SUMMARY OF IMPORTANT THEOREMS
AND FORMULAE
1. Differentiation
Chain Rule for Functions of Several
Variables:
If
with corresponding
fomulae for uxy and uyy.
Implicit Functions:
If F(x, y) = 0,
Jacobians:
If x = f(x, y), u = y(x, y),
Rules for Jacobians:.
2. CONVERGENCE OF DOUBLE SEQUENCES
Convergence Test for Double Sequences:
The sequence anm converges, or,
in symbols,
3. UNIFORM CONVERGENCE AND INTERCHANGE OF
INFINITE OPERATIONS
Dini's Theorem:
If a series of positive
continuous functions converges to a continuous limit function
in a closed region, it converges uniformly
to that limit.
Interchange of Differentiation and
Integration:
Differentiation of an integral with respect to a parameter
provided that f(x, y) and fx(x,
y) are continuous in the interval under consideration
Interchange of Differentiation and
Integration in Improper Integrals:
provided that fx(x, y)
is continuous in the interval under consideration and the
integrals
converge uniformly in that interval.
Interchange of Two Integrations:
If f(x, y) is continuous and a, b,
a, b
are constants,
The order of integration may also be reversed when the limits
are not constants, provided that both integrations are performed
over entire the
region concerned and corresponding new
limits arc introduced
Interchange
of Two Integrations in Improper Integrals.
provided that the integral 
converges uniformly in
the interval a £ x £
b.
4. SPECIAL DEFINITE INTEGRALS
Fresnel's Integrals:
Fourier's
Integral Theorem:
If f(x) is sectionally smooth and 
converges and if f(x+ 0)+ f(x - 0) =
2f(x), then
The Gamma Function:
If x > 0, the gamma function G(x)
is defined by the equation
It satisfies the functional equation
hence, if x is a positive integer n,
For all values of x other than 0, -1, -2, ··· , it
may- he expressed by the formulae
where 
is Euler's constant. More over, for every integer m
³ 2,
Again,
, Extension
Theorem.
Hence, in particular,
The beta
function B(x, y)
is defined as follows for positive values of x and y:
The beta and gamma functions are connected by the
relation
For any complex z,
where C denotes a path
which surrounds the positive real axis and approaches it
asymptotically on either side.
5. MEAN VALUE THEOREMS
Mean
Value Theorem for Functions of Two Variables:
Taylor's
Theorem for Funtions of Two Variables:
where the remainder Rn (in the symbolical
notation is given by
If, as n increases, this remainder tends to zero, we
have the infinite Taylor series
Mean
Value Theorems for Multiple lntegrals :
where DR is the area
of R and m a value
intermediate between the maximum and minimum of f(x,
y) in R.
Similarly, if p(x, y) ³ 0,
6.
VECTORS
Definition
of a vector
Notation: A
vector v in three dimensions has the
components v1, v2, v3.
Length of a Vector:
Addition of Vectors:
is the vector with the components
Scalar (inner) Product:
where d is the angle
between u and v.
Vector (outer) Product:
is the vector with the components
Differentiation:
If the coordinate axes are rotated, the vector components are
transformed in the same way as x, y, z,
the components of the position vector.
By the derivative of the function f(x,
y)
in the direction of
the unit vector n
with the components cos a, sin a we mean the limit
Hence
In particular,
and hence, in general,
In the same way, in three dimensions, the derivative
in the direction of the vector n with
components cos a, cos b, cos g
is
The Differential Operations: With every scalar
function f(x1, x2,
x3) is associated a vector grad f
with the components fx , fy,
fz. The derivative of f in
the direction of the unit vector n is n
gradf.
There is associated with every vector field u(x1,
x2, x3) a vector curl
u with the components
and a scalar
function
Using the symbolic
vector 
with the components
we have
Moreover,
7. MULTIPLE INTEGRALS
Definition of multiple integral:
The
rules for the addition of integrands and combination of regions
of integration are the same as for ordinary integrals.
Transformation of a Multiple Integral: If the
oriented region R of the xy-plane is mapped onto
a correspondingly oriented region R' of the uv-plane
by a reversible (1.1) transformation the Jacobian
of which does not vanish anywhere,
then
An analogous
formula holds for any number of dimensions.
In particular, transformations to polar co-ordinates
or
yield
and
Reduction of a Multiple Integral to
Ordinary Integrals:
Let a £ y £ b in R and for every y
a = a(y) £ x
£ b(y) = b;
then
8. INTEGRAL THEOREMS OF GAUSS, GREEN AND
STOKES
Definition of curvilinear (line) integral.
1. Two Dimensions:
If the region R is simply connected, the line integral
![]()
is independent of the path C joining two points in R
if, and only if, the condition of
integrability holds at every point of R.
In this case, if the initial point is fixed, the integral is a
function U(x, h) of the end-point, such that the
vector A with components a, b
satisfies the relation
Gauss' Theorem:
Let R be a simply-connected region and C its
boundary. Then
or, in vector
notation,
where n is the unit
vector in the direction of the outward normal, An
the normal component of the vector A
with components f, g and ds the arc
element of the boundary curve.
Green's
Theorem:
In vector notation, the first form of the theorem is
where
and 
denotes differentiation in the direction of the
outward normal.
2. Three Dimensions:
The necessary and sufficient condition that the line integral
shall be independent of the path C joining two points
in a simply-connected region R is
or, in full,
Surface Integral: This is given by
or
if x = x(u, v), y
= y(u, v), z = z(u,
v) and the oriented region B in the xy-plane
corre»ponds to the surface S.
Gauss' Theorem:
Let n be the unit vector in the direction
of the outward normal and An the normal
component of the vector A with components
a, b, c; moreover, let denote differentiation
in the direction of the outward normal. Then
or, in
vector notation,
the integrals on the right hand side being taken over the
closed surface S bounding the region R.
Green's
Theorem:
where and S have the same meaning as before and
Stokes' Theorem: Let the oriented
surface S be bounded by the correspondingly oriented curve
C. Then
In vector notation:
Let At be the tangential component of
the vector A = (f, y, c) in the direction in which
the curve C is described, (curl A)n
the component of curl A in the direction
of the outward normal and ds the arc element on C,
measured in the direction in which the curve is described: Then
9. MAXIMA AND MINIMA
The following rules hold only for maxima and minima inside
region under consideration.
Free Maxima and Minima of a
function of Two Variables:
The necessary conditions for an extreme value of
the function u = f(x, y) are
If these conditions are satisfied and if
there is» an extreme value
at the point in question. It is a maximum or a minimum according
to whether fxx(and hence also fyy)
is negative or positive. If
the point is a saddle
point.
Maxima and Minima subject to Subsidiary
Conditions (Method of undetermined Multipliers):
If in the function u = f(x1,
··· , xn) the n variables
are linked by the m subsidiary conditions (m <
n)
we introduce m multipliers l1,
··· , lm
and form the function
Then, the m conditions and the n additional equations
yield (m + n) necessary
conditions for the extreme points.
10. CURVES AND SURFACES
In what follows (x, h) or (x,
h, z)
are current co-ordinates.
1. Plane Curves.
Equation of the curve:
Equation of the tangent
at the point (x, y) (Volume
I, 5.1.3, 3.2.1):
Equation of the normal
at the point (x, y) (Volume
I, 5.1.3; 3.2.1):
Curvature (Volume I,
5.2.6, 3.2.3):
Radius of curvature (Volume I,
5.2.6; 3.2.3):
Evolute (locus of centres of
curvature) (Volume I,
5.2.6, A5.1:
Involute (Volume I,
5.2.6):
where a is an arbitrary constant and s the arc
length measured from a given point (s being the
parameter).
Point of inflection (Volume
I,
3.5.1,
5.1.5; 3.2.3):
The necessary
condition for a point of inflection is
Angle between two curves
(Volume
I, 5.1.3; 3.2.3):
In particular, the curves are
orthogonal if
the curves touch if
Two curves y
= f(x),
y = g(x
have contact of order n
at a point x, if
2. Curves in Space:
Equation of the curve:
Direction cosines of the tangent:
Curvature:
where ds is the arc element.
3. Surfaces:
Equation of the surface:
Equation
of the tangent plane:
Direction
cosines of the normal as well
where
Angle
between two surfaces
in particular, the condition that the surface are orthogonal
is
4.
Envelopes
In order to obtain the envelope
of the family of plane curves
or of the family of surfaces
we calculate the discriminant by
eliminating c from the equations
The discrimmant contains the envelope
and also the geometrical locus of the singular points.
If the family of curves is given by the parametric equations x
= f (t,c), y
= y(t, c),
the discriminant is obtained by eliminating c and t from
the equations
The envelope of a two-parameter family of surfaces
is contained in the equation obtained by eliminating the two
parameters c1, c2 from
the equations
11. ARC LENGTH, AREA, VOLUME
Arc length:
Let a plane curve be given by the equations
The length of arc is
The length of arc of the three-dimensional curve
is
Area of Plane Surface:.
The area bounded by the curve
and two radius vectors q
0, q1, where r,
ql are polar co-ordinates, is
given
by
The area enclosed by the curve
the two co-ordinates x
= x0,
x =
x1
and the x-axis
is
Let R be a positively-oriented plane surface and C
its boundary (orientation
and sign of an area ). Then the area of the surface is
Area
of Curved Surface Let the equation of the surface be
In the case (c), let E, F, G be the
so-called fundamental quantities of the surface, i.e.,
let
Then
The arc length of the curve
drawn on the surface is then
The area of the curved surface
lying vertically above the region R in the xy-plane
is
the last integral being taken over the region B of the uv-plane
which corresponds to the region R .
The area of the surface of
revolution
produced when the curve
is rotated about the z-axis, is
where is the arc
of the meridian curve z = f(x).
Volume I, 5.2.8.
The surface wn
of the unit sphere in n dimensions
is given by
Volumes:
The volume bounded below by the region R and above by
the surface S with the equation
is given by
with
sign.
If the surface S is closed
and forms the entire boundary of the region V,
the volume of this region is given
by
In polar co-ordinates,
the same
volume is given by
where R is the region of rqf
-space corresponding to the region V.
The volume of the surface of
revolution
which is produced, when the curve
is rotated about the z-axis,
is (also Volume
I, 5.2.8)
The volume vn
of the unit sphere in n dimensions
is
given by
The volume swept out by a moving
plane area P of area A
is
where dn/dt is the component of the
velocity of the mean centre of P perpendicular to the
plane of P.
12. CAICULUS OF VARIATIONS
The necessary and sufficient condition for the integral
to be stationary is Euler's equation
or
If F involves several functions u1(x),
u2(x), ··· , un(x)
and their derivatives, then a necessary
and sufficient condition that the integral
shall he stationary is that u1(x),
u2(x), ··· , un(x)
satisfy the system of Euler's equations
If F depends on x, u(x), u'(x),
u"(x), Euler's
equation is
If 
is to be made stationary
subject to the subsidiary condition G(x, y, z)
= 0, then a necessary
condition is
where l is Lagrange's multiplier.
13. ANALYTIC FUNCTIONS
Definition:
The necessary and sufficient
condition for
to be analytic in
a regionR is that in R the Cauchy-Riemann
differential equations
hold.
Cauchy's
theorem: If f(t) is analytic
in a simply-connected region R, then
if C is a closed curve inside R.
Cauchy's formula:
Under the same condition as Cauchy's
theorem, the formula
holds if z is a point in side C.
If f(z) is analytic
inside and on the boundary of a circle |z - z0|
£ R, it can
be expanded in a power series
which converges inside the circle. Here
