# Normal Derivative

## normal derivative

[′nȯr·məl di′riv·əd·iv] (mathematics)

The directional derivative of a function at a point on a given curve or surface in the direction of the normal to the curve or surface.

## 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).

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.