## 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 \[u_3\]

.