Chapter 12: Functions of Several
Variables
Section 12.1
Introduction to Functions of Several Variables
Notation for Functions of Several Variables
Previously we have studied functions of one variable, y = f(x) in
which x was the independent variable and y
THE CHAIN RULE (CASE 1) Suppose that z=f (x, y)
is a differentiable function of x and y, where x=g (t)
and y=h (t) and are both differentiable functions of t.
Then z is a differentiable function of t and
dz f dx f dy
dt x dt y dt
dz z dx
d
d
z ddy
dt x dt
DEFINITION If f is a function of two variables
x and
d y , then
th
the
th gradient
di t off f iis the
th vector
t
function f defined by
f
f
f ( x, y) f x ( x, y),
) f y ( x, y) i
j
x y
Du f ( x, y) f ( x, y) u
DEFINITION The directional derivative of f
a
TANGENT PLANES
Suppose a surface S has equation z = f (x, y),
)
where f has continuous first partial derivatives,
and let P(x0, y0, z0) be a p
point on S. Let C1 and C2
be the two curves obtained by intersection the
vertical planes y = y0 and x = x0 with
Chapter
p
12: Partial
Derivatives
Section 12.3
Partial
P
ti l derivatives
d i ti s are d
defined
fi d ass d
derivatives
i ti s of
fa
function of multiple variables when all but the variable of
interest are held fixed during the differentiation.
Definition
DEFINITION
A function of two variables has a local
maximum at (a, b) if
f (x,
(x y) f (a,
(a b) when (x
(x, y) is near (a,
(a b).
b)
[This means that
f (x, y) f (a, b) for all points (x, y) in some
disk with center (a, b).]
The number f (a, b) is
called a
MAGNETIC EFFECT OF CURRENT - I
1. Magnetic Effect of Current Oersteds Experiment
2. Amperes Swimming Rule
3. Maxwells Cork Screw Rule
4. Right Hand Thumb Rule
5. Biot Savarts Law
6. Magnetic Field due to Infinitely Long Straight Current
carrying Conducto
College of Arts and Sciences
Department of Physics
General Physics II- PHY 102.01
Third Midterm (15%)
Sunday: July 11, 2004
Summer Semester
Dr. Nasser Hamdan
Name:
ID:
Question
1
2
3
4
5
6
Total grade
Total grade
Full grade
20
20
10
10
20
20
Grade
/100
/1
The American University of Sharjah
College of Arts and Sciences
Physics Department
PHY102-General Physics II
First Midterm (10%)
March 13, 2005
5:00 p.m. to 6:15 p.m.
Write your name and ID in ink.
Name: .
Instructors:
Nidhal Guessoum
ID:
time of your sec
College of Arts and Sciences
Department of Physics
General Physics II PHY 102.01
Second Midterm
Tuesday November 24, 2009
Professor Nasser Hamdan
Name:
ID:
Question
1
2
3
4
5
Total grade
1.
2.
3.
4.
5.
6.
There are 5 questions and 4 pages.
Read each quest
The American University of Sharjah
College of Arts and Sciences
Physics Department
PHY102.02-General Physics II
Exam II
Monday April 13, 2015
5:00 p.m. to 6:15 p.m.
Write your name and ID in ink.
Name: .
ID:
Instructors:
Nasser Hamdan
Problem
1
2
3
4
5
To
The American University of Sharjah
College of Arts and Sciences
Physics Department
PHY102.02-General Physics III
Exam III
Sunday May 10, 2015
5:00 p.m. to 6:15 p.m.
Write your name and ID in ink.
Name: .
ID:
Instructors:
Nasser Hamdan
Problem
1
2
3
4
5
To
1) Calculate the maximum charge stored in the capacitor.
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12V 2) Calculate the magnitude and direction ofthe total magnet
American University of Sharjah
College of Arts and Sciences
Department of Physics
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American University of Sharjah
Coriege DfAn‘s And Sciences
PHY 1'02 {GENERAL PHYSICS H) = Fa” RIDGE
Section G4
Dr. Nasser Hamﬂan
Secund Midterm
Wednesday, h-Tmremher 5, 2045’?
HALE. . . .
l. D.
.h;
5011": the problems ncath'. showing all thE slaps nyﬂur
a) 3 marks
b) 4 marks
1) The capacitors in the circuit shown were initially
uncharged.
a) Determine the charge on each capacitor after the
switch is closed for a long time.
b) What is the rate of charge buildup on each
capacitor at time t = 6 ms.
= 2 k
C
Class practice Problems (Chapter 3)
Problem: While exploring a cave, a student starts at the entrance and moves the
following distances. She goes 75.0 m north, 250 m east, 125 m at an angle 30.0 north of
east, and 150 m south. Find the resultant displacem
Class practice Problems (Chapter 2)
Problem 21: Along a straight road, a car moving with a speed of 130 km/h is
brought to a complete stop in a distance of 210 m.
a) Find the magnitude of the deceleration (assumed uniform)
b) How long does it take to stop
E
X
A
M
P
L
E
1A
Examples 1A and
1B are almost the
same. What is the
difference?
Solution
Spring 2013 Exam III,
Problem II
E
X
A
M
P
L
E
1B
Examples 1A
and 1B are
almost the
same. What is
the difference?
Example 2
Solution
Example 3
Solution
E
X
A
M
P
L
E
For
v > E/B
implies vB > E
implies qVB > qE
implies FB > FE
v < E/B
implies vB < E
implies qVB < qE
implies FB < FE
A particle with a
greater speed will
be deflected in the
direction of
magnetic force
A particle with a
lesser speed will be
deflected in th
General Physics II/FALL 2014
_ 5 U
0" W5
"um/10O
Exam Rules:
0 Exam is strictly 75 minutes long.
0 Document your work or earn no credit. Use the back of the sheet if you run out of
space. Crossover any parts that correspond tolany given up t
Questionl Questions
./25 . ./25 ./100
Exam Rules:
0 Exam is strictly 60 minutes long.
0 Document your work or earn no credit. Use the back of the sheet if you run out of
space. Crossover any parts that correspond to any given up thoughts. Always
10.1.IDENTIFY: Use Eq. (10.2) to calculate the magnitude of the torque and use the right-hand rule illustrated in
Figure 10.4 in the textbook to calculate the torque direction.
(a) SET UP: Consider Figure 10.1a.
EXECUTE:
Figure 10.1a
This force tends to p
11.5.IDENTIFY: Apply to the ladder.
SET UP: Take the axis to be at point A. The free-body diagram for the ladder is given in Figure 11.5. The torque
due to F must balance the torque due to the weight of the ladder.
EXECUTE:
EVALUATE: The force required is
PHY101 section 8,9
HW chapter 8
8.3.
IDENTIFY and SET UP:
EXECUTE: (a) and
(b) and the result from part (a) gives The baseball has the greater magnitude of momentum.
(c) so gives so
The woman has greater kinetic energy.
8.15.IDENTIFY: The player imparts a
Suggested problems from chapter 14
14.4.IDENTIFY: The period is the time for one cycle and the amplitude is the maximum displacement from
equilibrium. Both these values can be read from the graph.
SET UP: The maximum x is 10.0 cm. The time for one cycle i
7.9.
PHY 101 sections 8, 9
HW chapter 7
IDENTIFY: The forces on the rock are gravity, the normal force and friction.
SET UP: Let at point B and let be upward. The work done by friction is negative; The free-body diagram
for the rock at point B is given in
PHYSICS 101.05
HW Chapter 9
9.2.IDENTIFY: since the angular velocity is constant.
SET UP:
EXECUTE: (a)
(b)
EVALUATE: In we must use the same angular measure (radians, degrees or revolutions) for both and
9.5.IDENTIFY and SET UP: Use Eq. (9.3) to calculate