Math 21a Third Old Final Exam Spring, 2009

1 True-False questions. Circle the correct letter. No justifications are required.

T F The curve r(t) = (1− t)A+ tB, t ∈ [0, 1] connects the point A with the point B.

T F For every c, the function u(x, t) = (2 cos(ct) + 3 sin(ct)) sin(x) is a solution to the waveequation utt = c2uxx.

T F Let (x0, y0) be the maximum of f(x, y) under the constraint g(x, y) = 1. Thenfxx(x0, y0) < 0.

T F The function f(x, y, z) = x2− y2− z2 decreases in the direction 〈2,−2,−2〉/√

12 at thepoint (1, 1, 1).

T F Assume F is a vector field satisfying |F(x, y, z)| ≤ 1 everywhere. For every curve Cgiven by r(t), 0 ≤ t ≤ 1, the line integral

∫C F · dr is less or equal than the arc length

of C.T F Let F be a vector field which coincides with the unit normal vector n for each point on

a curve C. Then∫C F · dr = 0.

T F If for two vector fields F and G one has curl F = curl G, then F = G + 〈a, b, c〉, wherea, b, c are constants.

T F For every vector field F the identity grad(div F) = 0 holds.

T F If div F(x, y, z) = 0 for all (x, y, z), then curl F = 〈0, 0, 0〉 for all (x, y, z).

T F For every function f(x, y, z), there exists a vector field F such that div F = f .

2 Indicate with a check in the column below “conservative” if a vector fields is conservative (that is ifcurl F(x, y, z) = 〈0, 0, 0〉 for all points (x, y, z)). Similarly, mark the fields which are incompressible(that is if div F(x, y, z) = 0 for all (x, y, z)). No justifications are needed.

conservative incompressibleVector Field curlF = 0 div F = 0

F(x, y, z) = 〈−5, 5, 3〉

F(x, y, z) = 〈x, y, z〉

F(x, y, z) = 〈−y, x, z〉

F(x, y, z) = 〈x2 + y2, xyz, x− y + z〉

F(x, y, z) = 〈x− 2yz, y − 2zx, z − 2xy〉

3 Let E be a parallelogram in three dimensional space defined by two vectors u and v.

(a) Express the diagonals of the parallelogram as vectors in terms of u and v.

(b) What is the relation between the length of the cross product of the diagonals and the area ofthe parallelogram?

(c) Assume that the diagonals are perpendicular. What is the relation between the lengths of thesides of the parallelogram?

4 Find the volume of the wedge shaped solid that lies above the xy-plane, below the plane z = x, andwithin the cylinder x2 + y2 = 4.

Math 21a Third Old Final Exam Spring, 2009

5 Match the equations with the objects. No justifications are needed.

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I II0 5 10 15

0

5

10

15

III IV

V

0

5

10

15

0

5

10

15

0

5

10

15

VI4 6 8 10 12 14

2

4

6

8

10

12

14

VII

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................................................................................................

VIII

Equations:

(a) g(x, y, z) = cos(x) + sin(y) = 1 (b) y = cos(x)− sin(x)

(c) r(t) = 〈cos(t), sin(t)〉 (d) r(u, v) = 〈cos(u), sin(v), cos(u) sin(v)〉

(e) F(x, y, z) = 〈cos(x), sin(x), 1〉 (f) z = f(x, y) = cos(x) + sin(y)

(g) g(x, y) = cos(x)− sin(y) = 1 (h) F(x, y) = 〈cos(x), sin(x)〉

6 Consider the surface parameterized by

x = uv cos v y = uv sin v z = v,

where 0 ≤ u ≤ 1 and 0 ≤ v ≤ π Which of the following integrals is an integral for the surface areaof this surface?

(a)∫ π

0

∫ 1

0

√u2 + u2v4 du dv (b)

∫ π

0

∫ 1

0

√v2 + u4v2 du dv

(c)∫ π

0

∫ 1

0

√v2 + u2v4 du dv (d)

∫ π

0

∫ 1

0

√u2 + u4v4 du dv

(e) None of the above.

Math 21a Third Old Final Exam Spring, 2009

7 Let the curve C be parameterized by r(t) = 〈t, sin t, t2 cos t〉 for 0 ≤ t ≤ π. Let f(x, y, z) =z2ex+2y + x2 and F = ∇f . Find

∫C F · dr.

8 Evaluate the line integral of the vector field F(x, y) = 〈y2, x2〉 in the clockwise direction around thetriangle in the xy-plane defined by the points (0, 0), (1, 0) and (1, 1) in two ways:

(a) By evaluating the three line integrals.

(b) Using Green’s theorem.

9 Evaluate∫ 8

0

∫ 2

y1/3

y2ex2

x8dx dy.

10 Match each of the following iterated integrals with its domain of integration.

(a)∫ 1

0

∫ 1

y

∫ y

0f(x, y, z) dz dx dy (b)

∫ 1

0

∫ y

0

∫ 1

yf(x, y, z) dz dx dy

(c)∫ 1

0

∫ 1

y

∫ x

0f(x, y, z) dz dx dy (d)

∫ 1

0

∫ y

0

∫ 1

xf(x, y, z) dz dx dy

(e)∫ 1

0

∫ y

0

∫ y

xf(x, y, z) dz dx dy

0

0.5

1

y0

0.5

1

x

0

0.5

1

z

Region (i)

00.5

1

y0

0.5

1

x

0

0.5

1

z

Region (ii)

0

0.5

1

y

0

0.5

1x

0

0.5

1

z

Region (iii)

00.5

1

y0

0.5

1

x

0

0.5

1

z

Region (iv)

00.51

x

00.5

1y

0

0.5

1

z

Region (v)

Math 21a Third Old Final Exam Spring, 2009

11 Find the volume of the largest rectangular box with sides parallel to the coordinate planes that can

be inscribed in the ellipsoid x2

4 + y2

9 + z2

25 = 1.

12 (a) Find all the critical points of the function f(x, y) = −(x4 − 8x2 + y2 + 1).

(b) Classify the critical points.

(c) Locate the local and absolute maxima of f .

(d) Find the equation for the tangent plane to the graph of f at each absolute maximum.

13 Use Stokes’ theorem to evaluate the line integral of F(x, y, z) = 〈−y3, x3,−z3〉 along the curver(t) = 〈cos(t), sin(t), 1− cos(t)− sin(t)〉 with 0 ≤ t ≤ 2π.

14 Let S be the graph of the function f(x, y) = 2−x2−y2 which lies above the disk {(x, y) : x2+y2 ≤ 1}in the xy-plane. The surface S is oriented so that the normal vector points upwards. Compute theflux

∫∫S F · dS of the vector field

F =⟨−4x+

x2 + y2 − 11 + 3y2

, 3y, 7− z − 2xz1 + 3y2

⟩through S using the divergence theorem.