Stokes Theorem
Recall the vector form of Greens Theorem:
F ds =
curl F k d(x, y )
D
D
where D is the positively oriented boundary of a (possibly multiply
connected) region D in 2 and F is a vector eld (of class C 1 ) dened on
an open superset of D and D

not parallel to any of
parallelogram
a
of
area
(AII,12.6.1) Let S be the
area"sof the Piojec
St' Sz uod Sabe the
Let
the coordinate Planes'
planes' Show that
the three coordinate
on
tions
"i:ffi.of the Parallelogram
usetheAreaCosine Pnnci'Ple)'
(Hint:
12+

Graphs of Exponential and Logarithmic Functions
Graphics borrowed from mathwarehouse.com
Ameer Athavale (IIT Bombay)
MA 105: Calculus
August 24, 2015
1/1

Geometric Illustrations
Extreme Value Theorem: Continuous f attains the maximum value at c and
the minimum value at d
Intermediate Value Theorem: Continuous f attains the intermediate value u at
c
Ameer Athavale (IIT Bombay)
MA 105: Calculus
August 3, 201

Geometric Illustrations
Rolles Theorem: The tangent to the graph of f at (c, f (c) has slope zero
Lagranges Mean Value Theorem: The tangent to the graph of f at
(c, f (c) is parallel to the straight line joining (a, f (a) and (b, f (b)
Ameer Athavale (IIT

MA 105: Calculus
Lecture 15
Prof. R. P. Kulkarni
IIT Dharwad
Tuesday, 13th September 2016
R. P. Kulkarni
MA 105: Lec-15
Area between Curves
Let f1 , f2 : [a, b] R, f1 f2 . Let
R := cfw_(x, y ) R2 : a x b and f1 (x) y f2 (x).
Define
b
Z
f2 (x) f1 (x) dx.
A

MA 105: Calculus
Lecture 2
R. P. Kulkarni
IIT Dharwad
Thursday, 04 August 2016
R. P. Kulkarni
MA 105: Lec-02
R, the set of all real numbers
Let R denote the set of all real numbers. R satisfies
Algebraic Properties:
a, b R = a + b, ab R.
Order Properties:

MA 105: Calculus
Lecture 2
R. P. Kulkarni
IIT Dharwad
Friday, 05 August 2016
R. P. Kulkarni
MA 105: Lec-02
Definition of lim an = a
n
Definition
lim an = `
n
if for every > 0, there is an integer n0 such that
n n0 = |an `| < , that is, an (` , ` + ).
R. P

MA 105: Calculus
Lecture 14
Prof. R. P. Kulkarni
IIT Dharwad
Monday, 12th September 2016
R. P. Kulkarni
MA 105: Lec-14
Tutorial 5
Tutorial Sheet No. 4: 6, 7 (i), (v), 8 (a), 10
Tutorial Sheet No. 5: 1 (ii), 3
R. P. Kulkarni
MA 105: Lec-14
Logarithmic Func

MA 105: Calculus
Lecture 4
Prof. R. P. Kulkarni
IIT Dharwad
Monday, 8 August 2016
R. P. Kulkarni
MA 105: Lec-04
Quiz 1
Quiz 1: during 29th August - 2nd September 2016,
Syllabus: Tutorial Sheets 1, 2 and 3,
Weightage: 10 percent,
Objective type.
R. P. Kulk

MA 105: Calculus
Lecture 16
Prof. R. P. Kulkarni
IIT Dharwad
Wednesday, 14th September 2016
R. P. Kulkarni
MA 105: Lec-16
Volume of a solid
D : bounded subset of R3 . A cross-section of D by a plane in
R3 is called a slice of D.
Let a < b, D lies between

MAlo4
s)
CnaAiftonaLPt.ttlezro
L
(AT., t2,11s) Let I 3,5t2) = C., o, xv) ' FinJ a'
tffe\efnhatle vecf"t $ai G of ]*'tef*n
Corofi.!,ruous
I d'
q ("t\ 4) = (t(x,t,<), M(a,9t.), o) sttcb +qt ,F= cu.vLt
o14tR3. hJhd*is the ?nost 3anenoL E of *+.is Sr* ?
5oln.

'(A7if,n.et.tt)
wztr)o',ti* tR3Lol,o.re
houn4aryis a.ciose,J s,rnrfarcS all let y berrlnzc,ni't
arlw novmol * S. Let F a/il q k |'r,roCI
v(*a fr14s or Wt'15 sqcb+h"t
A W,
CwvLy= <wtG a^J Aivf =dtrG
L
LntW Ho|i"vex
Ar"J SuJ +l^t
G ,l
= f .?1
evr^aulv',e,o>

One-Variable Taylors Theorem
Theorem
Let a < b, let f : [a, b] , let m be a non-negative integer, let f (m) be
continuous on the closed interval [a, b], and let f (m+1) (t) exist for every t
in the open interval (a, b). Let , be distinct points of [a, b],

Continued from where we left o.
Cauchys Mean Value Theorem
Let a, b with a < b. If f , g : [a, b] are continuous on [a, b] and
dierentiable on (a, b), then there exists some c (a, b) such that
g (c)(f (b) f (a) = f (c)(g (b) g (a).
LHopitals Rule for
0
0

Continuity of functions
Denition
Let D , let f : D , and let c D.
We say that f is continuous at c if for any sequence cfw_xn in D
converging to c we have the sequence cfw_f (xn ) converging to f (c).
If f is not continuous at c, then f is said to be dis

Lagrange Multiplier Method
Theorem
Let D n and let c be an interior point of D. Let f , g : D have
continuous rst partial derivatives in an open ball centered at c. Let
C = cfw_x D : g (x) = 0. Suppose the following three conditions are
satised:
(1) c C ,

Second Derivative Test or Discriminant Test
Theorem
Let D 2 , let c = (c1 , c2 ) be an interior point of D, let f : D have
continuous second partial derivatives in an open ball centered at c, and let
f (c) = 0.
(1) If f (c) > 0 and fxx (c) < 0, then f has

Fubinis Theorem
Theorem
Let f R([a, b] [c, d]) and let I =
f.
d [a,b][c,d]
(1) If, for each xed x,
the integral) c f (x, y )dy exists, then the
b ( d
iterated integral a
c f (x, y )dy dx exists and equals I .
b
(2) If, for each xed y(, the integral) a f

When is F a curl eld?
Denition
A subset D of 3 will be said to be special if it is path-connected and every closed
surface in D (of the type described in Lecture32) bounds a region in D, that is, it is the
entire boundary of a region in D.
We know that th

You should be aware of the following conventions:
a
b
If a < b and f R[a, b], then b f = a f .
a
If g (a) makes sense, then a g = 0.
Theorem
Let a, b with a < b, and let f R[a, b]. If m, M are respectively a
lower bound and an upper bound for f on [a, b],

Gauss (Divergence) Theorem
Gauss Theorem basically states that the ux of a vector eld out of a
closed surface (that is, a compact surface without a boundary) in 3 is
equal to the triple integral of the divergence of the vector eld over the
region enclosed

Vector Fields and Line Integrals
Denition
Let A n . A function F : A n is called a vector eld in n if n 2.
A function f : A is called a scalar eld .
Examples
(1) F(x, y , z) = (xy , yz, zx) for (x, y , z) A = 3 .
(2) Velocity eld V of a uid passing throug

:
IfC isasimpleclosedregularC1curveinlR2givenbVq(t) (*(t),A(t),
:
t e la,b],th; unit outernormalnofC isdefinedbytheequationn(t)
fieldonc, thenthenormalderi,ua
@,(i),_*,(t)llls,(r)ll. If @isascalar
i"fi;"e [email protected]
i

MA 101: Calculus
Lecture 1
Prof. R. P. Kulkarni
IIT Dharwad
Tuesday, 2nd August 2016
R. P. Kulkarni
MA 101: Lec-01
Text Book: G. B. Thomas and R. L. Finney,Calculus and
Analytic Geometry, 9th ed., Addison-Wesley/Narosa, 1998.
Evaluation Plan
Quiz I
10 mar

MA 105: Calculus
Lecture 6
Prof. R. P. Kulkarni
IIT Dharwad
Wednesday, 10 August 2016
R. P. Kulkarni
MA 105: Lec-05
Quiz 1
Quiz 1: Monday, 29th August 2016, 9 : 15 10 : 15
Syllabus: Tutorial Sheets 1, 2 and 3,
Weightage: 10 percent,
Objective type.
Maths

Name:
Date:
Stoichiometry Worksheet
JUST NEED ANSWERS FOR THE PROBLEMS IN RED
PLEASE!
PROBLEMS 6-10 only!
Directions: You are strongly encouraged to print out this worksheet and
work with paper & pencil and a calculator for these problems. Then,
either

Math 160
Spring 2017
TENTATIVE SCHEDULE
Instructor: Irene Palacios
Section # 7987
( CHANGES WILL BE ANNOUNCED on Discussion Board)
Week 1
Tues, Jan 31
Tues, Jan 31
Thurs, Feb 2
Thurs, Feb 2
Week 2
Tues, Feb 7
Thurs, Feb 9
Thurs, Feb 9
Intro, Ch 1, 2.1, 2.

MA 105 D1 Lecture 23
Ravi Raghunathan
Department of Mathematics
Autumn 2014, IIT Bombay, Mumbai
An application to electromagnetism
The proof of Stokes Theorem for a graph
The proof of Stokes theorem for parametrised surfaces
Gauss Divergence Theorem
Appli

MA 105 D1 Lecture 13
Ravi Raghunathan
Department of Mathematics
Autumn 2014, IIT Bombay, Mumbai
Higher Derivatives
Maxima and Minima
The second derivative
Higher partial derivatives
Just as we repeatedly differentiated a function of one variable to
get hi

MA 105 D1 Lecture 3
Ravi Raghunathan
Department of Mathematics
Autumn 2014, IIT Bombay, Mumbai
Cauchy sequences: the definition
Limits of functions
Odds and ends about limits
Continuity
Cauchy sequences
As we saw last time, it is not easy to tell whether