Differential and Integral Calculus
Index
Affine transformation, definition
Affine transformation, uniqueness of inverse
AFFINE TRANSFORMATIONS AND THE MULTIPLICATION OF DETERMINANTS
Affine Transformations of Plane and Space
Affine Transformations, Combination
Affine transformations, theorems
Algebraic curve, isolated /Conjugate point
Algebraic function
Analytical geometry
Analytical Geometry, Preliminary Remarks
Anchor ring
Anchor ring, line element
Angle between curves
Angle between surfaces
Angle between two curves
Angle between two planes
Applications of Differential Calculus
Arbitrary sequence of polyhedra
Area and Volume of n-dimensional unit sphere
Area element of surface
AREA IN SPACE OF ANY DIMENSION
AREA OF A TRIANGLE
Area of anchor ring
Area of Curved Surface
Area of curved surface
Area of given surface
Area of hemisphere
Area of n-dimensional unit sphere
Area of region bounded by curve
Area of spherical surface
Area of surface in parametric representation
Area of surface of revolution
Area, concept
Areal distortion
Areas of Surfaces in more than Three Dimensions
Argument point
Argument, discontinuous/continuous
Arithmetic mean
Artin, book on G-function
Average density
B
Base of triangle
Beta Function
Bohr's theorem
Bohr's Theorem, proof
Bolzano's principle of point of accumulation
Boundary point
Bounded infinite set
Branch, single-valued
C
Calculation of Volumes
Calculus of Errors
Canonical form of equation
Canonical form of straight line
Cauchy-Riemann equations
Caustic
Cavalieri's 'principle
Cell
Cell, area
Centre of curvature
Centre of Mass
Centre of mass
Centre of mass of hemispherical surface
Centroid
Chain Rule
Change of Independent Variables
Change of Order of Differentiation
Change of Order of Integration
CHAPTER V
Circular neighbourhood
Circulation
Circumscribed polygon
Closed curve
Closed region
Closed residue
Closed set
Combination of Affine Transformations
Combination of Mappings and Transfomations
Combination of two transformations
Comparison function
Component of vector in the direction of l
Compound function, differentiation
Compound function, differentiation
COMPOUND FUNCTIONS
Compound Pendulum
Compound Pendulum, definition
Compound pendulum, period
Concentric circles
CONCEPT OF FUNCTION OF SEVERAL VARIABLES
Condition of integrability
Confocal conics
Confocal parabolas
Conformal Representation in General
Conformal transformation
Conjugate point of algebraic curve
Connected open set
Connected set
CONNECTION BETWEEN LINE INTEGRALS AND DOUBLE INTEGRALS IN THE PLANE
Connectivity
Constant
Content
Content of Plane Region
Content of Region of Higher Dimension
Content, concept
CONTINUITY
Continuity
Continuity and Differentiability of an Integral with respect to the Parameter
Continuity of convex function
Continuity of Implicit Functions
Continuous Dependence on a Parameter
Continuously differentiable function
Continuously differentiable primitive transformations
Continuum mechanics
Continuum, moving
Contour lines
Convergent integral
Convex downwards
Convex function
Convex function, continuity
Convex Functions
Convex region
Convex upwards
Convexity
Co-ordinate Axes
Co-ordinate surfaces
Corners
Cross-curves
Curl of a Vector field
Curvature vector
Curve of zero length
Curve, length of
Curve, multiple point
Curve, regular point
Curve, sectionally smooth
Curve, singular point
CURVED SURFACED, EFINITION OF AREA
Curves touch
CURVES, FAMILIES
Curves, family of
Curves, singular points
Curvilinear co-ordinate surfaces
Curvilinear co-ordinates
Curvilinear integral
Curvilinear integral
Curvilinear mesh, replacement by parallelograms
Curvi-linear net
Cusp
Cusp-locus
Cusps
Cylindrical co-ordinates
D
Definition of Functions
Definition of Line Integral Notation
DEFINITION OF THE AREA OF A CURVED SURFACE
Deformation
Degenerate hyperbola
Degenerate hyperbola
Degenerate transformation
Degenerate transformation
Density
Density
Dependent function
Determinant of third order
Determinant, column
Determinant, diagonal rule
Determinant, evaluation
Determinant, expansion by column
Determinant, expansion by row
Determinant, product of rows
Determinant, row
Determinant, second order
Determinant, two-row
DETERMINANTS, SIMPLE THEOREMS
Determinants, theorems
Developments of Differential Calculus
Difference between scalar product of two vectors and ordinary product of two numbers
Differentiability
Differentiability
Differentiability, geometrical expression
Differential
Differential and integral calculus, fundamental link
Differential Equations
Differential notation
Differential notation
Differential of a function
Differential, total
Differentiating an integral with respect to a parameter
Differentiation and integration, generalizationConcepts of
Differentiation Formulae for Inverse Functions
Differentiation in Given Direction
Differentiation of compound function
Differentiation of implicit function
Differentiation under integral sign
Differentiation under Integral Sign
Differentiation with respect to outward normal
DIFFERENTIATION/INTEGRATION TO FRACTIONAL ORDER
Dini's Theorem on Uniform Convergence of Monotonic Sequences of Functions
Direction
Direction cosines
Direction cosines in case of parametric representation
Direction cosines of normal
Direction cosines of normal to a surface
Direction cosines of tangent
Direction of normal
Directions and Vectors
Dirichlet discontinuous factor
Dirichlet's formula
Dirichlet's limit formula
Discontinuities
Discriminant
Discriminant
Discriminant curve
Distance
Divergence of a Vector field
Divergence of velocity field
Divergent integral
Domain
Domain Integral as Volume
Double integral
Double Integral as Volume
Double integral of function over region
Double integral, estimation formula
Double integral, extension
Double integral, fundamental rules
Double integral, fundamental rules
Double integral, general rule
Double integral, notation
Double Integral, Transformation
Double limit
Double limiting processes
Double Limits in Case of Continuous Variables
Double point
Double point
Double root
Double Sequences and their Limits
E
Electric charge
Electro-magnetism
Elementary Calculation of Volume
Elementary cone
Ellipse
Ellipse, tangent
Ellipse. curvature
Ellipsoid
Envelopes of Families of Surfaces
Envelopes of One-Parameter families of Curves
Epicycloid
Equation of tangent
Equations of Straight Line and Plane
Equatorial plane
Equilateral triangle
Estimation formula for double integral
Euler relation
EULERIAN INTEGRALS
EULER'S / LAGRANGE'S REPRESENTATIONS OF FLUID MOTION
Euler's beta function
Euler's representation of fluid motion
Evaluation of a determinant
Even function
Evolute, definition
Existence Implicit Functions
EXISTENCE OF MULTIPLE INTEGRAL
Existence of Multiple Integral of Continuous Function
Existence of Partial Derivative
Existence theorem
Existence theorem for integral
Extension Theorem
Extreme Values, Sufficient Conditions
F
Face angle
FAMILIES OF CURVES, SURFACES AND THEIR ENVELOPES
Families of Vectors
Family of curves
Family of parametric curves
Family of surfaces
Field of force, work done
Flow intensity
Flow, Divergence and Intensity
Flow, irrotational
Flow, total
Fluid motion, Euler representation
Fluid motion, Lagrange representation
Focal co-ordinates
Folium of Descartes
Folium of Descartes
Force field
Formulae for Transforming Axes
FOURIER INTEGRAL
Fourier integral formula
Fourier integral theorem
Fourier Integral Theorem, Proof
Fourier theorem examples
Free extreme value
Frenet's formulae
Fresnel Integral
Function log G(x) and its Derivatives
Function, algebraic
Function, continuous
Function, continuously differentiable
FUNCTION, DERIVATIVES
Function, differential of
Function, discontinuous
Function, linear fractional
Function, linear homogeneous
Function, multi-valued
Function, rational integral
Functional determinant
Functional value
Functions and their Ranges of Definition
Functions of Several Variables and their Derivatives
Functions with Isolated Infinite Discontinuities
Functions, geometrical representation
FUNCTIONS, SYSTEMS OF
G
Gamma (x), Gauss's infinite product
GAMMA FUNCTION
Gauss' theorem
Gauss' theorem in general form
Gauss' Theorem, remarks
Gauss' Theorem, statement and proof
Gauss' theorem, vector form
Gauss, repeated integral
Gaussian fundamental entities of surface
Gauss's infinite product for Gamma (x)
Gauss's Theorem, Interpretation
General Analytical Concept of Integral
General convergence conditions for integral
General co-ordinates
General ellipsoid
GENERAL FORMULA FOB AREA (OR VOLUME) OF REGION BOUNDED BY SEGMENTS OF STRAIGHT LINES OR PLANE AREAS
General power function
General theorem for functions of several variables
General Theorem on Inversion of Transformations
Generalization of concepts of differentiation and integration
Generalized summation
Geographical longitude
Geographical longitude
Geometrical expression of differentiability
Geometrical Meaning of Determinant of Transformation
GEOMETRICAL MEANING OF TOTAL DIFFERENTIAL
G-function, book
G-Funktion, Definition
G-Funktion, Functional Equation
Gradient
Gradient
Gradient field
Gradient of a potential
Gradient of a Scalar
Gravitating particle
Gravitational constant
Gravitational potential
Green's first theorem
Green's second theorem:
Green's Theorem
GULDIN'S FORMULA
Guldin's rule
H
Height of triangle
Heine-Borel Covering Theorem
Hemispherical surface, centre of mass
Hill-top of surface
Hölder's inequality
Holdich's theorem
Homogeneous function of degree h
Homogeneous function of first degree
HOMOGENEOUS FUNCTIONS
Homogeneous polynomials
Homogeneous quadratic expression
Horizontal tangent plane
Hydrodynamics
Hyperbolic paraboloid
Hyperboloid of one/ two sheets
I
Identical transformation
Identities
Image curve
Image point
Image region
Image tetrahedron
Implicit function, differentiation
Implicit function, second derivatives
IMPLICIT FUNCTIONS
Implicit Functions, theorem of
Implicit representation
Implicit Representation of Surfaces
Improper double integral, criterion for convergence
Improper extreme value
Improper integral
Improper integral
Improper integral
Improper Integral, Integration and Differentiation of with respect to Parameter
IMPROPER INTEGRALS
IMPROPER INTEGRALS AS FUNCTIONS OF A PARAMETER
Improper integrals with finite intervals of integration
Improper repeated integral
Improper repeated integral
Improper, repeated integral
Incompressible fluid, steady flow
Independent variables
Infinite integral, summary and Extensions
Infinite product of Weierstrass
Infinite Regions of Integration
Initial solution
Inscribed net
Integrability condition
Integral calculus, mean value theorem
Integral Estimates
Integral of function over (n - 1)-dimensional surface
Integral of function over a region
Integral of function with lines of infinite discontinuity
Integral of functions with jump discontinuities
Integral of Jacobian
Integral over n-dimensional region
Integral over three dimensional region
INTEGRAL THEOREMS FOR THE PLANE, lNTERPRETATION AND APPLICATIONS
INTEGRAL THEOREMS OF GAUSS, STOKES AND GREEN
Integral transformed to new variables
Integral, continuity and differentiability with respect to Parameter
Integral, convergent
Integral, curvilinear
Integral, divergent
Integral, double
Integral, existence theorem
Integral, general analytical concept
Integral, general convergence conditions
Integral, geometrical point of view
Integral, monotonic convergence
Integral, multiple
INTEGRAL, MULTIPLE, REDUCTION TO SINGLE INTERGRALS
Integral, repeated
Integral, transformation formula
Integral, triple
INTEGRALS AS FUNCTIONS OF A PARAMETER
Integrals over curved surfaces
Integrals over Rectangle
Integrals over Regions in Three and More Dimensions
Integrals over surfaces
Integrals, general transformations
Integrals, iransformation theory
Integrand with infinite discontinuity
Integration in two dimensions
Integration of Total Differential
Integration over Infinite Regions
Integration over Regions in Several Dimensions
Integration over Surfaces in more than Three Dimensions
Integration variable, notation
Integration, Extension to Regions in Several Dimensions
INTEGRATION/ DIFFERENTIATION TO FRACTIONAL ORDER
Integrals, GEOMETRICAL APPLICATIONS
Interior point
Internal connecting lines
Internal point
Intersection of neighbouring surfaces
INTRODUCTION OF NEW INDEPENDENT VARIABLES
Inverse functions, partial derivatives
Inverse funotions, second derivatives
Inversion in three dimensions
Inversion of mapping
Inversion of Transformations, General Theorem
Inverted uniquely
Irrotational flow
Isolated exceptional point
Isolated point
Isolated point of algebraic curve
Isolated point of indeterminacy
Isoperimetric ineqnality
Isosceles triangle
J
Jacobian
Jacobian
Jacobian in n dimensions
Jacobians, negative
K
Kinetic energy, definition
L
Lagrange multiplier
Lagrange representation of fluid motion
LAGRANGE'S / EULER'S REPRESENTATIONS OF FLUID MOTION
Lagrange's identity
Lagrange's method of undetermined multipliers
Lagrange's rule
Laplace equation
Laplace operator
Law of conservation of energy
Left-handed
Lemniscate
Lemniscate
Length of a vector
Length of arc
Length of curve
Length of equivalent simple pendulum
Limit for Functions of Several Variables
Limit function
Limit in Case of Several Variables
Line element
Line integral of function along curve
Line Integral, definition
Line integral, estimate
Line integral, general
Line Integral, Interpretation in Mechanics
Line Integral, main theorem
Line integral, sign property
Line integral, vector notation
LINE INTEGRALS
Line integrals in the plane
Line integrals in the plane
Line integrals in three dimensions
Line integrals of functions of n variables
Line integrals, fundamental rules
Line of discontinuity
Line of support
Linear equations
Linear equations, homogeneous system
Linear equations, two homogeneous one with three unknowns
Linear function
Linear homogeneous function
Linear part of increment
LINK BETWEEN EULER'S AND LAGRANGE'S REPRESENTATIONS OF MOTION OF A FLUID
Locus of intersections of neighbouring curves
log G(x), difference equation
log G(x), functional equation
Lower semi-continuity
Lower sums
Luminous point
M
Magnetic declination
Main Theorem on Line Integrals
Manifold, one-dimensional
Manifold, two-dimensional
Mapping
Mapping of plane onto itself
Mapping, affine
Mapping, inversion
Mapping, resultant
Mappings, resolution and combination
MAPPINGS, SYSTEMS OF
Mass
MAXIMA AND MINIMA
Maxima and Minima with Subsidiary Conditions
Mean density
mean geometric mean arithmetic
Mean Value Theorem
MEAN VALUE THEOREM FOR FUNCTIONS OF SEVERAL VARIABLES
Mean value theorem of integral calculus
Mechanics
Meridian
Meridian curve
Method of displacement
Method of undetermined mutipliers for arbitray numbers of variables and subsidiary conditions
Method of Undetetermined Multipliers in the Simplest Casem Proof
Mixed partial derivative
Model
Model
Moment
Moment of inertia
Moment of Inertia
Moment of inertia of beam
Moment of inertia of distributed mass
Moment of several particles
Moment of the surface distribution
Moment of the volume-distribution
Moment, definition
Monotonic convergence of integral
Monotonic sequence
Motion, tangential /normal Components
Moving continuum
Multi-parameter family
Multiple integral
Multiple Integral of Continuous Function, Existence
MULTIPLE INTEGRAL, EXISTENCE
Multiple integrals, resolution
MULTIPLE INTEGRALS, TRANSFORMATION
MULTIPLE INTEGRALS, TRANSFORMATION
Multiple point
Multiple point of a curve
Multiplication of determinants
Multiplication Theorem
Multiply-connected region
Multiply-connected region
Multi-valued function
N
Nabla
Necessary conditions for extreme values
Necessary conditions for maximum or minimum
Negative Jacobians
Neighbourhood
Neighbourhood
Neighbourhood
Neighbourhood, rectangular
Net over region
Newton's law of gravitation
Node
Nomography
Non-convex regions
Normal acceleration
Normal form of straight line
Normal, direction cosines of
Normal, equation of
North pole
Notation, integration variable
Notation, line integral
O
Odd function
One-dimensional manifold
(one-one) (1,1) mapping
One-Parameter families of Curves, envelopes
One-parameter family
One-parameter family of surfaces
One-to-one mapping
Open set
Optics
Order of magnitude
Order of vanishing
Order to which a Function vanishes
ORDINARY INTEGRALS AS FUNCTIONS OF A PARAMETER
Oriented curve
Orthogonality condition
Orthogonality condition for surface
Osculating plane
P
Parabola
Parabolic co-ordinates
Paraboloid
Parallel cubical parabolas
Parallels of latitude
Parameter
Parametric curve
Parametric curves on a surface
Parametric equations (with parameter t) of closed plane curve
Parametric form of equation of straight line
Parametric representation
Parametric representation, direction cosines
Parametric Representations
Partial derivative
Partial derivative, mixed
Partial derivative, notations
Partial derivatives of higher order
Partial derivatives of inverse functions
PHYSICAL APPLICATIONS
Physical meaning of scalar product
Physics
Plane, affine transformations
Plane, equation
Planimeter of Amsler
Poimt, neighbourhood
Point of inflection
Point-to-point correspondence
Point-to-point correspondence
Polar co-ordinate net
Polar co-ordinate net
Polar co-ordinates
Polar co-ordinates in plane
Polar co-ordinates in space
Polar co-ordinates in space
Polar Co-ordinates, Transformation of Du
Polar distance
Polar distance
POLAR PLANIMETER
Polar reciprocal
Polygonal path
Polyhedron, surface area
Polynomial
Position vector
Positive rotation about vector
Positively homogeneous function
Potential at external point
Potential equation
Potential function
Potential of Attracting Masses
Potential of distributed mass
Potential of field
Potential of force
Potential of solid sphere
Potential of vector field.
Potentials at internal, external points
Preliminary Remarks, Analytical Geometry / Vector Analysis
Primitive transformation
Primitive transformation
Primitive transformations, continuously differentiable
Principal normal vector
Principal normal vector
PRINCIPLE OF POINT OF ACCUMULATION
Products of rows of determinant
Q
Quadratic
Quadratic cone
Quadratic differential form
Quadratic form
Quadratic form, definite
Quadratic form, indefinite
Quadratic form, positive/ negative definite
Quadratic form, semi-definite
Quadrilateral mesh
Quotient rule
R
Radius of curvature
Ranges of Definition of Functions
Rational integral function
Reciprocal equations
Reciprocal formulae
Reciprocal formulae
Rectangular co-ordinate net
Rectangular Co-ordinates and Vectors
Rectangular hyperbolas
Rectangular neighbourhood
Rectangular net
REDUCTION OF MULTIPLE INTEGRAL TO SINGLE INTERGRALS
Reflected ray
Region
Region, circular
Region, closed
Region, connectivity
Region, multiply-connected
Region, open
Region, rectangular
Region, simply-connected
Region, simply-connected
Region, spherical
Regions of more than Two Dimensions
Remainder of integral
Remainder of integral
Repeated differentiation
Repeated integral
Representation, parametric
Residual region
Resolution
Resolution in three-dimensional space
Resolution of a Motion into Tangential and Normal Components
Resolution of General Affine Transformation
Resolution of Mappings and Transfomations
Resolution of Multiple Integrals
Resultant mapping
Resultant pf affine transformations
Resultant transformation
Reversibility of order of differentiation
Right circular cylinder
Right-handed screw
Ring-shaped region
Rot
Rotation
S
Saddle value
Scalar
Scalar field
Scalar function
Scalar multiplication
Scalar Multiplication of Vectors
Scalar product
Scalar product of gradient and unit vector
Scalar product, physical meaning
Schuler
Screw surface
Second derivatives of implicit function
Second derivatives of inverse functions
Second order determinant
Sectionally smooth
Sectionally smooth curve
Sectionally smooth curve
Semi-continuity
Sequences of points
Shrinking sequences of closed sets
SIMPLE THEOREMS ON SECOND AND THIRD ORDER DETERMINANTS
Simply-connected region
Simply-connected region
Single stationary system
Single-valued branch
Single-valued branches
Singular point
Singular point of a curve
Singular Points of Curves
SINGULAR POINTS OF PLANE CURVES
SINGULAR POINTS OF SURFACES
Singularity
Sinks
Slope of surface
Smooth arcs of curves
Smooth Arcs, Theorem
Solid angle
Sourc
Source-free flow
Space Differentiation
Space differentiation
Space, affine transformations
Space, n-dimensional
Space-differentiation
Specific circulation
Sphere
Spherical region
Steady flow
Steiner's theorem
Stereographic image
Stereographic projection
Stokes' Theorem
Stokes' Theorem, Interpretation
Straight Line, equation
Straight line, normal /canonical form
Straight line, parametric form of equation
Strophoids
Sum of two vectors
Surface Area in parametric representation
Surface area of polyhedron
Surface in three-dimensional space
Surface integrals
Surface of revolution
Surface, direction cosines of normal
Surface, orthogonality condition
Surfaces family ¥ 2
SURFACES, FAMILIES
Surfaces, Implicit Representation
SURFACES, SINGULAR POINTS
Symbol dxdy
Symbolic notation
SYSTEMS OF FUNCTIONS, TRANSFORMATlONS AND MAPPINGS
Systems of Implicit Functions, General Theorem on Inversion
T
Tangent Plane
Tangent plane
Tangent plane
Tangent, direction cosines of
Tangent, equation of
Tangential /Normal Components of a Motion
Tangential equation
Tangential function
TANGENTIAL REPRESENTATION OF A CLOSED CURVE
Taylor expansion
TAYLOR'S THEOREM FOR FUNCTIONS OF SEVERAL VARIABLES
THE VOLUME OF TETRAHEDRON
Theorem of Implicit Functions
Theorem of Implicit Functions for more than Two Independent Variables
Theorem on reversibility of order of differentiation
Theory of Curves in Space
Theory of differential equations
Three-rowed determinant
Torsion
Torus
TOTAL DIFFERENTIAL
Total differential
Total Differential of a Function
Total Differential, Integration of
Total flow
Total mass.
Transcendental function
Transfomations, resolution and combination
Transformation formula for integral
Transformation formula for polar co-ordinates
Transformation of Du to Polar Co-ordinates
TRANSFORMATION OF MULTIPLE INTEGRALS
TRANSFORMATION OF MULTIPLE INTEGRALS
Transformation, affine
Transformation, conformal
Transformation, degenerate
Transformation, degenerate
Transformation, primitive
Transformation, resultant
TRANSFORMATlONS, SYSTEMS OF
Triple integral
Triple integral
Tube surface
Tube-surface
Two planes, angle
Two-dimensional manifold
Two-row determinant
U
Uniform continuity
Uniform convergence
Uniform Convergence
Uniform convergence
Unique inverse
Unit sphere, n-dimensional
Unit vector
Upper sums
V
Variable parameter
Vector
Vector Analysis, Preliminary Remarks
Vector components
Vector field in n-dimensional space
Vector Fields
Vector fields, source-free
Vector manifold
VECTOR METHODS, APPLICATIONS
VECTOR MULTIPLICATION
Vector Multiplication of two Vectors
Vector product, geometrical construction
Vector Product, rules of calculation
Vector projection theorem
Vector representing moment
Vector, free
Vector, inner product
Vector, inner product, commutative law
Vector, inner product, distributive law
Vector, notation
Vector,bound
Vectors
Vectors, associative law
Vectors, commutative law
Vectors, Scalar Multiplication
Vectors, scalar multiplication
Vectors, scalar product
Velocity field of solid body rotating about an axis
Velocity field, divergence
Velocity vector
Volume below surface
VOLUME IN SPACE OF ANY DIMENSION
Volume of a solid of revolution
Volume of general ellipsoid
Volume of n-dimensional unit sphere
Volume of parallelepiped
Volume of pyramid
Volume of solid of Revolution
Volume of Tetrahedron
Volume of unit-hemi-sphere
Volume, ellipsoid of revolution
Volume, sign
Volumes in Polar Co-ordinates
W
Weierstrass' infinite product
Weierstrass's principle of point of accumulation
Work done by field of force
Z
Zero vector