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Math 292 – Final Exam

DVC Spring 2020

Name: ___________________________________________________________

Score: ___________ /120

6 of the 7 problems will be graded. Each problem is worth 20 points.

You must drop (omit) 1 problem.

Place the # you are omitting in the blank. # _________

Also place an “X” through the problem, or simply do not submit the problem you are omitting.

You will be graded on logic, use of notation and accuracy. Follow the directions for each problem and be sure to show

your work. No credit will be given to answer without explanations or reasonings.

Allowed Materials

• calculator

• Lecture notes

• Your personal homework solutions

• Scratch Paper

• Quiz solutions

__________________________________________________________________________________________________

By signing below, you are confirming that the only resources you have used for this exam are from the list given above.

You are also confirming that no other person has helped you with your solutions. If these requirements are not satisfied,

you will be given a zero on this exam and will fail the course.

X_________________________________________________________________________________

1) Consider the gravitational field 𝑭𝑭 represented by the function

𝑭𝑭(𝑥𝑥, 𝑦𝑦, 𝑧𝑧) =

−𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚

−𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚

−𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚

𝒊𝒊

+

𝒋𝒋

+

𝒌𝒌, 𝑚𝑚, 𝑀𝑀, 𝐺𝐺 𝑎𝑎𝑎𝑎𝑎𝑎 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐

(𝑥𝑥 2 + 𝑦𝑦 2 + 𝑧𝑧 2 )3/2

(𝑥𝑥 2 + 𝑦𝑦 2 + 𝑧𝑧 2 )3/2

(𝑥𝑥 2 + 𝑦𝑦 2 + 𝑧𝑧 2 )3/2

a) What is the domain of the gravitational field?

b) Sketch the vector field 𝑭𝑭

c) Explain why each term of 𝑭𝑭 has a negative sign.

d) Given that �𝒙𝒙⃑ =< 𝑥𝑥, 𝑦𝑦, 𝑧𝑧 > is a position vector, write

�⃑)

𝑭𝑭(𝑥𝑥, 𝑦𝑦, 𝑧𝑧) in compact form 𝑭𝑭(𝒙𝒙

e) Find the work done by 𝑭𝑭 in moving an object from (1,0,0) to (−1,0,3𝜋𝜋) along the helix as pictured.

2) Below is a theorem from your textbook (p.1089).

a) Use the picture (and your textbook for all given assumptions) to show that

𝑓𝑓𝑦𝑦 (𝑥𝑥, 𝑦𝑦) = 𝑄𝑄(𝑥𝑥, 𝑦𝑦)

Be sure to explain your reasoning, do not simply just copy the steps given in the textbook.

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b) Show that ∮𝐶𝐶 𝑭𝑭 ∙ 𝑑𝑑𝒓𝒓 = 0 if 𝑭𝑭 = 〈2𝑥𝑥𝑥𝑥 − 𝑦𝑦 2 , 𝑥𝑥 2 − 2𝑥𝑥𝑥𝑥〉 and 𝐶𝐶 is the ellipse

integral directly.

𝑥𝑥 2

4

+

𝑦𝑦 2

9

= 1 without calculating the line

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3) Solve the line integrals. If you use a theorem, be sure to state assumptions needed to use the theorem.

𝑡𝑡

a) Given 𝐹𝐹⃗ = sin 𝑥𝑥 𝒊𝒊 + cos 𝑦𝑦 𝒋𝒋 + 𝑥𝑥𝑥𝑥 𝒌𝒌 , where 𝐶𝐶 is represented by the equation 𝑟𝑟⃗(𝑡𝑡) = 𝑡𝑡 3 𝒊𝒊 − 𝑡𝑡 2 𝒋𝒋 + 𝑡𝑡 𝒌𝒌 , 0 → 1

Find ∫𝐶𝐶 𝑭𝑭 ∙ 𝑑𝑑𝒓𝒓. Leave answer in exact form. No decimal answers.

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b) Solve ∮𝐶𝐶 �𝑒𝑒 √𝑥𝑥

2 +1

+ 𝑦𝑦 + 𝜋𝜋� 𝑑𝑑𝑑𝑑 + �𝑥𝑥 2 +

1

� 𝑑𝑑𝑑𝑑

𝑦𝑦 2 +𝑦𝑦+1

where 𝐶𝐶 is boundary of the rectangle with vertices

(1,2), (5,2), (5,4) and (1,4) oriented in the clockwise direction.

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4) Follow the directions for each part.

a) Find ∇ ∙ 𝑭𝑭(1,2,3) where 𝑭𝑭 = 〈

field 𝑭𝑭 at (1,2,3)?

𝑥𝑥

,

𝑦𝑦

cos(𝑥𝑥 2 𝑦𝑦𝑦𝑦) , 𝑥𝑥𝑒𝑒 𝑦𝑦𝑦𝑦 〉 . What does the value you calculated indicate about the vector

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b) Determine if 𝐹𝐹⃗ = < 𝑦𝑦 2 𝑧𝑧 3 , 2𝑥𝑥𝑥𝑥𝑧𝑧 3 , 3𝑥𝑥𝑦𝑦 2 𝑧𝑧 2 > is a conservative vector field or not. Defend your claim mathematically.

_____/10

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c) Let 𝑓𝑓 be a scalar field and 𝐹𝐹 is some vector field. State whether each expression is meaningful.

If so, state whether it is a scalar field or a vector field. (example: ∇ ∙ 𝑭𝑭 is a scalar field.)

∇(∇ ∙ 𝑭𝑭)

∇(𝐹𝐹) × (∇ ∙ 𝐹𝐹)

∇ × (∇ × (∇ × 𝐹𝐹))

∇ ∙ (∇(∇ × 𝐹𝐹))

_____ /4

5) a-b Set-up the flux integrals for the given surfaces in the variables indicated. Your final answer should be a scalarvalued double integral. That is, the double integral should does not contain any vector quantities. The differential is

given. Do not solve the integrals you setup in a. and b. No work is needed for a-b.

a. 𝑭𝑭(𝑥𝑥, 𝑦𝑦, 𝑧𝑧) = 5𝚤𝚤̂ + 10𝚥𝚥̂ + 15𝑘𝑘� over the hemi-sphere with radius 8, 𝑧𝑧 ≥ 0.

_____/5

b. 𝑭𝑭(𝑥𝑥, 𝑦𝑦, 𝑧𝑧) = 5𝑥𝑥𝑥𝑥𝚤𝚤̂ + 4𝑧𝑧𝚥𝚥̂ over part of the cylinder with radius 2 and 𝑥𝑥 ≥ 0, 0 ≤ 𝑦𝑦 ≤

1

2

, 0 ≤ 𝑧𝑧 ≤ 2.

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_____/5

c. Evaluate ∬𝑆𝑆(𝑥𝑥 2 𝑧𝑧 + 𝑦𝑦 2 𝑧𝑧)𝑑𝑑𝑑𝑑, where 𝑆𝑆 is part of the plane 𝑧𝑧 = 9 + 𝑥𝑥 + 𝑦𝑦 that lies inside the cylinder 𝑥𝑥 2 + 𝑦𝑦 2 = 9.

_____/10

6) Stokes’ Theorem is given without hypothesis.

� 𝑑𝑑𝑑𝑑

� 𝑭𝑭 ∙ 𝑑𝑑𝒓𝒓 = �(∇ × 𝑭𝑭) ∙ 𝒏𝒏

𝜕𝜕𝜕𝜕

𝐶𝐶

a. What does 𝜕𝜕𝜕𝜕 and 𝐶𝐶 represent? What does the circle mean on the single integral?

Draw a general picture and give a brief explanation.

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b. What does the left-hand side of this equation represent? Give a brief explanation.

_____/5

_____/5

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c. Calculate 𝑛𝑛� 𝑑𝑑𝑑𝑑 if 𝑆𝑆 is given by the equation 𝑧𝑧 = 𝑥𝑥 2 + 3𝑦𝑦 2 . Explain what 𝑛𝑛� 𝑑𝑑𝑑𝑑 represents. Draw a picture to help

explain.

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d. What does ∇ × 𝐹𝐹 represent? Explain. Draw a diagram explaining what (∇ × 𝑭𝑭) ∙ 𝑛𝑛� 𝑑𝑑𝑑𝑑 is describing.

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7) Find the value of the flux ∬𝐷𝐷 𝐹𝐹⃗ ∙ 𝑛𝑛� 𝑑𝑑𝑑𝑑 through the cube bounded by the planes 𝑥𝑥 = 0, 𝑥𝑥 = 𝜋𝜋, 𝑦𝑦 = 0, 𝑦𝑦 = 𝜋𝜋, 𝑧𝑧 = 0

and 𝑧𝑧 = 𝜋𝜋 in the vector field 𝐹𝐹⃗ = 〈 3𝑥𝑥, 4𝑥𝑥𝑥𝑥 − 𝑥𝑥𝑥𝑥, 2𝑥𝑥𝑥𝑥 + 𝜋𝜋𝜋𝜋𝜋𝜋 〉. Your answer should be a scalar (no decimal answers).

If you use a Theorem, you must show that all assumptions of the Theorem are met.

_____/20

Extra Credit: A correct response will earn you a 5% boost on this exam. Partial credit is rare here.

Let 𝐵𝐵 ⊂ ℝ3 be a solid homogeneous object. Let 𝑢𝑢(𝑥𝑥, 𝑦𝑦, 𝑧𝑧) represent the heat at position (𝑥𝑥, 𝑦𝑦, 𝑧𝑧) in 𝐵𝐵 .

Fourier’s Law states that the velocity 𝑽𝑽(𝒙𝒙, 𝒚𝒚, 𝒛𝒛) of heat in an object is proportional to the gradient of 𝑢𝑢. That is,

𝑽𝑽 = −𝑘𝑘 ∇𝑢𝑢 ,

𝑘𝑘 > 0

where 𝑘𝑘 is a proportionality constant and 𝑽𝑽 is described as a vector (velocity) field.

Using Gauss’ Theorem, derive the heat equation (partial differential equation)

Hints

•

•

•

𝑘𝑘 2

𝜕𝜕𝜕𝜕

=

∇ 𝑢𝑢

𝜕𝜕𝜕𝜕 𝜎𝜎𝜎𝜎

Heat = 𝜎𝜎(𝑥𝑥, 𝑦𝑦, 𝑧𝑧) ∙ 𝜌𝜌(𝑥𝑥, 𝑦𝑦, 𝑧𝑧) ∙ 𝑢𝑢(𝑥𝑥, 𝑦𝑦, 𝑧𝑧) ∙ (volume) , where 𝜎𝜎 is the specific heat of the body and 𝜌𝜌 is the density

of the body at the position (𝑥𝑥, 𝑦𝑦, 𝑧𝑧)

∇2 denotes the Laplacian operator

Think about flux

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Tags:

gravitational field

F represented by the function

domain of the gravitational field

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