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Directional Derivatives

    Partial derivatives find the rate of change of directional derivatives _gr_1.gif] in the directions of the directional derivatives _gr_2.gif] and directional derivatives _gr_3.gif] axis; that is in the direction of the unit vectors directional derivatives _gr_4.gif] and directional derivatives _gr_5.gif], respectively. More precisely, if directional derivatives _gr_6.gif], then directional derivatives _gr_7.gif] and if directional derivatives _gr_8.gif], then directional derivatives _gr_9.gif] To find the rate of change of directional derivatives _gr_10.gif] at directional derivatives _gr_11.gif] in the direction of directional derivatives _gr_12.gif], we use the vertical plane that passes through directional derivatives _gr_13.gif] in the direction directional derivatives _gr_14.gif] which intersects directional derivatives _gr_15.gif] in a curve directional derivatives _gr_16.gif]. The slope of the tangent line directional derivatives _gr_17.gif] to directional derivatives _gr_18.gif] at directional derivatives _gr_19.gif] is the rate of change of directional derivatives _gr_20.gif] in the direction of directional derivatives _gr_21.gif] If directional derivatives _gr_22.gif] is another point on directional derivatives _gr_23.gif] and directional derivatives _gr_24.gif] are the projections of directional derivatives _gr_25.gif] on the directional derivatives _gr_26.gif]plane, then the vector directional derivatives _gr_27.gif] is parallel to directional derivatives _gr_28.gif] and so

directional derivatives _gr_29.gif]

for some scalar directional derivatives _gr_30.gif] Therefore, directional derivatives _gr_31.gif] and

directional derivatives _gr_32.gif]

If we take the limit as directional derivatives _gr_33.gif], we obtain the rate of change of directional derivatives _gr_34.gif] (with respect to distance) in the direction of directional derivatives _gr_35.gif], which is called the directional derivative of directional derivatives _gr_36.gif] in the direction of directional derivatives _gr_37.gif]

Definition (Directional Derivative) Let directional derivatives _gr_38.gif] be a function of two variables, and let directional derivatives _gr_39.gif] be a unit vector. The directional derivative of directional derivatives _gr_40.gif] at directional derivatives _gr_41.gif] in the direction of directional derivatives _gr_42.gif] is given by

directional derivatives _gr_43.gif]

provided the limit exists.

Proposition (Directional Derivative) Let directional derivatives _gr_44.gif] be a function that is differentiable at directional derivatives _gr_45.gif] Then directional derivatives _gr_46.gif] has a directional derivative in the direction of the unit vector directional derivatives _gr_47.gif] which is given by

directional derivatives _gr_48.gif]

    Proof. We define a function directional derivatives _gr_49.gif] of a single variable directional derivatives _gr_50.gif] by directional derivatives _gr_51.gif] so that

directional derivatives _gr_52.gif]

directional derivatives _gr_53.gif]

directional derivatives _gr_54.gif]

Applying the chain rule with directional derivatives _gr_55.gif] and directional derivatives _gr_56.gif]

directional derivatives _gr_57.gif]

directional derivatives _gr_58.gif]

When directional derivatives _gr_59.gif] we have directional derivatives _gr_60.gif] and directional derivatives _gr_61.gif] so that  

directional derivatives _gr_62.gif]

directional derivatives _gr_63.gif]
directional derivatives _gr_64.gif]

Example (Directional Derivative)

(a) Find the directional derivative directional derivatives _gr_65.gif] if  

directional derivatives _gr_66.gif]

and directional derivatives _gr_67.gif] is the unit vector given by the angle directional derivatives _gr_68.gif] What is directional derivatives _gr_69.gif]  

    Solution. We have, directional derivatives _gr_70.gif] and

directional derivatives _gr_71.gif]

directional derivatives _gr_72.gif]

directional derivatives _gr_73.gif]

directional derivatives _gr_74.gif]

Therefore,
directional derivatives _gr_75.gif]

directional derivatives _gr_76.gif]

directional derivatives _gr_77.gif]

(b) Let directional derivatives _gr_78.gif] be defined by directional derivatives _gr_79.gif] Find the directional derivative of directional derivatives _gr_80.gif] at directional derivatives _gr_81.gif] in the direction toward the origin.

    Solution. A vector in the direction from directional derivatives _gr_82.gif] to directional derivatives _gr_83.gif] is directional derivatives _gr_84.gif] so a unit vector in this direction is therefore directional derivatives _gr_85.gif] We find that

directional derivatives _gr_86.gif]

and

directional derivatives _gr_87.gif]

Therefore, the directional derivative is given by

directional derivatives _gr_88.gif]

(c) Let directional derivatives _gr_89.gif] be defined by directional derivatives _gr_90.gif] Find the directional derivative of directional derivatives _gr_91.gif] at directional derivatives _gr_92.gif] in the direction toward the origin.

    Solution. A vector in the direction from directional derivatives _gr_93.gif] to directional derivatives _gr_94.gif] is directional derivatives _gr_95.gif] so a unit vector in this direction is therefore directional derivatives _gr_96.gif] We find that

directional derivatives _gr_97.gif]

and

directional derivatives _gr_98.gif]

Therefore, the directional derivative is given by

directional derivatives _gr_99.gif]
directional derivatives _gr_100.gif]

directional derivatives _gr_101.gif] Recommended Reading
directional derivatives _gr_102.gif]functions of several variables
directional derivatives _gr_103.gif]graphs of functions
directional derivatives _gr_104.gif]polynomial functions
directional derivatives _gr_105.gif]rational functions
directional derivatives _gr_106.gif]level curves
directional derivatives _gr_107.gif]level surfaces
directional derivatives _gr_108.gif]limits of multivariate functions
directional derivatives _gr_109.gif]continuity of multivariate functions
directional derivatives _gr_110.gif]partial derivatives
directional derivatives _gr_111.gif]higher order partial derivatives
directional derivatives _gr_112.gif]tangent planes
directional derivatives _gr_113.gif]total differential
directional derivatives _gr_114.gif]linear approximation with multivariate functions
directional derivatives _gr_115.gif]differentiability
directional derivatives _gr_116.gif]chain rule with one independent parameter
directional derivatives _gr_117.gif]chain rule with two independent parameters
directional derivatives _gr_118.gif]chain rule with several independent parameters
directional derivatives _gr_119.gif]directional derivatives
directional derivatives _gr_120.gif]the gradient
directional derivatives _gr_121.gif]the gradient and directional derivatives
directional derivatives _gr_122.gif]steepest ascent and steepest descent
directional derivatives _gr_123.gif]normal property of the gradient
directional derivatives _gr_124.gif]tangent planes and normal lines
directional derivatives _gr_125.gif]relative extrema
directional derivatives _gr_126.gif]critical points
directional derivatives _gr_127.gif]second partials test
directional derivatives _gr_128.gif]absolute extrema
directional derivatives _gr_129.gif]lagrange multipliers with one parameter
directional derivatives _gr_130.gif]lagrange multipliers with two parameters

directional derivatives _gr_131.gif] Recommended Math Books
directional derivatives _gr_132.gif] Thomas' Calculus, Early Transcendentals, Media Upgrade (11th Edition)
 directional derivatives _gr_133.gif] Thomas' Calculus, Media Upgrade (11th Edition)
 directional derivatives _gr_134.gif] Thomas' Calculus Early Transcendentals; Student's Solutions Manual; Part One
directional derivatives _gr_135.gif] Calculus (With Analytic Geometry)(8th edition)
directional derivatives _gr_136.gif] Calculus (Stewart's Calculus Series)
 directional derivatives _gr_137.gif] Applied Calculus
directional derivatives _gr_138.gif] Calculus Textbooks
directional derivatives _gr_139.gif] Elementary Calculus
directional derivatives _gr_140.gif] Advanced Calculus
directional derivatives _gr_141.gif] Supplementary Resources

directional derivatives _gr_142.gif] Recommended Math Gifts
 directional derivatives _gr_143.gif] Math Happy
directional derivatives _gr_144.gif] Calculus Happy
directional derivatives _gr_145.gif] Homework Happy
directional derivatives _gr_146.gif] Limits Happy
directional derivatives _gr_147.gif] I Love Math
directional derivatives _gr_148.gif] I Love Calculus
directional derivatives _gr_149.gif] I Love Homework
directional derivatives _gr_150.gif] I Love Multivariate Calculus
directional derivatives _gr_151.gif] I Love Limits

Cite this as:
Directional Derivatives
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/directional-derivatives.html
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