Chapter 4: Partial Differentiation
Section 4.4: Directional Derivative
Example 4.4.1
At the point P:1,1,0, and in the direction v=2 i−3 j+6 k, obtain the directional derivative of fx,y,z=x3−x y2−z.
Solution
Mathematical Solution
Let u=v/7 be the unit vector in the direction of v, and let R=P+t u be the line through P in the direction defined by u. The parametric equations of this line are
x=1+2 t/7,y=1−3 t/7,z=6 t/7
Along this line the function values of f are given by
wt=fxt,yt,zt=47t+1549t2−10343t3
The requisite directional derivative is then w′0=4/7.
Alternatively, obtain the vector ∇fx=a|f(x)P = fx i+fy j+fz kx=a|f(x)P = 2−2−1 and compute its dot product with u so that
DufP=2−2−1·172−36=47
Maple Solution - Interactive
Initialize
Tools≻Load Package: Student Multivariate Calculus
Loading Student:-MultivariateCalculus
Context Panel: Assign Name
f=x3−x y2−z→assign
v=2,−3,6→assign
u=v/v2→assign
Instant answer via the Context Panel
Type the name f. Context Panel: Evaluate and Display Inline
Context Panel: Student Multivariate Calculus≻Differentiate≻Directional Derivative Fill in the "Variables, Point, and Vector" dialog as shown in Figure 4.4.1(a) Click the OK button.
Figure 4.4.1(a) Variables, Point, and Vector dialog
f = x3−xy2−z→directional derivative47
Obtain ∇fx=a|f(x)P
Context Panel: Student Multivariate Calculus≻Differentiate≻Gradient (See Figure 4.4.1(b).)
Context Panel: Select Element≻1
Context Panel: Assign to a Name≻Gf
Figure 4.4.1(b) Gradient dialog
x3−x y2−z→gradient →select entry 1 →assign to a nameGf
Obtain DufP=(∇fx=a|f(x)P)·u
Common Symbols palette: Dot product operator
Context Panel: Evaluate and Display Inline
Gf·u = 47
Solution from first principles:
Obtain R=P+t u, the line through P with direction u
Write a sequence of point P and unit vector u.
Context Panel: Student Multivariate Calculus≻Lines & Planes≻Line
Context Panel: Student Multivariate Calculus≻Lines & Planes≻Representation≻parametric (See Figure 4.4.1(c).)
Context Panel: Assign to a Name≻L
Figure 4.4.1(c) Line representation dialog
1,1,0,u→make line<< Line 2 >>→representationx=1+2t7,y=1−3t7,z=6t7→assign to a nameL
Obtain wt=fxt,yt,zt and DufP=w′0
Expression palette: Evaluation template Press the Enter key.
Context Panel: Simplify≻Simplify
Context Panel: Differentiate≻With Respect To≻t
Context Panel: Evaluate at a Point≻t=0
fx=a|f(x)L
1+27t3−1+27t1−37t2−67t
= simplify
47t+1549t2−10343t3
→differentiate w.r.t. t
47+3049t−30343t2
→evaluate at point
47
Maple Solution - Coded
Install the Student MultivariateCalculus package.
withStudent:-MultivariateCalculus:
Define f.
f≔x3−x y2−z:
Define v as a list.
v≔2,−3,6:
Obtain the directional derivative
Apply the DirectionalDerivative command.
DirectionalDerivativef,x,y,z=1,1,0,v = 47
There is a DirectionalDerivative command in the Student VectorCalculus package, and a DirectionalDiff command in the Physics:-Vectors package. These alternatives will not be explored further; instead, the following two computations are provided.
Compute DufP=(∇fx=a|f(x)P)·u
Apply the Gradient command, evaluating the resulting vector at point P.
Gf≔Gradientf,x,y,z=1,1,0
Apply to the list v the convert/Vector command, then apply the Normalize command to normalize the resulting vector, thereby obtaining the unit vector u.
V≔convertv,Vector:u≔NormalizeV
Invoke the DotProduct command.
DotProductGf,u = 47
Obtain the directional derivative from first principles
Use the Line and GetRepresentation commands to obtain the parametric form of the line through P in the direction of u.
L≔GetRepresentationLine1,1,0,u,form=parametric
x=1+27t,y=1−37t,z=67t
Use the eval command to obtain the value of f along line L, then apply the simplify command.
w≔simplifyevalf,L
Apply the diff command to wt, then the eval command to obtain w′0.
evaldiffw,t,t=0
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