Charge density from electric flux density

[freezer]

Homework Statement

Determine the charge density due to the following electric flux density:

$$\overrightarrow{D} = \hat{r}4rsin(\phi ) + \hat{\phi}2rcos(\phi)+\hat{z}2z^{2}$$

Homework Equations

$$\rho _{v} = \triangledown \cdot \vec{D}$$

$$\rho _{v} = \frac{\partial }{\partial r} + \frac{\partial }{\partial \phi}+ \frac{\partial }{\partial z}$$

The Attempt at a Solution

$$\rho _{v} = \frac{\partial }{\partial r} 4rsin(\phi) + \frac{1}{r}\frac{\partial }{\partial \phi}2rcos(\phi)+ \frac{\partial }{\partial z}2z^{2}$$

$$\rho _{v} = 4sin(\phi)- 2sin(\phi) + 4z$$

$$\rho _{v} = 2sin(\phi) + 4z$$

does this look correct?

 

[rude man]

Homework Statement

Determine the charge density due to the following electric flux density:

$$\overrightarrow{D} = \hat{r}4rsin(\phi ) + \hat{\phi}2rcos(\phi)+\hat{z}2z^{2}$$

Homework Equations

$$\rho _{v} = \triangledown \cdot \vec{D}$$

$$\rho _{v} = \frac{\partial }{\partial r} + \frac{\partial }{\partial \phi}+ \frac{\partial }{\partial z}$$

The Attempt at a Solution

$$\rho _{v} = \frac{\partial }{\partial r} 4rsin(\phi) + \frac{1}{r}\frac{\partial }{\partial \phi}2rcos(\phi)+ \frac{\partial }{\partial z}2z^{2}$$

does this look correct?

There is a term missing here.

 

[freezer]

There is a term missing here.

I am not seeing it. For cylindrical I am only seeing the 1/r on the phihat term

 

[rude man]

I am not seeing it. For cylindrical I am only seeing the 1/r on the phihat term

Stick a term 4sinø in there somewhere, anywhere. No partial derivative. Just that term. (It's part of the rhat coefficient: 4rsinø/r).