Let c : [a, b] n be smooth, and let h : [, ] be such that h is
smooth, h (u) = 0 for any u in [, ], and h([, ]) = [a, b]. If d = c h
and F : c([a, b]) n , then we verify (for suciently smooth c and F)
that
cfw_
F ds
if h (u) > 0, u [, ]
c
F ds =
c F ds

Denition
Let S be a non-empty bounded subset of . We dene the oscillation of S , denoted
(S), to be the number sup S inf S. If f : S is a bounded function then we
dene the oscillation of f on S , denoted (f , S), to be the number (f (S), that is,
the numb

Cauchy-Schwarz-Bunyakowski Inequality
For x, y in n , one has
x, y xy.
.
Proof.
If y = 0, then the inequality holds trivially. Suppose y = 0. Then
x,y
y, y =
0 and we consider the vector u = x y,y
y.
x,y
y, y = x, y x, y = 0. (That is, u is
Note that u,

Greens Theorem
Greens Theorem relates the line integral of a vector eld over the
boundary of a planar region to a certain double integral over that region.
Let D be a region in 2 such that its boundary consists of a nite number
of simple closed geometric

Applications of Integrals
Denition
Let a, b with a < b. If f , g : [a, b] are in R[a, b], then the area
A of the region between the curves given by y = f (x) and y = g (x) is
b
f (x) g (x) dx.
A=
a
If f (x) g (x), x [a, b], then
b
A = a (f (x) g (x) dx.
I

Examples
(Tut Sheet 13, 1 (a) Consider the vector eld F on 3 given by
F(x, y , z) = (x y , x + z, y + z). We verify Stokes Theorem for F where S is
the surface of the cone z 2 = x 2 + y 2 intercepted by
x 2 + (y a)2 + z 2 = a2 , z 0. The cone is parametri

Integrability Condition 1
Let a, b with a < b, and let f : [a, b] be bounded.
Then f is integrable on [a, b] if and only if given any > 0, there exists > 0 such that
P P[a, b], P < = U(P, f ) L(P, f ) < .
Proof of Necessity
Suppose f R[a, b] (that is, f i

Parametrized Surfaces
Let D 2 such that D has an area, and let : D 3 . We write
(u, v ) = (x(u, v ), y (u, v ), z(u, v )
for (u, v ) D, and we assume that is of class C 1 , that is, actually stands
dened on an open superset U of D such that the component

Continued from where we left o.
Denition
Let D be a bounded subset of 2 , and let f : D be integrable. If f 0
on D, then we dene
the volume under the surface given by z = f (x, y ) and
above D to be
Df.
Examples
2
Let us nd the volume under the surface g

We illustrate a couple of applications of Integrability Conditions.
Theorem
Let a, b, c, d with a c < d b, and let f : [a, b] be bounded. If f is
integrable on [a, b], then f is integrable on [c, d].
Proof.
f integrable on [a, b] = f bounded on [a, b] = f

Arc Length
Suppose a curve C consists of points of the form by (x(t), y (t) where x
and y are continuously dierentiable functions dened on an interval [, ].
If P = cfw_t0 ,
t1 , ., tn is a partition of [, ] and Pr = (x(tr ), y (tr ) , then
the length nr=

Vector Valued Functions and Paths
Denition
Let D and let f = (f1 , . . . , fn ) : D n . If n = 1, then f is referred to
as a scalar valued function ; otherwise, f is referred to as a
vector valued function .
If t0 D, then f is said to be continuous at t0

Euclidean Space n
Denition
For n , n is the set cfw_x = (x1 , ., xn ) : xi for 1 i n of
n-tuples of real numbers. We say that xi is the ith component of the
tuple x = (x1 , ., xn ). Tuples x are referred to as points in n . Points
x = (x1 , ., xn ) and y

Gauss (Divergence) Theorem continued.
Invariance of Surface Integrals
Let U 3 be open and let P, Q, R : U be of class C 1 and such that
Px + Qy + Rz = 0. Let S1 and S2 be closed surfaces of the type described in
Gauss Theorem and lying in U such that S2 l

Maxima, Minima, Saddle Points
Denition
Let D n . We say that D is bounded if there exists some R > 0 such
that the open ball R (0) contains D. We say that D is a closed subset of
n if the complement n D of D in n is an open subset of n . A point
c n is sa

Dierentiation
The idea of dierentiating a function has its roots in two classical
problems: (1) Determine the tangent to a curve (2) Determine the
instantaneous speed of a moving body
Denition
Let D . A point c in D is said to be an interior point of D if

Continued from where we left o.
Darbouxs Theorem
Let a, b with a < b, and let f : [a, b] be bounded.
(1) Given > 0, > 0 such that P P[a, b],
P < = U(P, f ) < U(f ) + .
(2) Given > 0, > 0 such that P P[a, b],
P < = L(P, f ) > L(f ) .
Integrability Conditio

Dierentiable Inverse Theorem
Let I be an open interval, let c I , and let f : I be strictly monotone and continuous. Let
f 1 : f (I ) I be the inverse function. Then f (c) is an interior point of f (I ); moreover, if f is
dierentiable at c and f (c) = 0,

When is F a gradient eld?
Denition
Let D n and let F : A 3 be a continuous vector eld. We say that
the line integrals of F are path independent in D if
F ds =
c1
F ds
c2
for any piecewise
smooth curves c1 and c2 in D. (This is equivalent to
saying that c

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

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],

'(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>

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.

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

:
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 onCasthedirectionalderivativeof@inthedirectionof
i

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

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

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],