## Proof that each gradient vector grad u_i is Parallel to the of Tangent Vector the Corresponding Coordinate Curve

Take a right handed coordinate system
$(u_1 , u_2 , u_3 )$
.
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
$u_3$
.