Normal Derivative

of a function defined in space (or in a plane), the derivative in the direction of the normal to some surface (or to a curve lying in the plane). Let S be a surface, P a point on S, and f a function in some neighborhood of P. Then the normal derivative of f at P is equal to the limit of the ratio of the difference f(A) — f(P) over the distance from A to P, where A is a point on the normal to S at P that approaches P from one side of S (see Figure 1).

Figure 1

We distinguish between the derivative of f with respect to the outward-drawn and the inward-drawn normals to S, depending on the direction from which A approaches P. Consideration of normal derivatives is particularly important in the theory of boundary value problems.