\[(u_1 , u_2 , u_3 )\]
.The normal vector to the surface
\[u_i = c_i\]
is the gradient \[\mathbf{\nabla} u_i = \frac{\partial u_i}{\partial x} \mathbf{i} + \frac{\partial u_i}{\partial y} \mathbf{j} + \frac{\partial u_i}{\partial z} \mathbf{k}\]
The tangent vector to the coordinate curve
\[u_i\]
is \[\mathbf{\nabla} r = \frac{\partial r}{\partial u_i} \mathbf{i} + \frac{\partial r}{\partial u_i} \mathbf{j} + \frac{\partial r}{\partial u_i} \mathbf{k}\]
Consider the coordinate curve
\[u_1\]
formed by the intersection of the surfaces \[u_2 = c_2\]
and \[u_3 = c_3\]
.The component of the tangent vector parallel to
\[u_1\]
is \[\frac{\partial r}{\partial u_i}\]
and this is also perpendicular to the surface normals \[\mathbf{\nabla} u_2 , \: \mathbf{\nabla} u_3\]
to the surfaces \[u_2 = c_2 , \:u_3 = c_3\]
respectively.Hence the vectors
\[\frac{\partial r}{\partial u_1} , \: \mathbf{\nabla} u_1\]
which both point in the direction of increasing \[u_1\]
are parallel.
Similarly for \[u_2\]
and \[u_3\]
.