Given the variables $(\theta,r,x)$ and the nondimensionalization (change of coordinates):
$$ \bar{\theta}=\frac{\theta}{\theta_0}\\ \bar{r}=\frac{r}{r_o}\\ \bar{x}=\frac{x}{h} $$
The gradient of a scalar field $\phi$ in this coordinate system is written as: $$ \nabla\phi=\frac{1}{r}\frac{\partial\phi}{\partial\theta}\hat{\theta}+\frac{\partial\phi}{\partial r}\hat{r}+\frac{\partial\phi}{\partial x}\hat{x} $$
How do I write the gradient in the new coordinate system? I am tempted to write it as $$ \nabla\phi = \frac{1}{r_o\bar{r}}\frac{1}{\theta_0}\frac{\partial\phi}{\partial\bar{\theta}}\hat{\bar{\theta}}+ \frac{1}{r_o}\frac{\partial\phi}{\partial \bar{r}}\hat{^\bar{r}}+ \frac{1}{h}\frac{\partial\phi}{\partial \bar{x}}\hat{\bar{x}} $$
Not sure if this is exactly true and, if it is, what is a good way to actually prove it.