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B3. Find the gradient of the following functions at the given point and calculate the directional derivative in the direction of the unit vector u. (i) f(x, y) = x² - y/x² at (1,2), u = i – j. - (ii) g(x, y) = x² + 2xy + y² at (1,1), u = si + tj. . In case (ii), find the values of s and t that make u Vg: (a) a maximum, (b) a minimum, and (c) zero. Hint: remember that u is a unit vector, so s2 + t² = 1. Interpret your results geometrically.

B3. Find the gradient of the following functions at the given point and calculate the directional derivative in the direction of the unit vector u. (i) f(x, y) = x² - y/x² at (1,2), u = i – j. - (ii) g(x, y) = x² + 2xy + y² at (1,1), u = si + tj. . In case (ii), find the values of s and t that make u Vg: (a) a maximum, (b) a minimum, and (c) zero. Hint: remember that u is a unit vector, so s2 + t² = 1. Interpret your results geometrically.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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B3. Find the gradient of the following functions at the given point and calculate the directional derivative in the direction of the unit vector u. (i) f(x, y) = x² - y/x² at (1,2), u= gi-gj (ii) g(x, y) = = x² + 2xy + y² at (1,1), u = si + tj. In case (ii), find the values of s and t that make u Vg: (a) a maximum, (b) a minimum, and (c) zero. Hint: remember that u is a unit vector, so s² + t2 = 1. Interpret your results geometrically.
Transcribed Image Text:B3. Find the gradient of the following functions at the given point and calculate the directional derivative in the direction of the unit vector u. (i) f(x, y) = x² - y/x² at (1,2), u= gi-gj (ii) g(x, y) = = x² + 2xy + y² at (1,1), u = si + tj. In case (ii), find the values of s and t that make u Vg: (a) a maximum, (b) a minimum, and (c) zero. Hint: remember that u is a unit vector, so s² + t2 = 1. Interpret your results geometrically.
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