review for exam 4.apr 30, 2001  · d = {x2 + y2 + z2 6 4, x > 0, y > 0, z > 0}. solution:...

Review for Exam 4.

I Sections 16.1-16.5, 16.7, 16.8.

I 50 minutes.

I 5 problems, similar to homework problems.

I No calculators, no notes, no books, no phones.

I No green book needed.

Review for Exam 4.

I (16.1) Line integrals.

I (16.2) Vector fields, work, circulation, flux (plane).

I (16.3) Conservative fields, potential functions.

I (16.4) The Green Theorem in a plane.

I (16.5) Surface area, surface integrals.

I (16.7) The Stokes Theorem.

I (16.8) The Divergence Theorem.

Conservative fields, potential functions (16.3).Example

Is the field F = 〈y sin(z), x sin(z), xy cos(z)〉 conservative?If “yes”, then find the potential function.

Solution: We need to check the equations

∂yFz = ∂zFy , ∂xFz = ∂zFx , ∂xFy = ∂yFx .

∂yFz = x cos(z) = ∂zFy ,

∂xFz = y cos(z) = ∂zFx ,

∂xFy = sin(z) = ∂yFx .

Therefore, F is a conservative field, that means there exists ascalar field f such that F = ∇f . The equations for f are

∂x f = y sin(z), ∂y f = x sin(z), ∂z f = xy cos(z).

Conservative fields, potential functions (16.3).

Example

Is the field F = 〈y sin(z), x sin(z), xy cos(z)〉 conservative?If “yes”, then find the potential function.

Solution: ∂x f = y sin(z), ∂y f = x sin(z), ∂z f = xy cos(z).Integrating in x the first equation we get

f (x , y , z) = xy sin(z) + g(y , z).

Introduce this expression in the second equation above,

∂y f = x sin(z) + ∂yg = x sin(z) ⇒ ∂yg(y , z) = 0,

so g(y , z) = h(z). That is, f (x , y , z) = xy sin(z) + h(z).Introduce this expression into the last equation above,

∂z f = xy cos(z) + h′(z) = xy cos(z) ⇒ h′(z) = 0 ⇒ h(z) = c .

We conclude that f (x , y , z) = xy sin(z) + c . C

Conservative fields, potential functions (16.3).

Example

Compute I =

∫C

y sin(z) dx + x sin(z) dy + xy cos(z) dz , where C

given by r(t) = 〈cos(2πt), 1 + t5, cos2(2πt)π/2〉 for t ∈ [0, 1].

Solution: We know that the field F = 〈y sin(z), x sin(z), xy cos(z)〉conservative, so there exists f such that F = ∇f , or equivalently

df = y sin(z) dx + x sin(z) dy + xy cos(z) dz .

We have computed f already, f = xy sin(z) + c .Since F is conservative, the integral I is path independent, and

I =

∫ (1,2,π/2)

(1,1,π/2)

[y sin(z) dx + x sin(z) dy + xy cos(z) dz

]I = f (1, 2, π/2)− f (1, 1, π/2) = 2 sin(π/2)− sin(π/2) ⇒ I = 1.

Conservative fields, potential functions (16.3).

Example

Show that the differential form in the integral below is exact,∫C

[3x2 dx +

z2

ydy + 2z ln(y) dz

], y > 0.

Solution: We need to show that the field F =⟨3x2,

z2

y, 2z ln(y)

⟩is conservative. It is, since,

∂yFz =2z

y= ∂zFy , ∂xFz = 0 = ∂zFx , ∂xFy = 0 = ∂yFx .

Therefore, exists a scalar field f such that F = ∇f , or equivalently,

df = 3x2 dx +z2

ydy + 2z ln(y) dz .

Review for Exam 4.

I (16.1) Line integrals.

I (16.2) Vector fields, work, circulation, flux (plane).

I (16.3) Conservative fields, potential functions.

I (16.4) The Green Theorem in a plane.

I (16.5) Surface area, surface integrals.

I (16.7) The Stokes Theorem.

I (16.8) The Divergence Theorem.

The Green Theorem in a plane (16.4).

Example

Use the Green Theorem in the plane to evaluate the line integral

given by

∮C

[(6y + x) dx + (y + 2x) dy

]on the circle C defined by

(x − 1)2 + (y − 3)2 = 4.

Solution: Recall:

∮C

F · dr =

∫∫S

(∂xFy − ∂yFx

)dx dy .

Here F = 〈(6y + x), (y + 2x)〉. Since ∂xFy = 2 and ∂yFx = 6,Green’s Theorem implies∮

C

[(6y + x) dx + (y + 2x) dy

]=

∮C

F · dr =

∫∫S

(2− 6) dx dy .

Since the area of the disk S = {(x − 1)2 + (y − 3)2 6 4} is π(22),∮C

F · dr = −4

∫∫S

dx dy = −4(4π) ⇒∮

C

F · dr = −16π.

Review for Exam 4.

I (16.1) Line integrals.

I (16.2) Vector fields, work, circulation, flux (plane).

I (16.3) Conservative fields, potential functions.

I (16.4) The Green Theorem in a plane.

I (16.5) Surface area, surface integrals.

I (16.7) The Stokes Theorem.

I (16.8) The Divergence Theorem.

Surface area, surface integrals (16.5).Example

Integrate the function g(x , y , z) = x√

4 + y2 over the surface cutfrom the parabolic cylinder z = 4− y2/4 by the planes x = 0,x = 1 and z = 0.

Solution:

R

z

4

1

4

x

y

S

We must compute: I =

∫∫S

g dσ.

Recall dσ =|∇f ||∇f · k|

dx dy , with k ⊥ R

and in this case f (x , y , z) = y2 + 4z − 16.

∇f = 〈0, 2y , 4〉 ⇒ |∇f | =√

16 + 4y2 = 2√

4 + y2.

Since R = [0, 1]× [−4, 4], its normal vector is k and |∇f · k| = 4.Then, ∫∫

S

g dσ =

∫∫R

(x√

4 + y2) 2

√4 + y2

4dx dy .

Surface area, surface integrals (16.5).

Example

Integrate the function g(x , y , z) = x√

4 + y2 over the surface cutfrom the parabolic cylinder z = 4− y2/4 by the planes x = 0,x = 1 and z = 0.

Solution:

∫∫S

g dσ =

∫∫R

(x√

4 + y2) 2

√4 + y2

4dx dy .

∫∫S

g dσ =1

2

∫∫R

x(4 + y2) dx dy =1

2

∫ 4

−4

∫ 1

0x(4 + y2) dx dy

∫∫S

g dσ =1

2

[∫ 4

−4(4+y2) dy

][∫ 1

0x dx

]=

1

2

(4y+

y3

3

)∣∣∣4−4

(x2

2

)∣∣∣10∫∫

S

g dσ =1

22(42 +

43

3

)1

2= 8

(1 +

4

3

)⇒

∫∫S

g dσ =56

3.

Review for Exam 4.

I (16.1) Line integrals.

I (16.2) Vector fields, work, circulation, flux (plane).

I (16.3) Conservative fields, potential functions.

I (16.4) The Green Theorem in a plane.

I (16.5) Surface area, surface integrals.

I (16.7) The Stokes Theorem.

I (16.8) The Divergence Theorem.

The Stokes Theorem (16.7).

Example

Use Stokes’ Theorem to find the flux of ∇× F outward throughthe surface S , where F = 〈−y , x , x2〉 andS = {x2 + y2 = a2, z ∈ [0, h]} ∪ {x2 + y2 6 a2, z = h}.

Solution: Recall:

∫∫S

(∇× F) · n dσ =

∮C

F · dr.

The surface S is the cylinder walls and its cover at z = h.Therefore, the curve C is the circle x2 + y2 = a2 at z = 0.That circle can be parametrized (counterclockwise) asr(t) = 〈a cos(t), a sin(t)〉 for t ∈ [0, 2π].∫∫

S

(∇× F) · n dσ =

∮C

F · dr =

∫ 2π

0F(t) · r′(t) dt,

where F(t) = 〈−a sin(t), a cos(t), a2 cos2(t)〉 andr′(t) = 〈−a sin(t), a cos(t), 0〉.

The Stokes Theorem (16.7).

Example

Use Stokes’ Theorem to find the flux of ∇× F outward throughthe surface S , where F = 〈−y , x , x2〉 andS = {x2 + y2 = a2, z ∈ [0, h]} ∪ {x2 + y2 6 a2, z = h}.

Solution: F(t) = 〈−a sin(t), a cos(t), a2 cos2(t)〉 andr′(t) = 〈−a sin(t), a cos(t), 0〉. Hence∫∫

S

(∇× F) · n dσ =

∫ 2π

0F(t) · r′(t) dt,

∫∫S

(∇× F) · n dσ =

∫ 2π

0

(a2 sin2(t) + a2 cos2(t)

)dt =

∫ 2π

0a2 dt.

We conclude that

∫∫S

(∇× F) · n dσ = 2πa2.

Review for Exam 4.

I (16.1) Line integrals.

I (16.2) Vector fields, work, circulation, flux (plane).

I (16.3) Conservative fields, potential functions.

I (16.4) The Green Theorem in a plane.

I (16.5) Surface area, surface integrals.

I (16.7) The Stokes Theorem.

I (16.8) The Divergence Theorem.

The Divergence Theorem (16.8).

Example

Use the Divergence Theorem to find the outward flux of the fieldF = 〈x2,−2xy , 3xz〉 across the boundary of the regionD = {x2 + y2 + z2 6 4, x > 0, y > 0, z > 0}.

Solution: Recall:

∫∫S

F · n dσ =

∫∫∫D

(∇ · F)dv .

∇ · F = ∂xFx + ∂yFy + ∂zFz = 2x − 2x + 3x ⇒ ∇ · F = 3x .∫∫S

F · n dσ =

∫∫∫D

(∇ · F)dv =

∫∫D

3x dx dy dz .

It is convenient to use spherical coordinates:∫∫S

F ·n dσ =

∫ π/2

0

∫ π/2

0

∫ 2

0

[3ρ sin(φ) cos(φ)

]ρ2 sin(φ) dρ dφ dθ.

The Divergence Theorem (16.8).

Example

Use the Divergence Theorem to find the outward flux of the fieldF = 〈x2,−2xy , 3xz〉 across the boundary of the regionD = {x2 + y2 + z2 6 4, x > 0, y > 0, z > 0}.

Solution:∫∫S

F ·n dσ =

∫ π/2

0

∫ π/2

0

∫ 2

0

[3ρ sin(φ) cos(φ)

]ρ2 sin(φ) dρ dφ dθ.

∫∫S

F · n dσ =[∫ π/2

0cos(θ) dθ

][∫ π/2

0sin2(φ) dφ

][∫ 2

03ρ3 dρ

]∫∫

S

F · n dσ =[sin(θ)

∣∣∣π/2

0

][1

2

∫ π/2

0

(1− cos(2φ)

)dφ

][3

4ρ4

∣∣20

]∫∫

S

F · n dσ = (1)1

2

(π

2

)(12) ⇒

∫∫S

F · n dσ = 3π.

Review for the Final Exam.

I Monday, December 13, 10:00am - 12:00 noon. (2 hours.)I Places:

I Sctns 001, 002, 005, 006 in E-100 VMC (Vet. Medical Ctr.),I Sctns 003, 004, in 108 EBH (Ernst Bessey Hall);I Sctns 007, 008, in 339 CSE (Case Halls).

I Chapters 12-16.

I Problems, similar to homework problems.

I No calculators, no notes, no books, no phones.

I No green book needed.

Review for Final Exam.

I Chapter 16, Sections 16.1-16.5, 16.7, 16.8.

I Chapter 15, Sections 15.1-15.4, 15.6.

I Chapter 14, Sections 14.1-14.7.

I Chapter 13, Sections 13.1, 13.3.

I Chapter 12, Sections 12.1-12.6.

Chapter 16, Integration in vector fields.Example

Use the Divergence Theorem to find the flux of F = 〈xy2, x2y , y〉outward through the surface of the region enclosed by the cylinderx2 + y2 = 1 and the planes z = −1, and z = 1.

Solution: Recall:

∫∫S

F · n dσ =

∫∫∫D

(∇ · F) dv . We start with

∇ · F = ∂x(xy2) + ∂y (x2y) + ∂z(y) ⇒ ∇ · F = y2 + x2.

The integration region is D = {x2 + y2 6 1, z ∈ [−1, 1]}. So,

I =

∫∫∫D

(∇ · F) dv =

∫∫∫D

(x2 + y2) dx dy dz .

We use cylindrical coordinates,

I =

∫ 2π

0

∫ 1

0

∫ 1

−1r2 dz r dr dθ = 2π

[∫ 1

0r3 dr

](2) = 4π

( r4

4

∣∣∣10

).

We conclude that

∫∫S

F · n dσ = π. C

Chapter 16, Integration in vector fields.

Example

Use Stokes’ Theorem to find the work done by the forceF = 〈2xz , xy , yz〉 along the path C given by the intersection of theplane x + y + z = 1 with the first octant, counterclockwise whenviewed from above.

Solution:

C

z

x + y + z = 11

1

x

1 y

x + y = 1

S

R

Recall:

∫C

F · dr =

∫∫S

(∇× F) · n dσ.

The surface S is the level surface f = 0 of

f = x + y + z − 1

therefore, ∇f = 〈1, 1, 1〉, |∇f | =√

3 and|∇f · k| = 1.

n =∇f

|∇f |=

1√3〈1, 1, 1〉, dσ =

|∇f ||∇f · k|

dx dy =√

3 dx dy .

Chapter 16, Integration in vector fields.

Example

Use Stokes’ Theorem to find the work done by the forceF = 〈2xz , xy , yz〉 along the path C given by the intersection of theplane x + y + z = 1 with the first octant, counterclockwise whenviewed from above.

Solution: n =1√3〈1, 1, 1〉 and dσ =

√3 dx dy .

We now compute the curl of F,

∇× F =

∣∣∣∣∣∣i j k

∂x ∂y ∂z

2xz xy yz

∣∣∣∣∣∣ = 〈(z − 0),−(0− 2x), (y − 0)〉

so ∇× F = 〈z , 2x , y〉. Therefore,∫∫S

(∇× F) · n dσ =

∫∫R

(〈z , 2x , y〉 · 1√

3〈1, 1, 1〉

)√3 dx dy

Chapter 16, Integration in vector fields.

Example

Use Stokes’ Theorem to find the work done by the forceF = 〈2xz , xy , yz〉 along the path C given by the intersection of theplane x + y + z = 1 with the first octant, counterclockwise whenviewed from above.

Solution:

I =

∫∫S

(∇× F) · n dσ =

∫∫R

(〈z , 2x , y〉 · 1√

3〈1, 1, 1〉

)√3 dx dy .

I =

∫∫R

(z + 2x + y) dx dy , z = 1− x − y ,

I =

∫ 1

0

∫ 1−x

0(1+x) dy dx =

∫ 1

0(1+x)(1−x) dx =

∫ 1

0(1−x2) dx .

I = x∣∣∣10− x3

3

∣∣∣10

= 1− 1

3=

2

3⇒

∫C

F · dr =2

3.

Chapter 16, Integration in vector fields.

Example

Find the area of the cone S given by z =√

x2 + y2 for z ∈ [0, 1].Also find the flux of the field F = 〈x , y , 0〉 outward through S .

Solution:

1

x

y

z

S

R

1

1

Recall: A(S) =

∫∫S

dσ. The surface S is the

level surface f = 0 of the functionf = x2 + y2 − z2. Also recall that

dσ =|∇f ||∇f · k|

dx dy .

Since ∇f = 2〈x , y ,−z〉, we get that

|∇f | = 2√

x2 + y2 + z2, z2 = x2 + y2 ⇒ |∇f | = 2√

2 z .

Also |∇f · k| = 2z , therefore, dσ =√

2 dx dy , and then we obtain

A(S) =

∫∫R

√2 dx dy =

∫ 2π

0

∫ 1

0

√2r dr dθ = 2π

√2r2

2

∣∣∣10

=√

2 π.

Chapter 16, Integration in vector fields.

Example

Find the area of the cone S given by z =√

x2 + y2 for z ∈ [0, 1].Also find the flux of the field F = 〈x , y , 0〉 outward through S .

Solution: We now compute the outward flux I =

∫∫S

F · n dσ.

Since

n =∇f

|∇f |=

1√2 z〈x , y ,−z〉.

I =

∫∫R

1√2 z

(x2 + y2)√

2 dx dy =

∫∫R

√x2 + y2 dx dy .

Using polar coordinates, we obtain

I =

∫ 2π

0

∫ 1

0r r dr dθ = 2π

r3

3

∣∣∣10

⇒ I =2π

3.

Review for Final Exam.

I Chapter 16, Sections 16.1-16.5, 16.7, 16.8.

I Chapter 15, Sections 15.1-15.4, 15.6.

I Chapter 14, Sections 14.1-14.7.

I Chapter 13, Sections 13.1, 13.3.

I Chapter 12, Sections 12.1-12.6.

Chapter 15, Multiple integrals.

Example

Find the volume of the region bounded by the paraboloidz = 1− x2 − y2 and the plane z = 0.

Solution:

y

D

Rx

z

1

1

1

So, D = {x2 + y2 6 1, 0 6 z 6 1− x2 − y2},and R = {x2 + y2 6 1, z = 0}. We know that

V (D) =

∫∫∫D

dv =

∫∫R

∫ 1−x2−y2

0dz dx dy .

Using cylindrical coordinates (r , θ, z), we get

V (D) =

∫ 2π

0

∫ 1

0

∫ 1−r2

0dz r dr dθ = 2π

∫ 1

0(1− r2) r dr .

Substituting u = 1− r2, so du = −2r dr , we obtain

V (D) = 2π

∫ 0

1u

(−du)

2= π

∫ 1

0u du = π

u2

2

∣∣∣10

⇒ V (D) =π

2.

Chapter 15, Multiple integrals.

Example

Set up the integrals needed to compute the average of the functionf (x , y , z) = z sin(x) on the bounded region D in the first octantbounded by the plane z = 4− 2x − y . Do not evaluate theintegrals.

Solution: Recall: f =1

V (D)

∫∫∫D

f dv .

D

z

x

yR

2x + y + z = 4

2x + y = 42

4

4

Since V (D) =

∫ 2

0

∫ 4−2x

0

∫ 4−2x−y

0dz dy dx ,

we conclude that

f =

∫ 2

0

∫ 4−2x

0

∫ 4−2x−y

0z sin(x) dz dy dx∫ 2

0

∫ 4−2x

0

∫ 4−2x−y

0dz dy dx

.

Chapter 15, Multiple integrals.Example

Reverse the order of integration and evaluate the double integral

I =

∫ 4

0

∫ 2

y/2ex2

dx dy .

Solution: We see that y ∈ [0, 4] and x ∈ [0, y/2], that is,

y = 2x

y

x2

4Therefore, reversing the integration ordermeans

I =

∫ 2

0

∫ 2x

0ex2

dy dx .

This integral is simple to compute,

I =

∫ 2

0ex2

x dx , u = x2, du = 2x dx ,

I =

∫ 4

0eu du ⇒ I = e4 − 1.

Review for the Final Exam.

I Monday, December 13, 10:00am - 12:00 noon. (2 hours.)I Places:

I Sctns 001, 002, 005, 006 in E-100 VMC (Vet. Medical Ctr.),I Sctns 003, 004, in 108 EBH (Ernst Bessey Hall);I Sctns 007, 008, in 339 CSE (Case Halls).

I Chapters 12-16.

I ∼ 12 Problems, similar to homework problems.

I No calculators, no notes, no books, no phones.

Plan for today: Practice final exam: April 30, 2001.

Remark on Chapter 16.

Remark: The normal form of Green’s Theorem is atwo-dimensional restriction of the Divergence Theorem.

I The Divergence Theorem:

∫∫S

F · n dσ =

∫∫∫D

(∇ · F) dv .

I Normal form of Green’s Thrm:

∮C

F · n ds =

∫∫S

(∇ · F) dA.

Remark: The tangential form of Green’s Theorem is a particularcase of the Stokes Theorem when C , S are flat (on z = 0 plane).

I The Stokes Theorem:

∮C

F · dr =

∫∫S

(∇× F) · n dσ.

I Tang. form of Green’s Thrm:

∮C

F · dr =

∫∫S

(∇× F) · k dA.

Practice final exam: April 30, 2001. Prbl. 1.

Example

Given A = (1, 2, 3), B = (6, 5, 4) and C = (8, 9, 7), find thefollowing:

I−→AB and

−→AC .

Solution:−→AB = 〈(6− 1), (5− 2), (4− 3)〉, hence

−→AB = 〈5, 3, 1〉. In the same way

−→AC = 〈7, 7, 4〉.

I−→AB +

−→AC = 〈12, 10, 5〉.

I−→AB ·

−→AC = 35 + 21 + 4.

I−→AB ×

−→AC =

∣∣∣∣∣∣i j k5 3 17 7 4

∣∣∣∣∣∣ = 〈(12− 7),−(20− 7), (35− 21)〉,

hence−→AB ×

−→AC = 〈5,−13, 14〉.

Practice final exam: April 30, 2001. Prbl. 2.

Example

Find the parametric equation of the line through the point(1, 0,−1) and perpendicular to the plane 2x − 3y + 5x = 35. Thenfind the intersection of the line and the plane.

Solution: The normal vector to the plane 〈2,−3, 5〉 is the tangentvector to the line. Therefore,

r(t) = 〈1, 0,−1〉+ t 〈2,−3, 5〉,so the parametric equations of the line are

x(t) = 1 + 2t, y(t) = −3t, z(t) = −1 + 5t.

The intersection point has a t solution of

2(1+2t)−3(−3t)+5(−1+5t) = 35 ⇒ 2+4t+9t−5+25t = 35

38t = 38 ⇒ t = 1 ⇒ r(1) = 〈3,−3, 4〉.

Practice final exam: April 30, 2001. Prbl. 3.

Example

The velocity of a particle is given by v(t) = 〈t2, (t3 + 1)〉, and theparticle is at 〈2, 1〉 for t = 0.

I Where is the particle at t = 2?

Solution: r(t) =⟨( t3

3+ rx

),( t4

4+ t + ry

)⟩. Since

r(0) = 〈2, 1〉, we get that r(t) =⟨( t3

3+ 2

),( t4

4+ t + 1

)⟩.

Hence r(2) = 〈8/3 + 2, 7〉.I Find an expression for the particle arc length for t ∈ [0, 2].

Solution: s(t) =

∫ t

0

√τ4 + (τ3 + 1)2 dτ .

I Find the particle acceleration.

Solution: a(t) = 〈2t, 3t2〉.

Practice final exam: April 30, 2001. Prbl. 4.Example

I Draw a rough sketch of the surface z = 2x2 + 3y2 + 5.

Solution: This is a paraboloid along the vertical direction,opens up, with vertex at z = 5 on the z-axis, and the x-radiusis a bit longer than the y -radius.

I Find the equation of the tangent plane to the surface at thepoint (1, 1, 10).

Solution: Introduce f (x , y) = 2x2 + 3y2 + 5, then

L(1,1)(x , y) = ∂x f (1, 1) (x − 1) + ∂y f (1, 1) (y − 1) + f (1, 1).

Since f (1, 1) = 10, and ∂x f = 4x , ∂y f = 6y , then

z = L(1,1)(x , y) = 4(x − 1) + 6(y − 1) + 10.

Practice final exam: April 30, 2001. Prbl. 5.

Example

Let w = f (x , y) and x = s2 + t2, y = st2. If ∂x f = x − y and∂y f = y − x , find ∂sw and ∂tw in terms of s and t.

Solution:

∂sw = ∂x f ∂sx+∂y f ∂sy = (x−y)2s+(y−x)t2 = (x−y)(2s−t2).

Therefore, ∂sw = (s2 + t2 − st2)(2s − t2).

∂tw = ∂x f ∂tx+∂y f ∂ty = (x−y)2t+(y−x)2st = (x−y)(2t−2st).

Therefore, ∂tw = (s2 + t2 − st2)2t(1− s).

Practice final exam: April 30, 2001. Prbl. 6.

Example

Find all critical points of the function f (x , y) = 2x2 + 8xy + y4

and determine whether they re local maximum, minimum of saddlepoints.

Solution:

∇f = 〈(4x + 8y), (8x + 4y3)〉 = 〈0, 0〉 ⇒

{x + 2y = 0,

2x + y3 = 0.

−4y + y3 = 0 ⇒

y = 0 ⇒ x = 0 ⇒ P0 = (0, 0)

y = ±2 ⇒ x = ∓4 ⇒

{P1 = (4,−2)

P2 = (−4, 2)

Since fxx = 4, fyy = 12y2, and fxy = 8, we conclude thatD = 3(16)y2 − 4(16).

Practice final exam: April 30, 2001. Prbl. 6.

Example

Find all critical points of the function f (x , y) = 2x2 + 8xy + y4

and determine whether they re local maximum, minimum of saddlepoints.

Solution:P0 = (0, 0), P1 = (4,−2), P2 = (−4, 2), D = 3(16)y2 − 4(16).

D(0, 0) = −4(16) < 0 ⇒ P0 = (0, 0) saddle point.

D(4,−2) = 12(16)− 4(16) > 0, fxx = 4 ⇒ P1 = (4,−2) min.

D(−4, 2) = 12(16)− 4(16) > 0, fxx = 4 ⇒ P1 = (−4, 2) min.

Practice final exam: April 30, 2001. Prbl. 7.

Example

Evaluate the integral I =

∫ 1

0

∫ √x

xy dy dx by reversing the order

of integration.

Solution: The integration region is the set in the square[0, 1]× [0, 1] in between the curves y = x and y =

√x . Therefore,

I =

∫ 1

0

∫ y

y2

y dx dy =

∫ 1

0y(y − y2) dy =

∫ 1

0(y2 − y3) dy

I =(y3

3− y4

4

)∣∣∣10

=1

3− 1

4⇒ I =

1

12.

Practice final exam: April 30, 2001. Prbl. 8.

Example

Find the work done by the force F = 〈yz , xz ,−xy〉 on a particlemoving along the path r(t) = 〈t3, t2, t〉 for t ∈ [0, 2].

Solution:

W =

∫C

F · dr =

∫ 2

0F(t) · r′(t) dt,

where F(t) = 〈t3, t4,−t5〉 and r′(t) = 〈3t2, 2t, 1〉. Hence

W =

∫ 2

0(3t5 + 2t5 − t5) dt =

∫ 2

04t5 dt =

4

6t6

∣∣∣20

=2

326.

Therefore, W = 27/3.

Practice final exam: April 30, 2001. Prbl. 9.

Example

Show that the force fieldF = 〈(y cos(z)− yzex), (x cos(z)− zex), (−xy sin(z)− yex)〉 isconservative. Then find its potential function. Then evaluate

I =

∫C

F · dr for r(t) = 〈t, t2, πt3〉.

Solution: The field F is conservative, since

∂xFy = cos(z)− zex = ∂yFx ,

∂xFz = −xy sin(z)− yex = ∂zFx ,

∂yFz = −x sin(z)− ex = ∂zFy .

The potential function is a scalar function f solution of

∂x f = y cos(z)− yzex , ∂y f = x cos(z)− zex , ∂z f = −xy sin(z)− yex .

Practice final exam: April 30, 2001. Prbl. 9.

Example

Show that the force fieldF = 〈(y cos(z)− yzex), (x cos(z)− zex), (−xy sin(z)− yex)〉 isconservative. Then find its potential function. Then evaluate

I =

∫C

F · dr for r(t) = 〈t, t2, πt3〉.

Solution: Recall:

∂x f = y cos(z)− yzex , ∂y f = x cos(z)− zex , ∂z f = −xy sin(z)− yex .

The x-integral of the first equation impliesf = xy cos(z)− yzex + g(y , z). Introduce f into the secondequation above,

x cos(z)− zex + ∂yg = x cos(z)− zex ⇒ ∂yg(y , z) = 0,

so we conclude g(y , z) = h(z), hence f = xy cos(z)− yzex + h(z).

Practice final exam: April 30, 2001. Prbl. 9.

Example

Show that the force fieldF = 〈(y cos(z)− yzex), (x cos(z)− zex), (−xy sin(z)− yex)〉 isconservative. Then find its potential function. Then evaluate

I =

∫C

F · dr for r(t) = 〈t, t2, πt3〉.

Solution: Recall: f = xy cos(z)− yzex + h(z).

Introduce f into the equation ∂z f = −xy sin(z)− yex , that is,

−xy sin(z)− ex + h′(z) = −xy sin(z)− yex ⇒ h′(z) = 0.

So, h(z) = c , a constant, hence f = xy cos(z)− yzex + c .

Finally

∫C

F · dr =

∫ (1,1,π)

(0,0,0)df = f (1, 1, π)− f (0, 0, 0).

So we conclude that

∫C

F · dr = −(1 + πe).

Practice final exam: April 30, 2001. Prbl. 10.

Example

Use the Green Theorem to evaluate the integral

∫C

Fx dx + Fy dy

where Fx = y + ex and Fy = 2x2 + cos(y) and C is the trianglewith vertices (0, 0), (0, 2) and (1, 1) traversed counterclockwise.

Solution: Denoting F = 〈Fx ,Fy 〉, Green’s Theorem says∫C

F · dr =

∫∫S

(∇× F) · k dA =

∫∫S

(∂xFy − ∂yFx) dA.

∫C

F · dr =

∫∫S

(4x − 1) dx dy =

∫ 1

0

∫ 2−y

y(4x − 1) dx dy .

A straightforward calculation gives

∫C

F · dr = 3.

Practice final exam: April 30, 2001. Prbl. 11.

Example

Find the surface area of the portion of the paraboloidz = 4− x2 − y2 that lies above the plane z = 0. Use polarcoordinates to evaluate the integral.

Solution:

A(S) =

∫∫S

dσ, dσ =|∇f ||∇f · k|

dx dy

where f = x2 + y2 + z − 4. Therefore,

∇f = 〈2x , 2y , 1〉 ⇒ |∇f | =√

1 + 4x2 + 4y2, ∇f · k = 1.

A(S) =

∫ 2π

0

∫ 2

0

√1 + 4r2 r dr dθ, u = 1 + 4r2, du = 8r dr .

The finally obtain A(S) = (π/6)(173/2 − 1).

Practice final exam: April 30, 2001. Prbl. 12.Example

Use the Stokes Theorem to evaluate I =

∫∫S

[∇× (y i)

]· n dσ

where S is the hemisphere x2 + y2 + z2 = 1, with z > 0.

Solution: F = 〈y , 0, 0〉. The border of the hemisphere is given bythe circle x2 + y2 = 1, with z = 0. This circle can be parametrizedfor t ∈ [0, 2π] as

r(t) = 〈cos(t), sin(t), 0〉 ⇒ r′(t) = 〈− sin(t), cos(t), 0〉,

and we also have F(t) = 〈sin(t), 0, 0〉. Therefore,∫∫S

(∇× F

)· n dσ =

∮ 2π

0F(t) · r′(t) dt = −

∫ 2π

0sin2(t) dt

∫∫S

(∇× F

)· n dσ = −1

2

∫ 2π

0

[1− cos(2t)

]dt.

Practice final exam: April 30, 2001. Prbl. 12.

Example

Use the Stokes Theorem to evaluate I =

∫∫S

[∇× (y i)

]· n dσ

where S is the hemisphere x2 + y2 + z2 = 1, with z > 0.

Solution:

∫∫S

(∇× F

)· n dσ = −1

2

∫ 2π

0

[1− cos(2t)

]dt.

Recall that ∫ 2π

0cos(2t) dt =

1

2

(sin(2t)

∣∣∣2π

0

)= 0.

Therefore, we obtain∫∫S

(∇× F

)· n dσ = −π.

C