In Euclidean 3-space $\mathbb{E}^3$, we have naturally defined frame $U_1(p)$, $U_2(p)$, $U_3(p)$ for each $p\in\mathbb{E}^3$, where $U_1=(1,0,0)$, $U_2(0,1,0)$, $U_3=(0,0,1)$. The frame $U_1$, $U_2$, $U_3$ (as vector fields) is called the natural frame. As a generalization of the natural frame, we can define

Definition. Vector fields $E_1$, $E_2$, $E_3$ on $\mathbb{E}^3$ constitute a frame field on $\mathbb{E}^3$ provided
$$E_i\cdot E_j=\delta_{ij},\ i,j=1,2,3$$
where $\delta_{ij}$ is the Kronecker’s delta.

There are two important examples of frame fields: the cylindrical frame field and the spherical frame field.

Example. [The Cylindrical Frame Field]

Let $(r,\theta,z)$ be the usual cylindrical coordinates on $\mathbb{E}^3$.

Fig. 1 The Cylindrical Frame

We find a unit vector field in the direction in which each coordinate increases. For $r$, this is
$$E_1=\cos\theta U_1+\sin\theta U_2.$$
For $\theta$, we find
$$E_2=-\sin\theta U_1+\cos\theta U_2.$$ Finally for $z$, it is clearly
$$E_3=U_3.$$

Example. [The spherical Frame Field]

Let $(\rho,\theta,\varphi)$ be the usual spherical coordinates.

Fig. 2 The Spherical Frame

One can find the spherical frame $F_1$, $F_2$, $F_3$ using the cylindrical frame $E_1$, $E_2$, $E_3$. Clearly
$$F_2=E_2=-\sin\theta U_1+\cos\theta U_2.$$

Fig 3. The Spherical Frame

As one can see in the Figure 3, $F_1$ and $F_3$ are obtained as
$$\begin{align*} F_1&=\cos\varphi E_1+\sin\varphi E_3\\ &=\cos\varphi(\cos\theta U_1+\sin\theta U_2)+\sin\varphi U_3,\\ F_3&=-\sin\varphi E_1+\cos\varphi E_3\\ &=-\sin\varphi(\cos\theta U_1+\sin\theta U_2)+\cos\varphi U_3. \end{align*}$$
Hence,
$$\begin{align*} F_1&=\cos\varphi\cos\theta U_1+\cos\varphi\sin\theta U_2+\sin\varphi U_3,\\ F_2&=-\sin\theta U_1+\cos\theta U_3,\\ F_3&=-\sin\varphi\cos\theta U_1-\sin\varphi\sin\theta U_2+\cos\varphi U_3. \end{align*}$$