Directional Derivative

3 Key excerpts on "Directional Derivative"

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  Calculus

  Late Transcendentals

  • Howard Anton, Irl C. Bivens, Stephen Davis(Authors)
  • 2021(Publication Date)
  • Wiley

    (Publisher)

  Thus, at a fixed point the slope of the surface may vary with the direction (Figure 13.6.3). Analytically, the Directional Derivative represents the instantaneous rate of change of f (x, y) with respect to distance in the direction of u at the point (x 0 , y 0 ). Figure 13.6.2 Example 1 Let f (x, y) = xy. Find and interpret D u f (1, 2) for the unit vector u = √ 3 2 i + 1 2 j Solution. It follows from Equation (2) that D u f (1, 2) = d ds  f  1 + √ 3s 2 , 2 + s 2  s=0 Since f  1 + √ 3s 2 , 2 + s 2  =  1 + √ 3s 2   2 + s 2  = √ 3 4 s 2 +  1 2 + √ 3  s + 2 we have D u f (1, 2) = d ds  √ 3 4 s 2 +  1 2 + √ 3  s + 2  s=0 =  √ 3 2 s + 1 2 + √ 3  s=0 = 1 2 + √ 3 Since 1 2 + √ 3 ≈ 2.23, we conclude that if we move a small distance from the point (1, 2) Figure 13.6.3 in the direction of u, the function f (x, y) = xy will increase by about 2.23 times the distance moved. The definition of a Directional Derivative for a function f (x, y, z) of three variables is similar to Definition 13.6.1. 13.6 Directional Derivatives and Gradients 857 13.6.2 DEFINITION If u = u 1 i + u 2 j + u 3 k is a unit vector, and if f (x, y, z) is a function of x, y, and z, then the Directional Derivative of f in the direction of u at (x 0 , y 0 , z 0 ) is denoted by D u f (x 0 , y 0 , z 0 ) and is defined by D u f (x 0 , y 0 , z 0 ) = d ds [ f (x 0 + su 1 , y 0 + su 2 , z 0 + su 3 )] s=0 (3) provided this derivative exists. Although Equation (3) does not have a convenient geometric interpretation, we can still interpret Directional Derivatives for functions of three variables in terms of instantaneous rates of change in a specified direction. For a function that is differentiable at a point, Directional Derivatives exist in every direction from the point and can be computed directly in terms of the first-order partial derivatives of the function.

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  Mathematical Methods in Engineering and Physics

  • Gary N. Felder, Kenny M. Felder(Authors)
  • 2015(Publication Date)
  • Wiley

    (Publisher)

  So you make another measurement and determine that f (3.1, 1.9, 7.02) = 0. You might reasonably conclude that the derivative of f at the point (3, 2, 7) is very high, since f changes so much over a short distance. But another measurement reveals that f (3.1, 1.9, 6.9) is exactly −4, suggesting a derivative of 0. Which one is right? Our point is that the question “how fast is this function changing as you move from (3, 2, 7) to another point?” is still ambiguous: you need to specify what direction you are moving in! For any given direction, the function is changing at a particular rate. The Meaning of the Directional Derivative Given a scalar field f (x , y, z), a point (x 0 , y 0 , z 0 ), and a direction specified by the vector  u, the “direc- tional derivative” D  u f (x 0 , y 0 , z 0 ) gives the rate of change of f as you start at the point (x 0 , y 0 , z 0 ) and move in the direction of  u. As always, don’t let the smooth-sounding phrase “rate of change” pass you by too quickly. We might say, somewhat loosely, that D  u f gives the amount that f will change if you move by precisely one unit in the  u-direction. It would be more accurate to say that if you take a small step (magnitude ds ) in the  u-direction, D  u f represents the change in f per unit change in position, df ∕ds . In the limit as ds → 0, this ratio becomes the actual Directional Derivative. This can be expressed in the following equation, where  u represents a unit vector in the direction of  u. D  u f ( r ) = lim h→0 f ( r + h  u) − f ( r ) h The partial derivatives we have seen are special cases of Directional Derivatives. For instance, for the direction  u =  i , D  u f = f ∕x . But a scalar field has an infinite number of Directional Derivatives—not just two or three pointing along the coordinate axes.

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  Vectors in Physics and Engineering

  • Alan Durrant(Author)
  • 2019(Publication Date)
  • CRC Press

    (Publisher)

  F varies in space.

  Fig 5.1 The change of field value depends on the direction of the displacement.

  In order to calculate these new fields we have to know how to differentiate field functions. The basic concept here is the Directional Derivative which describes the spatial rate of change of a scalar field in a specified direction. This is just a generalisation of the familiar derivative df/dx from one dimension to three. We begin Section 5.1 by introducing the Directional Derivatives at a point and the three Directional Derivatives in directions parallel to the cartesian axes, known as partial derivatives. In Sections 5.2 to 5.4 we introduce the gradient, divergence and curl fields, show how these fields can be expressed in terms of partial derivatives and how the field values are calculated. Many physical phenomena depend on the spatial variations of scalar and vector fields and we shall discuss a selection of physical laws and processes that depend on the gradient of a scalar field or the divergence or curl of a vector field.

  Because of the limited level and scope of this book, some of the important results are quoted and supported by plausibility arguments rather than derived formally. The emphasis is on an understanding of concepts, a mastery of calculation techniques and an appreciation of the relevance to scientific and engineering applications.

  5.1 Directional Derivatives and Partial Derivatives

  Let Φ(r) and Φ(r + Δr) be scalar field values at field points P and Q which have position vectors r and r + Δr (Fig 5.2 ). We shall specify the direction of the displacement Δr = PQ by a unit vector u , i.e. Δr = |Δr|u . The spatial rate of change of Φ in the direction specified by u is denoted by Φ’u

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