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This is the Exam of Mathematics which includes Plane Parallel, Specific Heat, Perpendicular, Unit Tangent Vector, Parametric Equations, Vector Parallel, Parameterization, Curve, Intersection etc. Key important points are: Minimum Values, Maximum, Function, Curl, Vector Field, Function, Parametrized, Conservative, Constraints, Subject
Typology: Exams
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The exam is worth a total of 100 points with a duration of 2.5 hours. No books, notes or calculators are allowed. Justify all answers, show all work and explain your reasoning carefully. You will be graded on the clarity of your explanations as well as the correctness of your answers. UBC Rules governing examinations: (1) Each candidate should be prepared to produce his/her library/ AMS card upon request. (2) No candidate shall be permitted to enter the examination room after the expiration of one half hour, or to leave during the first half hour of the examination. Candidates are not permitted to ask questions of the invigilators, except in cases of supposed errors or ambiguities in the examination questions. (3) Candidates guilty of any of the following or similar practices shall be immediately dismissed from the examination, and and shall be liable to disciplinary action: a) Making use of any books, papers or memoranda, other than those authorized by the examiners. b) Speaking or communicating with other candidates. c) Purposely exposing written papers to the view of other candidates. The plea of accident or forgetfulness will not be received. (4) Smoking is not permitted during examinations.
Find the maximum and minimum values of
f (x, y) = e−^2 x
(^2) −y 2 (x^2 + 2y^2 )
on the set D = {(x, y)| 2 x^2 + y^2 ≤ 4 }.
Find the maximum and minimum values of f (x, y, z) = 4x− 2 y subject to the constraints 2z−x−y = 4 and x^2 + z^2 = 1.
Use the divergence theorem to calculate the flux of
Fˆ (x, y, z) =< x(y + y^2 ) − x
3 3
, x^2 y −
y^2 −
y^3 3
, z + (x^3 + 1)y >
across the surface S given by z = 4 − x^2 − y^2 , z ≥ 0.
Let S be the part of the plane x + y + z = 1 that lies in the first octant (that is the region where x ≥ 0, y ≥ 0 and z ≥ 0), and let C be the boundary of S oriented counterclockwise when viewed from above. Moreover let
Fˆ (x, y, z) =< y cos(2πx), ex^ + 2, 1 − y(z − 1) >.
a) Find the curl of Fˆ. b) Use Stokes’ theorem to evaluate
C Fˆ · drˆ.
Evaluate (^) ∫ ∫ ∫
E
ez √ 4 − x^2 − y^2
dV,
where E is the subset of the solid sphere x^2 + y^2 + z^2 ≤ 4 which lies inside the cylinder x^2 + y^2 = 1 and above the xy-plane.