1
Lecture 4 : Continuity and limits
Intuitively, we think of a function f : R R as continuous if it has a continuous curve. The term
continuous curve means that the graph of f can be drawn without jumps, i.e., the graph can be
drawn with a continuous moti
ES211 Thermodynamics
Semester II, 201213
Tutorial 8
Name:
Group:
Roll No:
Date: Oct 24th, 2013
Marks are awarded solely on the basis of attempting the question/s. Marks will not be deducted
for an incorrect answer.
1. The latent heat of fusion of water
ES211 Thermodynamics
Semester II, 201213
Tutorial 9
Name:
Group:
Roll No:
Date: Oct 31st, 2013
Marks are awarded solely on the basis of attempting the question/s. Marks will not be deducted
for an incorrect answer.
1. A lump of ice with a mass of 1.5 kg
Technische Universitt Mnchen
a
u
Fakultt fr Mathematik
au
Prof. Dr. M. Mehl
B. Gatzhammer
ST 2013
May 27, 2013
Numerical Programming 2 CSE
Tutorial 6: PDE  Classication
1) Classication Examples
Classify the following linear second order partial dierentia
ES211 Thermodynamics
Semester II, 201213
Tutorial 10 + 11.5
Name:
Group:
Roll No:
Date: Nov 7th, 2013
Marks are awarded solely on the basis of attempting the question/s. Marks will not be deducted
for an incorrect answer.
1. Exhaust gases leave an inter
dl
I
?l
FI
=\
Cr
b
\
s
$1.^
'. . 'l
r,
'
I tt
a
ei
t
=cfw_
C
<rs
:
*"
^2
'\
F'. l ^
olcv\
.l
t
.
h. :
.
\,
,
tLr
.
( 'r
\
'
_
x
c+\
i
e.o

\
tl
LlI
\)
.?
*J
o
$
"q
VI
qo
UJ
l^
.1cr
q
.t'
a
5
l_
l
T
N
c1)
ca
sl
?
t
'?
J5
c.r

tt
ll
)(
\,o
\
9
tt
f,(
oo
I
.9
I
a
\o
ll
lr
Jd
sl
\./)
R
\^
\
*cfw_
=
3
v
\+
s(
\
!
I
r* ,
\
\
\<
\
l
r\.
c.l
'
\
'
'
l
' " ll\;
\l'
:
2l
Fl
)G
\
L
l\
lt
\
JA
It
c.l
$\*
)
t\
t'
ll
\5
vl
tr,
(l
,"
,
a
cfw_
q
\n
x
ll
b\F
d
_9
G
'
I
ccfw_
T
' f,
c .l
cfw_
I
t
r
!.)
t.
v,
\
t_1
(1
r.trl c .r
G
elt
) ^ cfw_l(\
'
S+
C.l
)a
s'l
.cfw_r,
,d
\J
vr
c.l
l
\,
It
h
cfw_
+
r
s
.*
acfw_
(
(
g\f
I
ll
lt
c\t
r
l
F

n
t
q
rr
ll
ll
qf
r.
Jl'
\
.<
.
g
s
cfw_f
h
.sl
rt
r
s
e.l
,j
s
.
tl
r
o
(r
.e
t7
cfw_
s!
r.9
\9
so
ll
o
ll
1^
Practice Problems 3 : Cauchy criterion, Subsequence
1. Show that the sequence (xn ) dened below satises the Cauchy criterion.
(a) x1 = 1 and xn+1 = 1 +
(b) x1 = 1 and xn+1 =
1
2+x2
n
1
xn
for all n 1
for all n 1.
(c) x1 = 1 and xn+1 = 1 (x2 + 8) for all n
Practice Problems 7 : Mean Value Theorem, Cauchy Mean Value Theorem, LHospital Rule
1. Use the mean value theorem (MVT) to establish the following inequalities.
(a) ex 1 + x for x R.
1
(b) 2n+1 < n + 1 n <
(c)
x 1
x
1
2n
for all n N.
< lnx < x 1 for x > 1
United States
Environmental Protection
Agency
Pesticides And
Toxic Substances
(7508W)
738F91107
September 1991
R.E.D. FACTS
Silicon Dioxide and
Silica Gel
All pesticides sold or used in the United States must be registered by
Pesticide
Reregistration E
THE PELL EQUATION
1. Introduction
Let d be a nonzero integer. We wish to nd all integer solutions (x, y ) to
x2 dy 2 = 1.
(1)
1.1. History.
Leonhard Euler called (1) Pells Equation after the English mathematician John
Pell (16111685). This terminology ha
Abinitio Electronic and Structural Properties of Rutile Titanium
Dioxide
Chinedu, E. Ekuma and Diola Bagayoko
1
Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA
70803, U.S.A.
2
Department of Physics, Southern University an
MTH203: Assignment6
1. Consider the equation (1 + x2 )y + 2xy 2y = 0.
(i) Find its general solution y =
an xn in the form y = a0 y2 (x) + a1 y1 (x), where
y1 (x) and y2 (x) are power series.
(ii) Find the radius of convergence for y1 (x) and y2 (x).
(iii
Chapter 7
Second Order Partial Dierential
Equations
7.1
Introduction
A second order linear PDE in two independent variables (x, y ) can be written as
2u
u
2u
u
2u
+ C (x, y ) 2 + D (x, y )
+ E (x, y )
+ F (x, y ) u = G(x, y )
+ B (x, y )
x2
xy
y
x
y
the s
Second Order Linear Partial Differential Equations
Part I
Second linear partial differential equations; Separation of Variables; 2point boundary value problems; Eigenvalues and Eigenfunctions
Introduction
We are about to study a simple type of partial dif
then, treating (x2 1)n = (x 1)n (x + 1)n as a product and using Leibnitz rule to dierentiate n
times, we have
1
v (x) = n (n!(x + 1)n + terms with (x 1) as a factor) ,
2 n!
so that
Chapter C
v (1) =
C2.3
Properties of Legendre Polynomials
n!2n
= 1.
2n n!
Introduction to Partial Dierential Equations
John Douglas Moore
May 21, 2003
Preface
Partial dierential equations are often used to construct models of the most
basic theories underlying physics and engineering. For example, the system of
partial dierenti
INDIAN INSTITUTE OF TECHNOLOGY GANDHINAGAR
DEPARTMENT OF MATHEMATICS
MA 201: Autumn, 201314
Tutorial Sheet No.10
1. Classify the following equations in terms of order, linear/semilinear/quasilinear/fully nonlinear:
(i) u2 + u2 = 1 (ii) ux + uy + u2 = 0.
INDIAN INSTITUTE OF TECHNOLOGY GANDHINAGAR
DEPARTMENT OF MATHEMATICS
MA 201: Autumn, 201314
Tutorial Sheet No.11
1. Let r = x2 + y 2 , (x, y ) R2 ; or r = x2 + y 2 + z 2 , (x, y, z ) R3 . Show that u = ln r is a
solution of the twodimensional potential
1
Lecture 18 : Improper integrals
b
We dened a f (t)dt under the conditions that f is dened and bounded on the bounded interval
[a, b]. In this lecture, we will extend the theory of integration to bounded functions dened on
unbounded intervals and also to
1
Lecture 3 : Cauchy Criterion, BolzanoWeierstrass Theorem
We have seen one criterion, called monotone criterion, for proving that a sequence converges without
knowing its limit. We will now present another criterion.
Suppose that a sequence (xn ) conver
1
Lecture 6 : Rolles Theorem, Mean Value Theorem
The reader must be familiar with the classical maxima and minima problems from calculus. For
example, the graph of a dierentiable function has a horizontal tangent at a maximum or minimum
point. This is not
1
Lecture 1: The Real Number System
In this note we will give some idea about the real number system and its properties.
We start with the set of integers. We know that given any two integers, these can be added, one
can be subtracted from the other and t
1
Lecture 2 : Convergence of a Sequence, Monotone sequences
In less formal terms, a sequence is a set with an order in the sense that there is a rst element,
second element and so on. In other words for each positive integer 1,2,3, . . . , we associate an
1
Lecture 7 : Cauchy Mean Value Theorem, LHospital Rule
LHospital (pronounced Lopeetal) Rule is a useful method for nding limits of functions. There
are several versions or forms of LHospital rule. Let us start with one form called 0 form which
0
(x)
deal
1
Lecture 9 : Sucient Conditions for Local Maximum, Point of Inection
In Lecture 6, we have seen a necessary condition for local maximum and local minimum. In this
lecture we will see some sucient conditions.
Sucient Conditions for Local Maximum and Local
1
Lecture 38: Stokes Theorem
As mentioned in the previous lecture Stokes theorem is an extension of Greens theorem to surfaces.
Greens theorem which relates a double integral to a line integral states that
D
N
x
M
y
dxdy =
C
M dx + N dy
where D is a plane
1
Lecture 39: The Divergence Theorem
In the last few lectures we have been studying some results which relate an integral over a
domain to another integral over the boundary of that domain. In this lecture we will study a
result, called divergence theorem
1
Lecture 36: Line Integrals; Greens Theorem
Let R : [a, b] R3 and C be a parametric curve dened by R(t), that is C (t) = cfw_R(t) : t [a, b].
Suppose f : C R3 is a bounded function. In this lecture we dene a concept of integral for the
function f . Note