Geometrical definition of Curl -- proof

[vgarg]

Homework Statement

Need help with understanding proof of geometrical form of Curl.

Relevant Equations

Please see the attachment.

Can someone please explain me the rationale for the terms circled in red on the attached copy of page 400 of "Riley, Hobson, Bence - Mathematical Methods for Physics and Engineering, 3rd edition"?
Thank you.

Mentor Note: approved - it is only a single book page, so no copyright issue.

 

[docnet]

Homework Statement: Need help with understanding proof of geometrical form of Curl.
Relevant Equations: Please see the attachment.

Can someone please explain me the rationale for the terms circled in red on the attached copy of page 400 of "Riley, Hobson, Bence - Mathematical Methods for Physics and Engineering, 3rd edition"?
Thank you.

Mentor Note: approved - it is only a single book page, so no copyright issue.

In curvilinear coordinates, the coordinate lines are not necessarily straight lines, and the scale factors (metric coefficients) $h_2$ and $h_3$ can vary with the coordinates $u_2$ and $u_3$. Consequently, the components of the vector field $\mathbf{a}$ can change with respect to the coordinates, leading to nonzero derivative terms.

$\frac{\partial}{\partial u_2} (a_3 h_3)$ and $\frac{\partial}{\partial u_3} (a_2 h_2)$ account for the the change in the components of the vector field $\mathbf{a}$ as we move along the coordinate directions.

Edit: I'm not an expert on Calculus in curvilinear coordinates, so take my comment with a grain of salt if you can.

 

[vgarg]

In curvilinear coordinates, the coordinate lines are not necessarily straight lines, and the scale factors (metric coefficients) $h_2$ and $h_3$ can vary with the coordinates $u_2$ and $u_3$. Consequently, the components of the vector field $\mathbf{a}$ can change with respect to the coordinates, leading to nonzero derivative terms.

$\frac{\partial}{\partial u_2} (a_3 h_3)$ and $\frac{\partial}{\partial u_3} (a_2 h_2)$ account for the the change in the components of the vector field $\mathbf{a}$ as we move along the coordinate directions.

Edit: I'm not an expert on Calculus in curvilinear coordinates, so take my comment with a grain of salt if you can.

But why in only 2 directions and not in the other 2 directions?

 

[docnet]

I think its because the planar surface $PQRS$ is locally perpendicular to $\hat{e}_1$ in the curvilinear system, so any variations in the $u_1$ direction don't affect the surface defined by the $u_2$ and $u_3$ coordinates.

 

[vgarg]

Can one of the mentors please move this thread to Mathematics, Calculus forum? May be someone in that form can help me with this. Thank you!