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3.2 Directional Derivative

We can use all the notions and results of ordinary one dimensional calculus in any higher dimensional space by choosing a particular straight line L in that space and restricting our attention to our field f on the line L. If we denote the restriction of f to the line L by f / L, f / L then is an ordinary function of one variable. We can choose that  variable to be (signed) distance, s, along L from some origin on it.

Given a point p on the line L, the directional derivative of f at p in the direction of L is the ordinary derivative of f / L with respect to the signed distance variable, s, on L.

Computation of directional derivatives


The directional derivatives of f in the directions of the three axes, (those of i, j and k) are called the partial derivatives of f with respect to x, y and z respectively and are denoted by

These particular directional derivatives are easy to calculate; on their lines L  the other two variables are constant. We need not solve for these other variables; we may simply ignore them and take the derivative of f with respect to the indicated variable treating the others as constants.