Download MATH 550 Spring 1995 Final Exam: Vector Calculus Problems and more Exams Vector Analysis in PDF only on Docsity!
MARK BOX
Problem Points
Total 100
MATH 550
SPRING 1995
FINAL EXAM
Prof. Girardi
NAME:
SSN:
Instructions:
(1) To receive credit you must work in a logical fashion, SHOW ALL YOUR WORK,
INDICATE YOUR REASONING, and when applicable put your answer on
the line (or in the box) provided.
(2) The “Mark Box” indicates the problems along with their points. Check
that your copy of the exam has all of the problems.
(3) Allowed are a calculator and the class handouts, as indicated on the syllabus.
Not allowed are other notes and books.
(4) This exam covers (from Intro. to Vector Analysis by Davis & Snider, 6
th
ed.)
sections:
- Let
A = 〈 1 , 2 , 5 〉 and
B = 〈− 4 , 1 , 0 〉. Let θ be the angle between
A and
B.
Let
A =
A
||
A⊥ where
A
||
is parallel to
B and
A⊥ is perpendicular to
B.
Find:
A| = cos θ =
B| = is 0 ≤θ ≤
Π
2
or
Π
2
< θ ≤ Π?
A
B =
A
||
A ×
B =
A⊥ =
- Write a parameterization of the line through (7, 8 , 9) that is perpendicular to
the plane given by 2x − y + 5z = 4.
Answer:
R(t) = 〈 , , 〉
for t
- Green’s Theorem: Consider the vector field
F (x, y) =
y + e
(x
2 )
, x
2
y
Find the line integral
Γ
F · d ~R where Γ is the square with vertices
(1, 2), (5, 2), (5, 4), (1, 4) , traversed counterclockwise.
Answer:
Γ
F · d
R =
- Stoke’s Theorem: Consider the vector field
F (x, y, z) =
− 3 y
2
, 4 z , 6 x
Find the line integral
C
F · d ~R where C is the triangle with vertices
(2, 0 , 0), (0, 2 , 1), (0, 0 , 0) oriented counterclockwise when viewed from above.
Answer:
C
F · d
R =
Hint: special case
- A puffo is running along the curve y = e
x in such a way that its abscissa (ie.
the x coordinate) is increasing in time. The puffo’s speed (i.e. scalar velocity)
is a constant 4 (ft/min) and his scalar acceleration is a constant 10 (ft/min
2
).
Express his velocity vector and acceleration vector as a function of the abscissa
(ie. as a function of x and not time t ).
ANSWER: ~v(x) = < , >
ANSWER: ~a(x) = < , >
Hints: Note that the derivative (with repect to time t ) of the position vector
(as a function of time) gives a vector tangent to the curve. The derivative
(with repect to x ) of the position vector (as a function of abscissa) also gives
a vector tangent to the curve. A vector is determined by a direction and a
length. Next note that if, as a function of time t , his velocity vector looks
like < v 1
(t), v 2
(t) > , then [v 1
(t)]
2
(t)]
2 = 16. Differeniate both sides
of this equation with respect to t to see what it says about v(t) and a(t).
PRESENT YOUR SOLUTION IN A LOGICAL ORDERLY WAY! MARKS
WHICH APPEAR TO ME TO BE RANDOM WILL RECEIVE NO POINTS!
The next page is blank to provide you with more space if so needed.
