Chapter 4: Partial Differentiation
Section 4.4: Directional Derivative
Example 4.4.7
From first principles, obtain the directional derivative of fx,y,z at the generic point a,b,c and in the arbitrary direction u=p i+q j+r k, where u is a unit vector.
Solution
Mathematical Solution
The line that passes through the point a,b,c and that has direction v=p i+q j+r k is given vectorially by R=P+t u. Hence, the line is given by
R=xyz=abc+t pqr
or parametrically, by xt=a+p t,yt=b+q t,zt=c+r t. Note that this parametrization puts the line at a,b,c when t=0.
The function along this line reduces to
wt=fxt,yt,zt=fa+p t,b+q t,c+r t
The rate of change of w at the given point is w′0, computed as follows. First obtain
w′t
=fxxt,yt,zt x′t+fyxt,yt,zt y′t+fzxt,yt,zt z′t
=fxxt,yt,zt p+fyxt,yt,zt q+fzxt,yt,zt r
then evaluate at t=0 to obtain
w′0
=fxa,b,c p+fya,b,c q+fza,b,c r
=fxfyfz·pqr
=DufP
The vector fx i+fy j+fz k arises naturally via the chain rule. The differential operator
∇≡i ∂∂x+j ∂∂y+k ∂∂z
is called the nabla or del operator, and the vector ∇f=fx i+fy j+fz k is called the gradient of f.
Maple Solution - Interactive
Initialize
Tools≻Load Package: Student Multivariate Calculus
Loading Student:-MultivariateCalculus
Obtain the parametric equations for the line through P with direction u
Write a sequence of the point and the vector.
Context Panel: Student Multivariate Calculus≻Lines & Planes≻Line
Context Panel: Student Multivariate Calculus≻Lines & Planes≻Representation≻parametric
Context Panel: Assign to a Name≻L
a,b,c,p,q,r→make line<< Line 1 >>→representationx=pt+a,y=qt+b,z=rt+c→assign to a nameL
Obtain wt,w′t, and w′0
Expression palette: Evaluation template
Context Panel: Differentiate≻With Respect To≻t
Context Panel: Evaluate at a Point≻t=0
Context Panel: Conversions≻to diff notation
fx,y,zx=a|f(x)L
fpt+a,qt+b,rt+c
→differentiate w.r.t. t
D1fpt+a,qt+b,rt+cp+D2fpt+a,qt+b,rt+cq+D3fpt+a,qt+b,rt+cr
→evaluate at point
D1fa,b,cp+D2fa,b,cq+D3fa,b,cr
→to diff
∂∂afa,b,cp+∂∂bfa,b,cq+∂∂cfa,b,cr
Maple Solution - Coded
Install the Student MultivariateCalculus package.
withStudent:-MultivariateCalculus:
To the result of the Line command, apply the GetRepresentation command.
L≔GetRepresentationLinea,b,c,p,q,r,form=parametric
x=pt+a,y=qt+b,z=rt+c
Obtain wt=fxt,yt,zt
Use the eval command.
w≔evalfx,y,z,L
Obtain wt and w′0
Obtain w′t with the diff command and w′0 with the eval command.
Use the convert/diff command to obtain an alternate notation for the partial derivatives.
DD≔evaldiffw,t,t=0
convertDD,diff
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