Given a scalar field f, its gradient is a vector normal to the surface on which the field is constant, of length given by f's directional derivative in that direction.
We can always write

df

or , after trivial manipulation,

We may identify ds_j  = u_jdw_j , and obtain

Suppose we set two of the ds_i's to zero; then the directional derivative of f in the third direction must be the component of  the gradient in that direction. We therefore obtain

Exercises