Scalar factors of parabolic cylindrical coords

[lazyluke]

Homework Statement

I have a question to find Scalar factors of parabolic cylindrical coords and element dV with provided tranformation equations. I know the values for both of them and that the product of the scalar factors is the dV, but how do i derive those scalar factors? I don't even know where to start, i know i can use them to find Laplacian, curl and divergence but how do i find those values? how do i show my work? Please help Thank You Luke

Homework Equations

The Attempt at a Solution

 

[lazyluke]

Ok, so i came up with this using the Jacobian, but how do i get the scalar factors?, i assumed from Jacobian that hz=1 i donno if that is a legit assumption, please help.
ThatYou Luke

http://img444.imageshack.us/my.php?image=picture005up7.jpg

 

[thomate1]

Dear Luke,

Thank you for reaching out with your question. To find the scalar factors of parabolic cylindrical coordinates, we first need to understand the transformation equations and how they relate to the Cartesian coordinates. The parabolic cylindrical coordinate system is a 2-dimensional coordinate system that is often used in cylindrical symmetry problems.

The transformation equations for parabolic cylindrical coordinates are:
x = u
y = v
z = u^2/2

To find the scalar factors, we need to take the determinant of the Jacobian matrix of the transformation equations. The Jacobian matrix is given by:
J = [∂x/∂u ∂x/∂v; ∂y/∂u ∂y/∂v; ∂z/∂u ∂z/∂v]

Taking the determinant of this matrix will give us the scalar factors. Once we have the scalar factors, we can use them to find the element dV, which is given by:
dV = scalar factors * du * dv

To show your work, you can write out the Jacobian matrix and take the determinant to find the scalar factors. You can also show your calculations for finding the element dV using the scalar factors.

I hope this helps you with your homework. Good luck with your studies!

Best,