformations, like the ones involved in substitutions usually requiring a Jacobian by default, *do* geometrically change how you view the object in a new coordinate system.

Is it just 100% chance (or, you know, a conveniently chosen option) that in between spherical, cylindrical, and rectangular coordinate systems, your object doesn't change, but your axes/ways of looking at it do? Is this by design?

Answer 2

okay well

Answer 3

jacobian is about taking a set of axis and changing into a setoof other axis that are a function of the previous axis and the area is changing by a factor of how the old axis are changing wrt to the new axis

Answer 4

if u have time u shud watch this http://ocw.mit.edu/courses/mathematics/18-02-multivariable-calculus-fall-2007/video-lectures/lecture-18-change-of-variables/

Answer 5

changing between coordinate systems involves a jacobian too
for example, you replace \(dxdy\) with \(rdrd\theta\) for changing from cartesian to polar

here jacobian = \(r\)

Answer 6

unless the transformation is linear, the shape of an object will change when you do change of variables

but why are we worrying about how the shape changes? jacobian gives you the area factor directly right ?

Answer 7

hmm i was thinking... like for linear changes of variables

Answer 8

another nice example (from that video) is transforming a rhombus to a square :