# 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

``````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
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``````(4y 3 x)dx+ (x 4 y)dy
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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
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### 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
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(^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
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``````F.ds
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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

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``````F.ds
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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 :
``````
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``````F.dS
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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

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``````
``````
``````Fds.
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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

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``````
``````
``````
``````FdA.
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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

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``````
``````
``````
``````(F)dA.
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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

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``````
``````
``````FdA.
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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

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``````
``````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

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``````
``````(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

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``````
``````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

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``````Fds
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``````
``````Gds.
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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

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``````
``````Fds.
``````

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

``````D
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``````cos(x+y) dA.
``````