Del operator in a Cylindrical vector fucntion

[TheColector]

Hi there

1. I'm having a hard time trying to understand how come ∂r^/∂Φ = Φ^ ,∂Φ/∂Φ = -r^ -> these 2 are properties that lead to general formula.
2. I've been thinking about it and I couldn't explain it. I understand every step of "how to get Divergence of a vector function in Cylindrical Coordinates" except for these formulas below.

A picture attached for visualization.
3.

General formula
Thanks for any help.

 

[tnich]

Hi there

1. I'm having a hard time trying to understand how come ∂r^/∂Φ = Φ^ ,∂Φ/∂Φ = -r^ -> these 2 are properties that lead to general formula.
2. I've been thinking about it and I couldn't explain it. I understand every step of "how to get Divergence of a vector function in Cylindrical Coordinates" except for these formulas below.

View attachment 226837 A picture attached for visualization.
3.
View attachment 226838 General formula
Thanks for any help.

If you apply the definition of derivative, these formulas will fall right out. For $\frac {\partial \hat r} {\partial \theta}$, start with $\hat r = (cos \theta, sin \theta)$.
By the way, you can easily write $\hat r$ as $\#\#\backslash hat~r\#\$#. For more cool formatting tricks, see the LaTex tutorial.

 

[jedishrfu]

Here’s an earlier discussion on PF on the derivation that should help

https://www.physicsforums.com/threads/del-operator-with-coordinate-transformations.465652/