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Instant-use add-on functions for the Wolfram Language

Function Repository Resource:

 DirectionalD

Source Notebook

Compute the directional derivative of a function

Contributed by: Wolfram Staff (original content by Alfred Gray)

ResourceFunction["DirectionalD"][f,r,vars]

computes the derivative of a function f in the direction r with variables vars.

Details and Options

The directional derivative of a function f at the point p along an arbitrary vector represents the instantaneous rate of change of the function at that direction.
ResourceFunction["DirectionalD"] is a generalization of a partial derivative in which the rate of change is taken with respect to one of the variables, considering the rest as constant.
The directional derivative is formally defined as , where is the vector that indicates the direction.
An alternative definition of the directional derivative is taken to be with respect to a normalized arbitrary nonzero vector .
Other notations for the directional derivative are , , , , , , and .

Examples

Basic Examples (2) 

Directional derivative of a function of two variables:

In[1]:=
ResourceFunction["DirectionalD"][3 x^2 y, {1, 1/2}, {x, y}]
Out[1]=

Directional derivative of a function of three variables:

In[2]:=
ResourceFunction["DirectionalD"][ Sin[x] Cos[y] Exp[z], {1, 2, 3}, {x, y, z}]
Out[2]=

Scope (6) 

Directional derivative with a zero component in one direction:

In[3]:=
ResourceFunction["DirectionalD"][3 x^2 y, {0, 1}, {x, y}]
Out[3]=

A unit vector along direction (3/2,1):

In[4]:=
ResourceFunction["DirectionalD"][ x Sin[y], Normalize[{3/2, 1}], {x, y}]
Out[4]=

Without normalization:

In[5]:=
ResourceFunction["DirectionalD"][ x Sin[y], {3/2, 1}, {x, y}]
Out[5]=

Evaluated at the point p=(3/2,1):

In[6]:=
{%, %%} /. {x -> -2, y -> \[Pi]}
Out[6]=

Define a function:

In[7]:=
g[x_, y_, z_] := x y z

Compute the directional derivative:

In[8]:=
ResourceFunction["DirectionalD"][g[x, y, z], {1, 2, 3}, {x, y, z}]
Out[8]=

Directional derivative in the direction (-1,-1/2) evaluated at the point (-1,3/2):

In[9]:=
ResourceFunction["DirectionalD"][ 3 x^2 y, {-1, -1/2}, {x, y}] /. {x -> -1, y -> 3/2}
Out[9]=

A plot of the directional derivative:

In[10]:=
With[{p = {x, y, 3 x^2 y} /. {x -> -1, y -> 3/2}}, Show[Plot3D[3 x^2 y, {x, -2, 0}, {y, 0, 2}, Mesh -> None, PlotStyle -> Opacity[.5]], Graphics3D[{Red, Point[p], Arrowheads[.005], Arrow[{p, p + .5 {-1, -(1/2), 15/2}}]}]]]
Out[10]=

Visualize directional derivatives over a surface:

In[11]:=
arrows = With[{d = {.5, 1}}, Graphics3D[{Arrowheads[.025], Table[Arrow[{{x0, y0, Sin[x0] Cos[y0]}, {x0, y0, Sin[x0] Cos[y0]} + .5 Normalize[{d[[1]], d[[2]], ResourceFunction["DirectionalD"][Sin[x] Cos[y], d, {x, y}] /. {x -> x0, y -> y0}}]}], {x0, 0, 5, .5}, {y0, 0, 5, .5}]}]]; Show[Plot3D[Sin[x] Cos[y], {x, 0, 5.5}, {y, 0, 5.5}, Mesh -> None, PlotStyle -> Opacity[.5]], arrows]
Out[12]=

The directional derivative (red) and the gradient (blue):

In[13]:=
Manipulate[ With[{pt2 = Append[pt, 2 x^2 - y^2] /. Thread[{x, y} -> pt]}, Show[Plot3D[2 x^2 - y^2, {x, -5, 5}, {y, -5, 5}, Mesh -> None, PerformanceGoal -> "Quality", PlotStyle -> Opacity[.5], PlotRange -> {{-5, 5}, {-5, 5}, {-26, 51}}], Graphics3D[{Arrowheads[.005], Blue, Arrow[{pt2, pt2 + {1, 1, 4 x - 2 y}}], Red, Arrow[{pt2, pt2 + {a, b, Normalize[ ResourceFunction["DirectionalD"][ 2 x^2 - y^2, {a, b}, {x, y}]]}}]} /. Thread[{x, y} -> pt]]]], {{pt, {2, -2}}, {-4, -4}, {4, 4}}, {a, -1, 1}, {b, -1, 1}]
Out[13]=

Publisher

Enrique Zeleny

Version History

  • 2.0.0 – 01 October 2020
  • 1.0.0 – 15 June 2020

Source Metadata

  • Citation:
    • Gray, A., Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, 1997.

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