Covariant and contravariant of curvilinear system with given vectors

Question

I am given a curvilinear coordinates $a$, $b$ and $\phi$ as follows: $$a=\frac{x^2+y^2+z^2}{2z},\ b=\frac{x^2+y^2+z^2}{2\sqrt{x^2+y^2}},\ \phi=\arctan\frac{y}{x}.$$ I am asked to find covariant and contravariant basis vectors for it. But I cannot get the relation between $x$, $y$ and $z$ based on $a$, $b$ and $\phi$ to calculate these vectors and I am not given sufficient examples to solve these problems. I would appreciate any suggestions.

Answer 1

The tangential $(\mathbf{t}_i$) and the normal $\mathbf{n}_i$ basis are given by

$$\begin{cases} \mathbf{t}_i&=\frac{\partial \mathbf{r}}{\partial u_i} \\ \mathbf{n}_i&=\nabla u_i\end{cases}$$

I suggest you start with the normal basis (sometimes called the covariant basis) $\mathbf{n}_i$. It is rather straightforward to calculate the gradient of a, b and $\phi$

In this case, you will eventually find that

$$\nabla u_1\cdot \nabla u_2=\nabla u_1\cdot \nabla u_3= \nabla u_2\cdot \nabla u_3=0$$

Which means that your basis vectors are orthogonal.

Establish the corressponding scale factors $h_i=\frac{1}{|\nabla u_i|}$

Finally, use

$$\frac{1}{h_i}\frac{\partial \mathbf{r}}{\partial u_i}=h_i\nabla u_i$$

to find the contravariant basis $\mathbf{t}_i=\frac{\partial \mathbf{r}}{\partial u_i}$