Differential and Integral Calculus Vol. I

Index

A B C D E F G H I J K L M N O P Q R S T U V W Z

A

Abscissa
Absolute and conditional convergence, difference
Absolute Convergence
Absolute convergence
Absolute value
Absolutely convergent
Acceleration
Acceleration
Acceleration, gravitational
Acoustics
Air Resistance
Air resistance
Algebra, fundamental theorem
Algebraic equation
Algebraic function
Algebraic functions
Algebraic functions
Almost periodic function
Alternating current resistance
Alternating Currents
Amplitude
Analysis, arithmetized
Analytic functions of a complex variable
Analytical Definition of Integral
Analytical geometry
Analytical geometry
Anchor ring
Aperiodic system
Applications to Natural Sciences
Applied optics
Applied optics
Approximate Expression of Functions by Polynomials
Approximating point
Approximation is uniform
Arc, length
Archimedian spiral
Area as an Integral, General formula
Area in Polar Co-ordinates
Area of Surface of Revolution
Area, absolute
Area, Example
Area, Orientation
Argument of function
Arithmetic mean
Arithmetic progression of first order
Arithmetic progression of second order
arsin x
arsin, branch
arsin, principal value
artan, error
Associative law of addition
Associative law of multiplication
Astroid
Astroid
Asymptotically equal
Asymptotically equal
Average velocity
Axiom of differential and integral calculus
Axiomatical reasoning

B

Beats
Beats
Bernoulli Equation
Bernoulli Numbers
Bernoulli polynomials
Bessel inequality
Bi-molecular Reaction
Binary expansion of real numbers
Binomial coefficient, general
Binomial Series
Binomial Series, proof of Convergence
Binomial Series, Taylor series
Binomial theorem
Binomial Theorem, proof of
Body, Freely Falling
Boyle's Law
Boyle's law
Briggian logarithms

C

Calculation of Logarithms
Calculus of Errors
Calculus of Errors
Cardioid
Cardioid
Catalytic action
Catenary
Catenary
Catenoid
Cauchy's Convergence Test, proof of
Cauchy's notation Df(x)
Cauchy-Schwarz inequality
Centre of curvature
Centre of Mass
Centre of Mass of a Curve
Centre of mass, height of
Chain Rule
Chain Rule
Chain rule of differential calculus
Chain Rule, use of
Change of Axes for Parametrically represented Curves
Change of Independent Variables
Mixed Exercises:
Charge
Chemical law of mass action
Circle of curvature
Circle of Curvature
Circular frequency
Closed interval
Coefficient of self-induction
Coefficient of thermal expansion
Commensurable
Common ratio of geometric series
Commutative law of addition
Commutative law of multiplication
Comparison Test
Comparison with Geometric Series
Comparison with Integral
Complementary function
Complete integral
Complete neighbourhood
Complex number amplitude
Complex number, absolute value
Complex number, conjugate
Complex number, Modulus
Complex numbers
Complex Numbers
Complex Representation of Superposition of Sinusoidal Vibrations
Complex resistance of a circuit
Complex Variable, General Theory
Complex voltage
Compound function
Compound Functions, differentiation
Compounding examples
Compounding of more than two functions
Concept of Limit
Concept of Limit of a Sequence
Concept of Order of Magnitude
Condenser
Conditional and absolute convergence, difference
Conditional Convergence
Conditional convergence
Conditionally convergent
Conditionally convergent
Connection between Integral and Derivative
Constant
Constants of integration
Continuity
Continuity
Continuity
Continuity, concept of
Continuous function
Continuous Functions, more Theorems
Continuous Functions, theorems
Continuous Monotonic Function, inverse of
Continuous variable
Continuum of Numbers
Contour lines
Convergence
Convergence interval
Convergence of a series, necessary condition
Convergence of Fourier Series
Convergence of Fourier Series of Sectionally Smooth Function
Convergence of Power Series
Convergence test of Leibnitz
Convergence test, Cauchy's
Convergence Tests
Convergence, Absolute
Convergence, Concepts
Convergence, Conditional
Convergence, conditional
Convergence, Definition of
Convergence, fundamental theorem
Convergence, general principle
Convergence, Intrinsic Definition
Convergent Series, Multiplication
Convergent, Absolutely
Convergent, Conditionally
Convexity / Concavity of Curves
Cooling / Heating of Body by Surrounding Medium
Co-ordinate system, rectangular
Corner
Cubical parabola
Current co-ordinates
Curvature, centre
Curvature, radius of
Curve continuous
Curve, behaviour in the small
Curve, Centre of Mass
Curve, Curvature
Curve, Length
Curve, sense of
Curve, smooth
Curves touch
Curves, Contact of
Curves, General Remarks
Curves, Limiting Processes
Curves, order of contact
Cycloid
Cycloid
Cycloid, common
Cylindrical surface
Cylindroid

D

Damped harmonic oscillations
De Moivre's Theorem
Definite quadratic expression
Definition of Exponential Function by Differential Equation
Definition of Functions by means of Integrals
Degree
Density
Density
Density of set of rational numbers
Dependent variable
Derivative
Derivative / Difference Quotient
Derivative and Tangent
Derivative as a Velocity

Derivative of a function of a function
Derivative of f(x)^g(x)
Derivative of Indefinite Integral
Derivative of product
Derivative of quotient
Derivative of Rational Function
Derivativ of sum
Derivatives of a Function of Several Variables
Derivatives of Parametrically Represented Curve
Derivatives of Trigonometric Functions
Derived curve
Derived curve
Derived function
Descartes' folium
Descartes' folium
Difference between absolute and conditional convergence
Difference quotient
Difference Quotient / Derivative
Differentiability
Differentiability / Continuity of Functions
Differentiability of Functions
Differentiable
Differential coefficient
Differential equation
Differential equation, homogenous
Differential equation, integral
Differential equation, non-homogeneous
Differential equation, particular solution
Differential equation, solution
Differential Equations
Differential Equations for Simplest Types of Vibrations
Differential Equations of First order, solvable by Simple Integration
Differential Equations of Order n with Constant Coefficients. Homogeneous Linear Homogeneous Linear Differential Equations of Order n with Constant Coefficients
Differential Equations of Second Order
Differentiation
Differentiation and Integration of Elementary Functions
Differentiation and Integration, comments
Differentiation of f(x)
Differentiation of Infinite Series
Differentiation of Inverse Functions
Differentiation of Power Series
Differentiation Rules
Differentiation, Compound Functions
Differentiation, Functions of Functions
Differentiation/Integration of Power Series
Diffraction of light
Direction cosines of normal
Direction cosines of tangent
Dirichlet Integral
Discontinuities
Discontinuity
Discontinuity, infinite
Distortion factor
Distributive law of multiplication
Divergence Tests
DivergenceConcepts
Division
Division of Series
Domain

E

e is Irrational, proof
Eccentric angle
Effective voltage
Elastic force
Elastic Vibration
Electrical Oscillations
Elementary Functions
Elementary Integrals
Elementary transcendental functions
Ellipse
Elliptic functions
Elliptic Integral
Elliptic Integrals
End-point
Epicycloid
Equations, Numerical Solution
Error
Error estimate during integration
Estimation of Integrals and mean Value Theorem of Integral Calculus
Euler formula
Euler's constant
Evaluation of p
Even function
Evolute
Exact mathematics
Examples of concept of function
Existence of Definite Integral of Continuous Function
Expansion of Functions in Power Series
Explicit definition
Exponential curve
Exponential function
Exponential Function
Exponential Function
Exponential Function and Logarithm
Exponential Function and Logarithm
Exponential function and Logarithm as Limits
Exponential Function, Order of Magnitude Logarithm
Expression of Numbers in Scales other than 10
Extension of the Concept of Integral. Improper Integrals
External electromotive force
External electromotive force
Extrema

F

Factorials
False Position Rule
False Position, rule of
Fejér kernel
Fermat's principle of least time
Finite sum, rearrangement of terms
Force
Force elastic
Forced motion
Formulation of Concept of Function
Fourier coefficients
Fourier expansion of cosh m x and sinh m x
Fourier expansion of f(x) = |sin x|
Fourier expansion of f(x) = |x|
Fourier expansion of f(x) = cos mx
Fourier expansion of f(x) = sin mx
Fourier expansion of f(x) = xcos x
Fourier expansion of j (x) = x²
Fourier expansion of Step Function
Fourier expansion of y (x) = x
Fourier partial sum
Fourier Series
Fourier Series
Fourier series
Fourier Series of Sectionally Smooth Function, Convergence
Fourier series, complex notation
Fourier Series, Convergence
Fourier Series, Examples
Fractional rational functions
Free motion
Free Oscillations
Fresnel integrals
Friction
Frictional constant
Frictional force
Function of a function
Function of Several Variables
Function of Several Variables, Derivatives
Function y = e^-1/x
Function y = e^-1/x²
Function y = tanh 1/x
Function y = x sin 1/x, y(0) = 0
Function y =x tanh 1/x
Function, almost periodic
Function, not expandable in Taylor Series
Function, sectionally continuous
Functional determinant
Functional equation
Functions of an Integral variable
Functions of Functions, differentiation
Functions of several variables, Ranges of Definition
Functions which are not Integrable in Terms of Elementary Functions
Functions, Differentiability
Functions, expressed approximately by Polynomials
Functions, Geometrical Representation
Functions, Limiting processes
Limiting Processes with functions
Functions, Periodic
Fundamental
Fundamental Ideas of Calculus
Fundamental Rules of Differentiation
Fundamental theorem of differential and integral calculus

G

Galvanometer, tangent
Gamma Function
Gas theory
General Definition of Indefinite Integral
General Exponential Function a^x
General Formula for Differentiation
General Integration Rules
General Power x^a
General rational functions
General solution
General Theory of functions of a Complex Variable
Generalized Mean Value Theorem
Generalized mean value theorem of differential calculus
Geometric progression
Geometric Series
Geometric series
Geometric series, infinite
Geometrical Illustration of Limits
Geometrical reasoning
Gradient
Graphical Integration
Graphical Representation
Gravitational acceleration
Gravity
Greatest and Least Values of Continuous functions
Greek mathematics
Gregory's series
Guldin's rule

H

Half-period of radioactive disintegration
Harmonic
Harmonic series
Harmonics
Height of the centre of mass
Hemi-sphere
Higher Derivatives
Higher function
Homogeneous Linear Differential Equations of Order n with Constant Coefficients
Homogenous differential equation
Hyperbolic Functions
Hyperbolic Functions, Addition Theorems and Differentiation
Hyperbolic Functions, Analytical Definition
Hyperbolic Functions, Inverse
Hyperbolic paraboloids
Hypercycloid

I

Impedance
Implicit definition
Implicit Functions
Implicit Functions, Geometrical Interpretation
Impressed electro-motive force
Improper integral
Incommensurable
Incommensurable segments
Indefinite Integral
Indefinite integral
Independent variable
Independent Variables, Change
Indeterminate expressions
Indirect proof
Inequalities
Infinite at -th order
Infinite decimal
Infinite derivative
Infinite Discontinuities
Infinite discontinuity
Infinite geometric series
Infinite Interval of Integration
Infinite open interval
Infinite Product for sin x
Infinite products
Infinite Products
Infinite Series
Infinite Series, Differentiation
Infinite Series, Improper integrals
Infinite Series, operations
Infinitely small quantities
Infinitesimal
Infinitesimals
Infinities
Inflection point
Inflectional tangent
Inflectional tangent
Initial Conditions
Initial position
Initial state of vibrating system
Initial velocity
Insoluble problem
Integral as Area
Integral as Function of the Upper Limit
Integral Calculus , more details
Integral diverges
Integral, definite
Integral, fundamental Rules
Integral, symbols
Integrals for multi-valued functions
Integrals, Double, Reduction to Repeated Single Integrals
Integrals, Improper
Integrals, Multiple
Integrals, Repeated
Integration
Integration and Differentiation of x^a, a Irrational
Integration by parts
Integration by parts
Integration by parts for definite integrals
Integration by parts, Examples
Integration formulae
Integration of Linear Function
Integration of Power Series
Integration of product
Integration of R(cos x, sin x)
Integration of R(cosh x, sinh x)
Integration of Rational Functions
Integration of Rational Functions, fundamental Types
Integration of Simplest Functions
Integration of sin x and cos x
Integration of sum
Integration of x²
Integration of x^a for any Irrational a
Integration of x^a, a positive integer
Integration of x^a, a rational
Integration variable
Integration, Method of Substitution
Integration, numerical
Integration, Partial fractions
Integration, proper fraction
Interest, Compounded Continuously
Interior point
Intermediate Expressions
Intermediate Value Theorem
Interval, open at one end
Intuition, pure a priori
Intuitive evidence
Inverse Function and its derivative
Inverse Functions
Inverse of exponential function
Inverse of Logarithm
Inverse of the Power Function
Inverse Tangent
Inverse Tangent
Inverse Trigonometric Functions
Irrational numbers
Isosceles triangle
Iteration
Iteration

J

Jacobian
Jump
Jump Discontinuities

K

Kinetic theory of gases

L
Lag, current
Laplace equation
Lattice point
Law of refraction
Least upper bound
Legendre Polynomials, recurrence relation
Leibnitz' series
Leibnitz's convergence test
Leibnitz's notation
Leibnitz's Rule:
Lemniscate
Length of arc
Length of arc of semi-cubical parabola?
Length of Curve
Limit curve
Limit from interior of interval
Limit function
Limit, Concept for Continuous Variable
Limit, concept of
Limit, one-sided
Limiting Processes
Limiting Processes with Curves
Limiting Processes with Functions
Limits of Bounded Monotonic Sequences, existence of
Limits of Sequences
Limits, operations with
Limits, products of
Linear function
Linear function
Linear function
Lines of support
Logarithm
Logarithm
Logarithm and Exponential Function
Logarithm and Exponential function as Limits
Logarithm to base a
Logarithm to base e
Logarithm, Addition Theorem
Logarithm, Definition
Logarithm, Differentiation
Logarithm, natural
Logarithmic decrement
Logarithmic function
Logarithms, calculation
Lower limit of set of numbers

M

Machines
Maclaurin's theorem
Magnetic needle
Making and Breaking an Electric Circuit
Mass Distribution
Mathematical induction
Maxima / Minima, examples
Maxima and Minima
Maxima and Minima
Mean Value Theorem
Mean value theorem of differential calculus
Mean Value Theorem of Integral Calculus
Mechanics , Vibration Problems
Mechanics, Fundamental Hypotheses
Mechanics, Problems of Particle
Method of exhaustion
Method of Undetermined Coefficients
Method of Undetermined Coefficients
Microphone
Minute
Modulus
Modulus of the system of logarithms to base a
Molecular reaction
Moment of a Curve
Moment of Inertia
Moment of Inertia
Moment of inertia of Catenary
Moment of inertia of Common Cycloid
Moment of inertia of Ellipse
Moment of inertia of Lemniscate
Moment of inertia of system of masses
Moment of system of particles
Moments of Mass
Monotonic decreasing function
Monotonic Function
Monotonic function
Monotonic increasing function
Monotonic Sequences
Multiple Integrals
Multiplication and Division of Power Series
Multiplication by constant
Multiplication of Absolutely Convergent Series
Multiplication of Series
Multiplication of Series
Multi-valued function
Multi-valued functions

N

Natural base e
Natural frequency
Natural logarithm
Natural numbers
Necessary condition for convergence of a series
Need for Extension of Rational Numbers
Negative fractions
Negative integers
Negative number
Neil's parabola / Semi-cubical parabola
Newton's law of cooling
Newton's Method
Newton's Method
Newton's second fundamental law
Non-differentiable
Non-homogeneous differential equation
Non-homogeneous Differential Equation, Solution
Normal, direction cosines
Notation for partial derivatives
n-th derivative f ⁽ⁿ⁾(x)
n-th differential coefficient
n-th partial sum
Number 0
Number axis
Number p as limit
Number p(n)
Numerical integration
Numerical integration
Numerical integration, Estimation of Error
Numerical integration, examples
Numerical integration, Rectangle Rule
Numerical integration, Simpson's Rule
Numerical integration, Trapezoid/ Tangent Formulae
Numerical Methods
Numerical Methods
Numerical Solution of Equations
Numerical value

O

Odd function
Ohm's law
Ohm's law for alternating currents
Open interval
Operations with Power Series
Optical principle of least time
Order of Magnitude
Order of Magnitude of a Function in Neighbourhood of an Arbitrary Point
Order of Magnitude of a Function tending to Zero
Order of Magnitude of Functions
Order of Magnitude, General Remarks
Origin of coordinates
Oscillation about limit
Oscillations, electrical
Oscillatory system
Osculating Circle
Osculating, parabolas
Overtone

P

p , evaluation
Parabaloid of revolution
Parabola
Parabola
Parabola, semi-cubical
Parabolas osculating
Parallel curve
Parallel curves
Parameter
Parameter, Change of
Parametrically represented Curves, Change of Axes
Partial derivative
Partial derivatives, mixed
Partial derivatives, notation
partial sum
partial sum, n-th
Particle
Particles, moment of System
Particular integral
Particular solution of differential equation
Pedal curve
Pedal curve
Pendulum
Periodic Functions
Periodicity
Perturbation
Phase
Phase displacement
Phase displacement
Physically infinitesimal quantities
Physics, Vibration Problems
Plane Curves, Theory of
Plane in space
Point 1
Point of Accumulation, Principle of
Point of inflection
Point of inflection
Points of accumulation, upper/lower
Polar Coordinates
Polar Co-ordinates
Polar Co-ordinates, Area
Polygonal path
Polynomials
Polynomials
Polynomials, Taylor's Theorem
Positive fractions
Positive number
Power Series
Power Series Convergence
Power Series with Complex Terms
Power Series, Differentiation/Integration
Power Series, Multiplication and Division
Power Series, Operations
Power Series, Uniqueness
Prime number
Prime numbers
Primitive Function
Primitive Function
Primitive function of f(x)
Primitive of f(x)
Probability Integral
Product form
Progress of Chemical Solution
Proof of the Binomial Theorem
Properties of binomial coefficients
Property of continuity

Q

Quadratic expression

R

Radian measure p
Radians
Radioactive Disintegration
Radius of curvature
Range (R) of definition
Range of definition
Ratio test
Rational fractional functions
Rational Functions
Rational integral functions
Rational number
Rational operations of calculation
Rational Representation of Trigonometric and Hyperbolic Functions
Rational, integral functions
Real Numbers and Infinite Domains
Recording Instruments
Rectangle Rule
Rectangular hyperbola
Rectangular region
rectifiability
Recurrence Formulae
Reduction of Double Integrals to Repeated Single Integrals
Reduction to Integrals of Rational Functions
Refraction, Snell's law
Region, Oriented
Relation between Mean Value Theorems of Differential Calculus and Integral Calculus
Relative extreme values
Remainder
Remainder Estimation
Remainder, Taylor series
Repeated Integrals
Representation of Curves
Representation of curves, parametric
Resistance of Air
Resistance of electric circuit
Resolution into Partial Fractions
Resolution of cotan x into Partial Fractions
Resonance
Resonance curve
Resonance frequency
Resonance phenomenon
Restoring force
Reverse process of differentiation
Root finding, Examples
Root test
Roots of unity
Rule of False Position

S

Schwartz's Inequality for integrals
Schwarz's Inequality
Secant
Second
Second derivative
Second difference
Second difference quotient
Second Mean Value Theorem of the Integral Calculus
Sectionally continuous function
Sectionally monotonic
Semi-cubical parabola
Semi-cubical parabola / Neil's parabola
Sense, negative
Sense, positive
Separable variables
Separation of variables
Sequence, bounded convergent
Sequence, divergent
Sequence, monotonic
Sequence, monotonic decreasing
Sequence, monotonic increasing
Sequences of curves
Sequences of Functions
Sequences of Numbers
Series 1/(1 + x)²
Series in Bernoulli Numbers
Series of Functions
Series, arsin x
Series, Division
Series, geometric
Series, Multiplication
Series, Multiplication
series, sum
Series, x = log {x + (1 + x²)^½}
Sexagesimal system
Sharp corner
Significance of Higher Derivatives
Simple harmonic oscillations
Simple harmonic vibrations
Simplest Differentiation Rules and Applications
Simpson's Rule
Simpson's Rule
Single-valued branch
Single-valued function
Sinusoidal
Sinusoidal vibrations
Sinusoidal Vibrations, Complex Representation of Superposition
Sinusoidal Vibrations, Superposition
Slope of surface
Snell's law of refraction
Solution of Homogeneous Equation
Special Functions
Specific heat
Specific mass
Specific probability
Specific Quantity
Stationary point
Stationary value
Step-function
Stirling's Formula
Strength of materials
Structnral engineering
Subscript notation
Substitution for convergent improper integrals
Substitution Formula
Substitution Formula, another Proof
Substitution, basic formula
Substitution, convergent improper integrals
Subtraction
sum of series
Sum, lower
Sum, upper
Summation of infinite series
Superposed
Superposition of Sinusoidal Vibrations
Superposition principle
Surface
Surface of Revolution, Area and Volume
Symbol ¥
Symbol for difference D
Symbol Öx
System of Rational Numbers
System of real numbers

T

Tangent and Derivative
Tangent Formula
Tangent galvanometer
Tangent galvanometer
Tangent, direction cosines
Tangent, inflectional
Taylor series
Taylor series, Applications
Taylor series, Binomial Series
Taylor series, expansion of elementary functions
Taylor series, Exponential function
Taylor series, Geometrical Applications
Taylor series, infinite
Taylor series, order of terms
Taylor series, remainder
Taylor series, sin x, cos x, sinh x, cosh x
Taylor's Theorem
Taylor's Theorem
Taylor's theorem
Taylor's Theorem, Arbitrary Function
Taylor's Theorem, Polynomials
Term-by-Term Integration
Terminating series
Theorem of continuity of functions of continuous functions
Theory of extreme values
Theory of Functions of Several Variables
Theory of Maxima and Minima
Theory of numbers
Theory of numbers
Theory of probability
Time
Torus
Total Quantity
Tractrix
Transcendental functions
Trapezoid Formula
Trapezoid/ Tangent Formulae, Numerical integration
Trigonometric Functions
Trigonometric functions
Trigonometric functions, addition theorems
Trigonometric functions, orthogonality relations
Trigonometric summation formula
calc3.htm - turning value, turning point
Turning value

U

Undamped system
Uniform Continuity of Continuous Functions, theorem
Uniform Convergence Test
Uniform motion
Uniformity of Continuity
Uniformly Convergent Series, Continuity of Sum
Uniformly Convergent Series, Integration
Uniqueness of Power Series
Uniqueness of the Solution
Unit interval
Upper bound
Upper limit of set of numbers
Upper partial
Use of Primitive Function in Evaluation of Definite Integrals

V

Variable, independent
Variation of Atmospheric Pressure with Height
Variation of parameters
Velocity at time t
Velocity of motion
Vertex
Vibrating system, initial state
Vibration phenomena
Vibration Problems
Vibration, Elastic
Voltage
Volume of Surface of Revolution

W

Wallis' Product
wave-crests
wave-troughs
Weierstrass' Approximation Theorem

Z

Zero point
Zeroes/ Infinities of Functions
zeta function