Jacobian

[jasoncurious]

Hi all, I am currently learning Jacobian and I wonder where can we apply this thing in the engineering field?

[elfmotat]

Well, it's certainly useful when doing multiple integrals in funky coordinates, because dV = det(J) dx₁dx₂...dx_n, where dV is the volume element.

 

For example, let's say we want to calculate the area of a circle (area is really just 2-dimensional volume). Now we could do this in Cartesian coordinates, but it's going to be a lot messier than if we do it in polar coordinates. Now, of course in Cartesian coordinates the 2-volume (area) element is just given by dA=dxdy. What is dA in polar coordinates? First we compute the Jacobian:

 

 

Computing the determinant, we find that det(J)=r. So now we know that dA=det(J)drdϕ=rdrdϕ, and we can now evaluate the following integral:

 

Edited December 13, 2012 by elfmotat

[ajb]

As elfmotat has said, the Jacobian (which I usually mean to be the determinant of the matrix of partial derivatives) arises in integration theory when changing coordinates. Basically, the Jacobian is needed to ensure that the integral does not depend on the coordinates employed. By picking coordinates suited to your problem you can make the expressions more "natural". (elfmotat's example is good here.)

 

As an aside, tensor densities, that is tensors weighted by powers of the Jacobian, are important in differential geometry and mathematical physics. These arise in the context of general relativity and quantum field theory, for example.

 

 

[elfmotat]

As an aside, tensor densities, that is tensors weighted by powers of the Jacobian, are important in differential geometry and mathematical physics. These arise in the context of general relativity and quantum field theory, for example.

 

Very true, [math]\sqrt{-g}=det(J)[/math] where [math]g=det(g_{\mu \nu })[/math]. This is useful, for example, in the Hilbert action for deriving General Relativity:

 

[math]S=k\int R\sqrt{-g}~d^4x[/math]