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13.3 The Gradient in General Orthogonal Coordinate System

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


or , after trivial manipulation,

We may identify ds= ujdwj , and obtain

Suppose we set two of the dsi'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