Calculus期末代考 Let S be the surface defined in cylindrical coordinates by r = 1 + z2, r 3, oriented with normals pointing towards the z-axis.

1.(20Points, Parts abc) Flux wave

Let K be the cylindrical surface (x z)2 + (y z)2 = 1 in R3. Let S be the portion of the surface z = sin x cos y inside the region (x  z)2 + (y  z)2 1. Let

(x, y, z) = (1 + x  y)(1, 1, 1).

a.Find a continuous unit normal vector field onK.

b.Sketch K and your chosen normalfield on the same diagram.

c.Calculate the  flux  of

through S oriented with normals pointing upwards.

2.(20Points, Parts ab) Double integral    Calculus期末代考

Let D denote the unit disk in R2given by: x2 + y2 < 1.

a.Consider the map  R2 defined by:

Show  that  T  is  a  di↵eomorphism,  that  is,  T  is  C1,  injective  (1-1),  surjective  (onto),  and  has C1 inverse.



3.(20Points, Parts ab) Loops and flux

a.Recall that (You may assume this without proof.)

Consider the vector field

(i)What is  the  (maximal)  domain  D of   ?  Is  conservative on D?

(ii)Is   conservative  on  the  upper  half-plane  {(x, y2 R2|y > 0} ?

(iii)LetCa,b be the circle in R2 of radius 2, centered at (a, b), traversed

Consider (a, b) such that Ca,b lies in the domain D of .  Determine the maximum value of and  describe  all  (a, b)  which  achieve  this  value.

b.Let     Calculus期末代考

Let S be the surface defined in cylindrical coordinates by r = 1 + z2, r 3, oriented with normals pointing towards the z-axis. Divide S into the portion S+ lying above the xy-plane, and the portion S lying below the xy-plane.

Determine  whether  the  flux  of  F~  through  S+ or  S is  larger.

4.(20 Points) Surfaceintegral

Let S be the portion of the surface x4 + y4 + z4 = 1 lying in the first octant in R3 (i.e. x  0,y   0,z   0),oriented  with  normals  pointing  away  from  the  origin. Calculate  where   (x, y, z) = (x2yz + sin(yz), xy2z + sin(xz), xyz2 + sin(xy)).

5.(Parts abc)AMGM  Calculus期末代考

(a)(4Points) Consider the map Φ(xi, · · , xn) = (y1, ·· · , yn) defined by (where this expression makes sense).

(i)Is the (maximal) domain of Φ closed and bounded? (You do not need to provide a proof.)

(ii)Isthe image of Ø closed and bounded? (You do not need to provide a proof.)

(b)(10Points) Using calculus techniques, prove that x2x2 · · for any x , ·· · ,x .

(c)(6 Points) Deduce the arithmetic mean – geometric mean inequality, that is, if a1, · · , an> 0,then

6.(20 Points, Parts abcde)True/False  Calculus期末代考

Circle whether or not the corresponding statement is TRUE or FALSE. Briefly justify your answer with a proof or counterexample. Important: Points will be awarded as follows:

(4 points) for correct answer with justification;

(3 points) for correct answer with promising but incomplete justification; (2 points) for correct answer;

(1 point) for no answer;

(0 points) for incorrect answer.    Calculus期末代考

a.If   is  a  nonconstant  flow  line  of  a  C2 vector  field  decreasing function of t.


b.LetD = R3 \ {x z = 0} denote 3D space with the y-axis removed.If   is a C2 vector field defined on D and  curl ≡ 0,  then   for  some  real  valued  function  f on  D.


c.Let f  :  R2 R be  a  C2 function  such  that   and detH(f )(0, 0)  0, where H(f ) is the Hessian (matrix of second partial derivatives).Then f has a local minimum at (0,0).


d.Usinglinear approximation of the function f (x, y, z) = ex+2yx+3y+z3 at (0, 0, 0), we can esti- mate f (0.01, 0.01, 0.03) by 1.


e.Define f :

Then  lim(x,y)→(0,0)f (x, y) = 0.



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