The Jacobian and coordinate transformations | Calculus in a Nutshell | LetThereBeMath |

Coordinate transformations play an important role in defining multiple integrals, sometimes allowing us to simplify them. The formal mathematical way to perform a coordinate transformation is through the Jacobian matrix.

[@maxpercer7119]

I think it's more clear to say that the absolute value of the Jacobian is the stretching factor when you map an infinitesimal square area element du dv in the u,v plane to an infinitesimal parallelogram area in the new coordinate system in the x,y plane. In this view, a change of coordinates can be thought as a re-parametrization of the entire x,y space in terms of parameters u,v. It is useful to draw the u,v plane separately (with orthogonal axes). So a change of coordinates in the x,y plane implies there is another u,v plane such that (x,y) = (x(u,v) , y(u,v)). A nice example is the change of polar coordinates. Draw a plane with orthogonal axes r and Θ and shade in some region, typically a rectangle 0

[@abnereliberganzahernandez6337]

you are a genius, I was looking forn an answer like that

[@ccuuttww]

how can we calculate j inverse with r=(x^2+y^2)^0.5 theta=arctan(y/x)
just do the same stuff?

[@irishuser2k11]

what is the scaling factor, what is it used for and where do you use it?

[@LetThereBeMath]

The scaling factor is the determinant of the Jacobian matrix, it is used to adjust the coordinate system when you are transforming variables. Its main application is for integrals in other coordinate systems

[@sriramvankudothu27]

What is physical significance of jacobian in general

[@cphVlwYa]

@@sriramvankudothu27 Whenever you have a non linear transformation, it will behave like a linear transformation as you zoom really far in at a given point. It then follows that there should be some matrix which represents this localized linear transformation; this is the Jacobian matrix. As a result, it's determinant will tell you how much an area at a given point gets scaled (as all determinants do for linear transformations)

[@william384]

Think you for the video, I learned a lot:-D

[@pavithra088]

Sir, please explain jacobian as transfer maping from one coordinate system to another coordinate system with few illustrations

[@annu_mishra1417]

Sister in our college also they gave it as assignment.. Sister if you had answer share chrome link Or youTube video link pls sister 🙏

[@stipepavic843]

thx alooottt, subbed :D