Directional Derivative

The directional derivative del _(u)f(x_0,y_0,z_0) is the rate at which the function f(x,y,z) changes at a point (x_0,y_0,z_0) in the direction u. It is a vector form of the usual derivative, and can be defined as

del _(u)f=del f·(u)/(|u|)

where del is called "nabla" or "del" and u^^ denotes a unit vector.

The directional derivative is also often written in the notation


where s denotes a unit vector in any given direction and partialf/partialx=f_x denotes a partial derivative.

Let u^^=(u_x,u_y,u_z) be a unit vector in Cartesian coordinates, so



 del _(u^^)f=(partialf)/(partialx)u_x+(partialf)/(partialy)u_y+(partialf)/(partialz)u_z.

See also

Derivative, Gradient, Vector Derivative

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Kaplan, W. "The Directional Derivative." §2.14 in Advanced Calculus, 4th ed. Reading, MA: Addison-Wesley, pp. 135-138, 1991.Morse, P. M. and Feshbach, H. "Directional Derivatives." In Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 32-33, 1953.

Referenced on Wolfram|Alpha

Directional Derivative

Cite this as:

Weisstein, Eric W. "Directional Derivative." From MathWorld--A Wolfram Web Resource.

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