Library of Math
Library of Math
Online Math Organized by Subject Into Topics
Subscribe to the Library of Math Feed
Math Online Math Books

The Gradient and Directional Derivatives

Proposition (Gradient and Directional Derivative) If the gradient and directional derivatives _gr_1.gif] is a differentiable function of the gradient and directional derivatives _gr_2.gif] and the gradient and directional derivatives _gr_3.gif], then the directional derivative of the gradient and directional derivatives _gr_4.gif] at the point the gradient and directional derivatives _gr_5.gif] in the direction of the unit vector the gradient and directional derivatives _gr_6.gif] is the gradient and directional derivatives _gr_7.gif]

    Proof. Because the gradient and directional derivatives _gr_8.gif] and the gradient and directional derivatives _gr_9.gif] we have

the gradient and directional derivatives _gr_10.gif]
the gradient and directional derivatives _gr_11.gif]

Example (Gradient and Directional Derivative) Find the following directional derivatives.

(a) Find the directional derivative of the function the gradient and directional derivatives _gr_12.gif] at the point the gradient and directional derivatives _gr_13.gif] in the direction of the vector the gradient and directional derivatives _gr_14.gif]

    Solution. We first compute the gradient vector at the gradient and directional derivatives _gr_15.gif]

the gradient and directional derivatives _gr_16.gif]

and  

the gradient and directional derivatives _gr_17.gif]

Note that the gradient and directional derivatives _gr_18.gif] is not a unit vector, but since the gradient and directional derivatives _gr_19.gif], the unit vector in the direction of the gradient and directional derivatives _gr_20.gif] is the gradient and directional derivatives _gr_21.gif] Therefore, we have

the gradient and directional derivatives _gr_22.gif]

the gradient and directional derivatives _gr_23.gif]

the gradient and directional derivatives _gr_24.gif]

the gradient and directional derivatives _gr_25.gif]

(b) Let the gradient and directional derivatives _gr_26.gif] and let the gradient and directional derivatives _gr_27.gif] be a unit vector perpendicular to both the gradient and directional derivatives _gr_28.gif] and the gradient and directional derivatives _gr_29.gif] Find the directional derivative of the gradient and directional derivatives _gr_30.gif] at the gradient and directional derivatives _gr_31.gif] in the direction the gradient and directional derivatives _gr_32.gif]

    Solution. The gradient is

the gradient and directional derivatives _gr_33.gif]

and the gradient and directional derivatives _gr_34.gif] Now since we are looking for a unit vector perpendicular to both the gradient and directional derivatives _gr_35.gif] and the gradient and directional derivatives _gr_36.gif] we find, the gradient and directional derivatives _gr_37.gif] the gradient and directional derivatives _gr_38.gif] and so the gradient and directional derivatives _gr_39.gif] Therefore,

the gradient and directional derivatives _gr_40.gif]
the gradient and directional derivatives _gr_41.gif]

the gradient and directional derivatives _gr_42.gif] Recommended Reading
the gradient and directional derivatives _gr_43.gif]functions of several variables
the gradient and directional derivatives _gr_44.gif]graphs of functions
the gradient and directional derivatives _gr_45.gif]polynomial functions
the gradient and directional derivatives _gr_46.gif]rational functions
the gradient and directional derivatives _gr_47.gif]level curves
the gradient and directional derivatives _gr_48.gif]level surfaces
the gradient and directional derivatives _gr_49.gif]limits of multivariate functions
the gradient and directional derivatives _gr_50.gif]continuity of multivariate functions
the gradient and directional derivatives _gr_51.gif]partial derivatives
the gradient and directional derivatives _gr_52.gif]higher order partial derivatives
the gradient and directional derivatives _gr_53.gif]tangent planes
the gradient and directional derivatives _gr_54.gif]total differential
the gradient and directional derivatives _gr_55.gif]linear approximation with multivariate functions
the gradient and directional derivatives _gr_56.gif]differentiability
the gradient and directional derivatives _gr_57.gif]chain rule with one independent parameter
the gradient and directional derivatives _gr_58.gif]chain rule with two independent parameters
the gradient and directional derivatives _gr_59.gif]chain rule with several independent parameters
the gradient and directional derivatives _gr_60.gif]directional derivatives
the gradient and directional derivatives _gr_61.gif]the gradient
the gradient and directional derivatives _gr_62.gif]the gradient and directional derivatives
the gradient and directional derivatives _gr_63.gif]steepest ascent and steepest descent
the gradient and directional derivatives _gr_64.gif]normal property of the gradient
the gradient and directional derivatives _gr_65.gif]tangent planes and normal lines
the gradient and directional derivatives _gr_66.gif]relative extrema
the gradient and directional derivatives _gr_67.gif]critical points
the gradient and directional derivatives _gr_68.gif]second partials test
the gradient and directional derivatives _gr_69.gif]absolute extrema
the gradient and directional derivatives _gr_70.gif]lagrange multipliers with one parameter
the gradient and directional derivatives _gr_71.gif]lagrange multipliers with two parameters

the gradient and directional derivatives _gr_72.gif] Recommended Math Books
the gradient and directional derivatives _gr_73.gif] Thomas' Calculus, Early Transcendentals, Media Upgrade (11th Edition)
 the gradient and directional derivatives _gr_74.gif] Thomas' Calculus, Media Upgrade (11th Edition)
 the gradient and directional derivatives _gr_75.gif] Thomas' Calculus Early Transcendentals; Student's Solutions Manual; Part One
the gradient and directional derivatives _gr_76.gif] Calculus (With Analytic Geometry)(8th edition)
the gradient and directional derivatives _gr_77.gif] Calculus (Stewart's Calculus Series)
 the gradient and directional derivatives _gr_78.gif] Applied Calculus
the gradient and directional derivatives _gr_79.gif] Calculus Textbooks
the gradient and directional derivatives _gr_80.gif] Elementary Calculus
the gradient and directional derivatives _gr_81.gif] Advanced Calculus
the gradient and directional derivatives _gr_82.gif] Supplementary Resources

the gradient and directional derivatives _gr_83.gif] Recommended Math Gifts
 the gradient and directional derivatives _gr_84.gif] Math Happy
the gradient and directional derivatives _gr_85.gif] Calculus Happy
the gradient and directional derivatives _gr_86.gif] Homework Happy
the gradient and directional derivatives _gr_87.gif] Limits Happy
the gradient and directional derivatives _gr_88.gif] I Love Math
the gradient and directional derivatives _gr_89.gif] I Love Calculus
the gradient and directional derivatives _gr_90.gif] I Love Homework
the gradient and directional derivatives _gr_91.gif] I Love Multivariate Calculus
the gradient and directional derivatives _gr_92.gif] I Love Limits

Cite this as:
The Gradient And Directional Derivatives
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/the-gradient-and-directional-derivatives.html
Online Math
about us contact us privacy policy terms of use mision statement lom help
The Library of Math - Online Math Organized by Subject Into Topics. © 2005 - 2008 www.LibraryOfMath.com All rights reserved.
Page copy protected against web site content infringement by Copyscape   Valid CSS! Valid HTML 4.01 Transitional