angular | Calculus – Vector Calculus 20E, Spring 2012, Lecture B, Final exam

Vector Calculus 20E, Spring 2012, Lecture B, Final exam

angular – 这是一个angular的practice, 考察angular的理解, 涉及了Vector Calculus等代写方面

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Three hours, eight problems. No calculators allowed.
Please start each problem on a new page.
You will get full credit only if you show all your work clearly.
Simplify answers if you can, but dont worry if you cant!
  1. Letbe the ellipsex^2 + 4y^2 = 4, oriented anticlockwise. Compute

(4y 3 x)dx+ (x 4 y)dy
  1. Find the integral

F.dswhereF=yi+xj+zkand the curveis the part of the parabola z=x^2 ,y= 0 going fromx=1 tox= 2.

  1. Find the integral

Rxyz dS, whereRis the rectangle inR

(^3) whose vertices are the points (0, 0 ,0),(1, 0 ,0),(0, 1 ,1),(1, 1 ,1).

  1. Find the area of the surface inR^3 described by (ucosv,usinv,u^2 ) 0 u2 0v 2 .
  2. Find the flux

F.dSof the vector fieldF=yixj+z

(^3) kthrough the surface inR (^3) which is oriented with an upward normal vector and described by (ucosv,usinv,v) 0 u2 0v 2 .

  1. Find the flux of the vector fieldF=x^3 i+y^3 j+z^3 kout of the unit sphere inR^3.
  2. Find the integral

F.dswhereF=xi+y

(^2) j+z (^3) kandis the oriented curve given by (sin^2 t,cos^3 t,sin^4 t) 0 t 2 .

  1. One of the two vector fields F=y^2 iz^2 j+x^2 k G= (x^3 3 xy^2 )i+ (y^3 3 x^2 y)j+zk

is conservative, and the other is not. Which is which? Find a potential for the conservative one.

1

Vector Calculus 20E, Spring 2013, Lecture A, Final exam

Three hours, eight problems. No calculators allowed.
Please start each problem on a new page.
You will get full credit only if you show all your work clearly.
Simplify answers if you can, but dont worry if you cant!
  1. Letbe the closed curve given by the equationsx=t^2 t, y= 2t^3 3 t^2 +tfor 0t1. Using Greens theorem, find the area enclosed by the curve.
  2. Find the integral

F.dswhereF=yi+xj+zkandis the helical curvex= 2 cost, y= 2 sint, z=tfor 0t 2 , oriented in the direction of increasingt.

  1. Find the integral

y

(^2) dA, where is the part of the cylinderx (^2) +y (^2) = 4 lying between the planesz= 0 andz=x+ 3.

  1. LetDbe the standard unit disc in thexy-plane, and let be the part of the graph of the functionz=xylying over the domainD. Find the surface area of .
  2. Let be the hemispherex^2 +y^2 +z^2 = 16,z0, oriented with the upward normal, and let LetFbe the vector field (x^2 +z)i+ 3xyzj+ (2xz)k. Compute the integral

(F).dA
  1. Find the flux of the vector fieldF=x^2 yi+z^8 j 2 xyzkout of the surface of the standard unit cube (0x,y,z1) inR^3.
  2. Find the integral

F.dswhereF=xi+yj+zkandis the oriented curve given by
(sin^2 tcost,cos^2 tsint,(t)^4 ) 0 t 2 .
  1. One of the two vector fields F= 3x^2 yi+x^3 j+ 5k G= (x+z)i+ (zy)j+ (xy)k

is conservative, and the other is not. Which is which? Find a potential for the conservative one.

1

Vector Calculus 20E, Fall 2014, Lecture A, Final Exam

Three hours, eight problems. No calculators allowed.
Please start each problem on a new page.
You will get full credit only if you show all your work clearly.
Simplify answers if you can, but dont worry if you cant!
  1. LetDbe the upper half of the unit disc, given byx^2 +y^2 1 ,y0. Find the average of the functionf(x,y) =yoverD.
  2. LetDbe the right-hand half of the unit disc, given byx^2 +y^2 1 ,x0. LetD=T(D), whereTis the map (u,v)7(u^2 v^2 , 2 uv). Calculate the area ofD.
  3. LetCbe the curve in the plane described byt7(cos^3 t,sint) for 0t 2 . Use Greens theorem to compute the area enclosed byC.
  4. Let be the part of the conez=

x^2 +y^2 lying above the standard unit square 0x,y1. Compute the surface area of .

  1. LetCbe the oriented tri angular path formed by travelling from (1, 0 ,0) to (0, 1 ,0) to (0, 0 ,1) and then back to (1, 0 ,0) along straight line segments. LetFbe the vector field given byF(x,y,z) = (y,x,x^2 ). Compute the circulation ofFaroundC:
C
F.ds
  1. Letbe the oriented patht7(

1 +t^2 ,^3

1 +t^3 ,^4

1 +t^4 ) for 0t1. LetFbe the vector field given byF(x,y,z) = (yz,xz,xy). IsFconservative? Calculate


F.ds
  1. Let be the part of the unit spherex^2 +y^2 +z^2 = 1 withx,y,z0, oriented outwards from the origin as usual. LetFbe the vector field given byF(x,y,z) = (y,x,1). Compute the flux of Fout of :

F.dS
  1. Let be the surface made by gluing the upper unit hemisphere (given byx^2 +y^2 +z^2 = 1,z0) onto the unit disc in thexy-plane (given byx^2 +y^2 1 ,z= 0); orient the whole surface outwards. LetFbe the vector field given byF(x,y,z) = (x^2 ,xz, 3 z). Compute the flux ofFout of :

F.dS

Vector Calculus 20E, Winter 2016, Lecture A, Final Exam

Three hours, eight problems. No calculators allowed.
Please start each problem on a new page.
You will get full credit only if you show all your work clearly.
Simplify answers if you can, but dont worry if you cant!

1.LetRbe the solid region between the spheres of radius 1 and 2, centred on theorigin. Compute

R
(x+y)^2 dV.

2.Let be the surface which is given byz= 1x^2 y^2 , z0. Compute the surface area of .

3.Let be the part of the unit spherex^2 +y^2 +z^2 = 1 which lies above the planez=^12. Compute the average of the functionf(x, y, z) =zover .

4.Letbe the oriented curve parametrised byt7(t, t^2 , t^3 ) for 0t1 and letFbe the vector field given byF(x, y, z) = (x+z, y^3 , 1 x). Compute



Fds.

5.LetRbe the regionx 0 , y 0 , z 0 , x+y+z1, and let be its boundary surface, oriented outwards. LetFbe the vector field given byF(x, y, z) = (4xz^2 , x+ 3z, 6 z). Compute



FdA.

6.Let be the part of the cone given byz=

x^2 +y^2 , 0 z1, oriented outwards. LetFbe the vector field given byF(x, y, z) = (z^2 y, z^2 x, z^4 ). Compute



(F)dA.

7.Let be the unit hemispherex^2 +y^2 +z^2 = 1, x0, oriented using the outward normal, and letFbe the vector field given byF(x, y, z) = (y, x, z). Compute the flux



FdA.

8.Find the potential functionwhich satisfies( 2 , 2 , 2 ) = 0 and=F, whereFis the vector field given byF(x, y, z) = (sinyzcosx, xcosy+ sinz, ycoszsinx).

Vector Calculus 20E, Winter 2017, Lecture B, Final Exam

Three hours, eight problems. No calculators allowed.
Please start each problem on a new page.
You will get full credit only if you show all your work clearly.
Simplify answers if you can, but dont worry if you cant!

1.Let be the part of the plane 2x+y+z= 6 wherex, y, z0. Compute


(x+z) dA.

2.Let be the piece of a cylinder given byx^2 +y^2 = 4,y0 and 0z1, oriented with the outward normal. LetFbe the vector field given byF(x, y, z) = (x, 1 , z^2 ). Compute


FdA.

3.Let be the hemisphere given by (x1)^2 +y^2 +z^2 = 1 andz0, oriented with the upward normal. LetFbe the vector field given byF(x, y, z) = (esinzy, x+ sinz,cosx). Compute


(F)dA.

4.LetRbe the solid pipe defined by 1x^2 +y^2 4 and 0z10, and let be its boundary surface, oriented out ofR. LetFbe the vector field given byF(x, y, z) = (x+yz, y+x^2 z^3 , xyz). Compute


FdA.

5.LetRbe the part of the unit ball given byx^2 +y^2 +z^2 1,x, y, z0. Find the average, over R, of the distance from the origin.

6.Letbe some curve which runs from (0, 0 ,0) to (1, 2 ,3). LetFandGbe vector fields given byF(x, y, z) = (3x^2 y^2 z, 2 x^3 yz, x^3 y^2 ) andG(x, y, z) = (3x^2 y^2 z, 2 x^3 yz, x^2 y^3. Evaluate, or say why it cant be evaluated without further information, the integrals



Fds


Gds.

7.Letbe the straight-line segment running from (1, 2 ,3) to (4, 0 ,2). LetFbe the vector field given byF(x, y, z) = (xz, 3 y,1). Calculate


Fds.

8.LetDbe the parallelogram whose vertices are (0,1),(1,0),(2,2),(3,1). Evaluate

D
cos(x+y) dA.