Chapter 4: Partial Differentiation
Section 4.4: Directional Derivative
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.
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
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
=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
=fxa,b,c p+fya,b,c q+fza,b,c r
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
Tools≻Load Package: Student Multivariate Calculus
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=p⁢t+a,y=q⁢t+b,z=r⁢t+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
→differentiate w.r.t. t
→evaluate at point
Maple Solution - Coded
Install the Student MultivariateCalculus package.
To the result of the Line command, apply the GetRepresentation command.
Use the eval command.
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.
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