Example 4-4-7 - Maple Help

All Products Maple MapleSim

Home : Support : Online Help : Education : Study Guides : MultivariateCalculus : Chapter 4 : Examples : Section 4-4 : Example 4-4-7

Chapter 4: Partial Differentiation

Section 4.4: Directional Derivative

Example 4.4.7

From first principles, obtain the directional derivative of $f (x, y, z)$ 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

$R = [\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} a \\ b \\ c \end{array}] + t [\begin{array}{c} p \\ q \\ r \end{array}]$

or parametrically, by $x (t) = a + p t, y (t) = b + q t, z (t) = c + r t$ . Note that this parametrization puts the line at $(a, b, c)$ when $t = 0$ .

The function along this line reduces to

$w (t) = f (x (t), y (t), z (t)) = f (a + p t, b + q t, c + r t)$

The rate of change of $w$ at the given point is $w' (0)$ , computed as follows. First obtain

$w' (t)$	$= f_{x} (x (t), y (t), z (t)) x' (t) + f_{y} (x (t), y (t), z (t)) y' (t) + f_{z} (x (t), y (t), z (t)) z' (t)$
	$= f_{x} (x (t), y (t), z (t)) p + f_{y} (x (t), y (t), z (t)) q + f_{z} (x (t), y (t), z (t)) r$

then evaluate at $t = 0$ to obtain

$w' (0)$	$= f_{x} (a, b, c) p + f_{y} (a, b, c) q + f_{z} (a, b, c) r$
	$= [\begin{array}{c} f_{x} \\ f_{y} \\ f_{z} \end{array}] \cdot [\begin{array}{c} p \\ q \\ r \end{array}]$
	$= D_{u} f (P)$

The vector $f_{x} i + f_{y} j + f_{z} k$ arises naturally via the chain rule. The differential operator

$\nabla \equiv i \frac{\partial}{\partial x} + j \frac{\partial}{\partial y} + k \frac{\partial}{\partial z}$

is called the nabla or del operator, and the vector $\nabla f = f_{x} i + f_{y} j + f_{z} k$ 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≻ $L$

$[a, b, c], ⟨p, q, r⟩$ $\overset{make line}{\to}$ $<< Line 1 >>$ $\overset{representation}{\to}$ $[x = p t + a, y = q t + b, z = r t + c]$ $\overset{assign to a name}{\to}$ $L$

Obtain $w (t), w' (t)$ , 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

$\binom{f (x, y, z)}{} | \binom{}{L}$

$f (p t + a, q t + b, r t + c)$

$\overset{differentiate w.r.t. t}{\to}$

$D_{1} (f) (p t + a, q t + b, r t + c) p + D_{2} (f) (p t + a, q t + b, r t + c) q + D_{3} (f) (p t + a, q t + b, r t + c) r$

$\overset{evaluate at point}{\to}$

$D_{1} (f) (a, b, c) p + D_{2} (f) (a, b, c) q + D_{3} (f) (a, b, c) r$

$\overset{to diff}{\to}$

$(\frac{\partial}{\partial a} f (a, b, c)) p + (\frac{\partial}{\partial b} f (a, b, c)) q + (\frac{\partial}{\partial c} f (a, b, c)) r$

Maple Solution - Coded

Initialize

•	Install the Student MultivariateCalculus package.

$with (Student :- MultivariateCalculus) &colon;$

Obtain the parametric equations for the line through P with direction u

•	To the result of the Line command, apply the GetRepresentation command.

$L ≔ GetRepresentation (Line ([a, b, c], ⟨p, q, r⟩), form = parametric)$

$[x = p t + a, y = q t + b, z = r t + c]$

Obtain $w (t) = f (x (t), y (t), z (t))$

•	Use the eval command.

$w ≔ eval (f (x, y, z), L)$

$f (p t + a, q t + b, r t + c)$

Obtain $w (t)$ 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.

$DD ≔ eval (diff (w, t), t = 0)$

$D_{1} (f) (a, b, c) p + D_{2} (f) (a, b, c) q + D_{3} (f) (a, b, c) r$

$convert (DD, diff)$

$(\frac{\partial}{\partial a} f (a, b, c)) p + (\frac{\partial}{\partial b} f (a, b, c)) q + (\frac{\partial}{\partial c} f (a, b, c)) r$

<< Previous Example Section 4.4 Next Example >>

© Maplesoft, a division of Waterloo Maple Inc., 2024. All rights reserved. This product is protected by copyright and distributed under licenses restricting its use, copying, distribution, and decompilation.

For more information on Maplesoft products and services, visit www.maplesoft.com

Download Help Document

Maple

Maple Add-Ons

Student Success Platform

Improving Retention Rates

MapleSim

MapleSim Add-Ons

Systems Engineering

Consulting Services

Maple T.A. and Möbius

Education

Industries

Automotive and Aerospace

Robotics

Machine Design & Industrial Automation

Other

Application Areas

Product Pricing

Purchasing

Institutional Student Licensing

Maplesoft Elite Maintenance (EMP)

Support

Product Training

Online Product Help

Webinars & Events

Publications

Content Hubs

Examples & Applications

Community

About Maplesoft

Media Center

User Community

Contact

Online Help

All Products Maple MapleSim

Maple

Powerful math software that is easy to use

Maple Add-Ons

Student Success Platform

Improving Retention Rates

MapleSim

Advanced System Level Modeling

MapleSim Add-Ons

Systems Engineering

Consulting Services

Maple T.A. and Möbius

Education

Industries

Automotive and Aerospace

Robotics

Machine Design & Industrial Automation

Other

Application Areas

Product Pricing

Purchasing

Institutional Student Licensing

Maplesoft Elite Maintenance (EMP)

Support

Product Training

Online Product Help

Webinars & Events

Publications

Content Hubs

Examples & Applications

Community

About Maplesoft

Media Center

User Community

Contact

Online Help

All Products Maple MapleSim