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1. Find minimum value of f (r, y) = 2x+3y-6ry on the unit square D = {(x, y) : 0 < r, y < 1}.

1. Find minimum value of f (r, y) = 2x+3y-6ry on the unit square D = {(x, y) : 0 < r, y < 1}.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section: Chapter Questions
Problem 12T
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help with theses problems please.

1. Find minimum value of f (r, y) = 2x+3y- 6ry on the unit square D = {(r, y) : 0 < r, y < 1}. 2. Let W be the bounded region within the cylinder r? + y? = 1 that is bounded from below by the ry-plane and from above by the plane z = 2.x. Use cylindrical coordinates (r, 0, z) to evaluate ffw 2x dV. Hint: 2cos? 0 = 1+ cos(20). 3. Let W be the part of the unit sphere centered at origin such that rectangular coordinates of each point in W are positive. Use spherical coordinates (p,0, 6) to evaluate fSfw 2z dV. 4. Evaluate fe fds where f(x, y, z) = 12r? and the path C is given by r(t) (t, 221, ) and 0 <t< 1. 5. Let F = (y, z, x). (a) Find curl(F). Decide if F is conservative. Justify your answer. (b) Evaluate fF. dr along the path r(t) = (t, sin t, – cos t), 0 <t < T. 6. Let F = (2ry, r² + z, y). (a) Find f(x, y, z) such that Vf = F. (b) Evaluate f F dr along the path r(t), 0<t < 1, such that r(0) = (1, 3, –2) and r(1) = (-1,2, 3).
Transcribed Image Text:1. Find minimum value of f (r, y) = 2x+3y- 6ry on the unit square D = {(r, y) : 0 < r, y < 1}. 2. Let W be the bounded region within the cylinder r? + y? = 1 that is bounded from below by the ry-plane and from above by the plane z = 2.x. Use cylindrical coordinates (r, 0, z) to evaluate ffw 2x dV. Hint: 2cos? 0 = 1+ cos(20). 3. Let W be the part of the unit sphere centered at origin such that rectangular coordinates of each point in W are positive. Use spherical coordinates (p,0, 6) to evaluate fSfw 2z dV. 4. Evaluate fe fds where f(x, y, z) = 12r? and the path C is given by r(t) (t, 221, ) and 0 <t< 1. 5. Let F = (y, z, x). (a) Find curl(F). Decide if F is conservative. Justify your answer. (b) Evaluate fF. dr along the path r(t) = (t, sin t, – cos t), 0 <t < T. 6. Let F = (2ry, r² + z, y). (a) Find f(x, y, z) such that Vf = F. (b) Evaluate f F dr along the path r(t), 0<t < 1, such that r(0) = (1, 3, –2) and r(1) = (-1,2, 3).
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