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Definition 3.32 If ƒ is a function of two variables r and y such that f. and fy exist, then the gradient of f, denoted by Vƒ (read as “del f"), is given by V/(1,v) = f.(x, y) î + f,(x, y) j. Note that the gradient Vf(x,y) is a vector and Dyf(r,y) = U - V f(r,y). Suppose a is the radian measure of the angle between U and Vf. Then D7f(r, y) = U -Vf(x, 9) = ||U|||V/(r,y)|| 0o a. (3.2) Equation (3.2) tells us that Df(x, y) will be a maximum when cos a = 1; so that we have ||V f(x, y)| gives the maximum value of D7f at the point (x, 9).

Definition 3.32 If ƒ is a function of two variables r and y such that f. and fy exist, then the gradient of f, denoted by Vƒ (read as “del f"), is given by V/(1,v) = f.(x, y) î + f,(x, y) j. Note that the gradient Vf(x,y) is a vector and Dyf(r,y) = U - V f(r,y). Suppose a is the radian measure of the angle between U and Vf. Then D7f(r, y) = U -Vf(x, 9) = ||U|||V/(r,y)|| 0o a. (3.2) Equation (3.2) tells us that Df(x, y) will be a maximum when cos a = 1; so that we have ||V f(x, y)| gives the maximum value of D7f at the point (x, 9).

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.2: Graphs Of Equations
Problem 45E
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Definition 3.32 If f is a function of two variables x and y such that f. and fy exist, then the gradient of f, denoted by Vƒ (read as “del f"), is given by V{(r,y) = f.(z. y) i + f,(r, y) j. Note that the gradient Vf(x, y) is a vector and Df(1,9) = U - V j(z, y). Suppose a is the radian measure of the angle between U and Vf. Then D7f(z, y) = U·Vf(r, y) = ||U|||Vf(r, y)|| o a. (3.2) Equation (3.2) tells us that D#f(x, y) will be a maximum when cos a = 1; so that we have ||Vf(x, y)|| gives the maximum value of D7f at the point (r, y).
Transcribed Image Text:Definition 3.32 If f is a function of two variables x and y such that f. and fy exist, then the gradient of f, denoted by Vƒ (read as “del f"), is given by V{(r,y) = f.(z. y) i + f,(r, y) j. Note that the gradient Vf(x, y) is a vector and Df(1,9) = U - V j(z, y). Suppose a is the radian measure of the angle between U and Vf. Then D7f(z, y) = U·Vf(r, y) = ||U|||Vf(r, y)|| o a. (3.2) Equation (3.2) tells us that D#f(x, y) will be a maximum when cos a = 1; so that we have ||Vf(x, y)|| gives the maximum value of D7f at the point (r, y).
5. The electric potential is V volts at any point (1, y) in the ry plane and V = e-2" cos(2y). Distance is measured in feet. (a) Find the rate of change of the potential at the point (0, r) in the direction of the unit vector cos n î+ sin ¿r ĵ. COS
Transcribed Image Text:5. The electric potential is V volts at any point (1, y) in the ry plane and V = e-2" cos(2y). Distance is measured in feet. (a) Find the rate of change of the potential at the point (0, r) in the direction of the unit vector cos n î+ sin ¿r ĵ. COS
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