# Partial derivative in Spherical Coordinates

Is partial derivative of ##u(x,y,z)## equals to
$$\frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}+\frac{\partial u}{\partial z}$$
Is partial derivative of ##u(r,\theta,\phi)## in Spherical Coordinates equals to
$$\frac{\partial u}{\partial r}+\frac{\partial u}{\partial \theta}+\frac{\partial u}{\partial \phi}$$

Thanks

## Answers and Replies

STEMucator
Homework Helper
I'm slightly confused about the question, but if you're just changing co-ordinates and taking partials it looks fine.

I'm slightly confused about the question, but if you're just changing co-ordinates and taking partials it looks fine.

Thanks for the reply. I just want to verify partial derivative in TRUE Spherical Coordinates system is $$\frac{\partial u}{\partial r}+\frac{\partial u}{\partial \theta}+\frac{\partial u}{\partial \phi}$$

Not the spherical amplitude representation of (x,y,z) where
$$\vec r=\hat x x +\hat y y+\hat z z\;\hbox { to which}\; x=r\cos\phi\sin\theta,\;y=r\sin\phi\sin\theta, \;z=r\cos\theta$$
In the ordinary Calculus III class.

Thanks

You are just differentiating wrt different functions, any calculations done by either method should give you same end results.

Thanks

vela
Staff Emeritus
Homework Helper
Is partial derivative of ##u(x,y,z)## equals to
$$\frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}+\frac{\partial u}{\partial z}$$
You need to get your terminology straight if you want people to understand you. The expression you wrote above is for the [strike]divergence[/strike] directional derivative of u in the (1,1,1) direction times the square root of 3. Each term is a partial derivative. So the answer to your question is no.

Is partial derivative of ##u(r,\theta,\phi)## in Spherical Coordinates equals to
$$\frac{\partial u}{\partial r}+\frac{\partial u}{\partial \theta}+\frac{\partial u}{\partial \phi}$$
No.

Last edited:
vanhees71
Gold Member
2021 Award
This expression is not a divergence. The divergence is defined for a vector field. In Cartesian coordinates it reads
$$\vec{\nabla} \cdot \vec{V}=\frac{\partial V_x}{\partial x}+\frac{\partial V_y}{\partial y}+\frac{\partial V_z}{\partial z}.$$

vela
Staff Emeritus