Calculus

Differential Calculus, Integral Calculus and Differential Equation

Authors: Dr. Vimal Saraswat, Dr. Anil Kumar Menaria, Dr. Chandrapal Singh Chouhan

This book includes the following topics

Pedal Equations and Derivative of the Length of an Arc

• Polar co-ordinates
• Relation between cartesian and polar co-ordinates
• Angle between radius vector and tangent
• Angle of intersection of two polar curves
• Polar tangent, subtangent, normal, subnormal and their lengths
• Perpendicular from pole to tangent and its length
• Pedal equation
• Differential coefficient of length of the arc

When the equation of curve is in cartesian form (x, y); When the equation of curve is in polar form (r, ፀ)

• Other formula

Mean Value Theorem

• Introduction
• Rolle’s Theorem
• Geometrical interpretation of Rolle’s theorem
• Algebraic interpretation of Rolle’s theorem
• Another form of Rolle’s theorem
• Lagrange’s mean value theorem
• Another form of Lagrange’s mean value theorem
• Geometrical interpretation of Lagrange’s mean value theorem
• Some important deduction from Lagrange’s mean value theorem
• Cauchy’s mean value theorem or first mean value theorem
• Another useful form of Cauchy’s mean value theorem
• General mean value theorem
• Second mean value theorem
• Generalised mean value theorem or Taylor’s theorem with Lagrange’s form of remainder
• Taylor’s theorem with Cauchy’s form of remainder
• Taylor’s theorem with Schlömilch and Roche form of remainder
• Maclaurin’s theorem with Schlömilch and Roche form of remainder
• Power series
• Taylor series
• Maclaurin’s series
• Power series expansion of some useful basic functions

Asymptotes

• Introduction
• Asymptote
• Condition for asymptote
• Asymptotes of general algebraic form of the curve
• Number of asymptotes of a curve
• Asymptotes parallel to the direction axes
• Alternative method for finding asymptotes
• Intersection of curve and its asymptotes
• Position of curve with respect to asymptote

Curvatures

• Introduction)
• Some important definitions
• A formula for radius of curvature)
• Cartesian formula for radius of curvature
• Parametric formula for radius of curvature
• Polar formula for radius of curvature
• Pedal formula for radius of curvature
• Tangential polar formula for radius of curvature
• Miscellaneous formula for radius of curvature
• Coordinate of centre of curvature
• Length of chord of curvatures

Curve Tracing

• Introduction
• Concavity and convexity
• Test of concavity or convexity with respect to coordinate axes
• Point of inflexion
• Point of undulation
• Singular points
• Multiple points
• Types of double points
• Types of cusp
• Species of cusp
• Necessary condition for the existence of double point
• Curve tracing
• Working method for tracing of cartesian curve
• Working method for tracing of polar curves
• Working method for tracing of parametric curve

Beta and Gamma Function

• Beta function
• Properties of beta function
• Gamma function
• Recurrence formula of gamma function
• Relation between beta and gamma functions
• Gamma formula
• Duplication formula
• Properties of gamma function

Quadrature

• Introduction
• Area bounded by cartesian curve and direction axes
• Common area of two cartesian curve
• Area and mass by double integration
• Area bounded by polar curves and radius vectors
• Area of closed curve

Rectification

• Introduction
• Length of arc of different form of curves
• Intrinsic equation of a curve
• To find the intrinsic equation of curve
• Length of arc of an evolute

Differential Equations of First Order & First Degree

• Differential equation
• Ordinary and partial differential equations
• Order and degree of a differential equation
• Solution of differential equation
• Equation of the first order and first degree
• Equations in which variables are separable
• Homogeneous differential equation
• Equations reducible to a homogeneous equation
• Linear differential equation
• Bernoulli’s differential equation reducible to the linear form
• Exact differential equation
• Method of finding out the general solution of an exact differential equation
• Equation reducible to an exact differential equation
• Methods of finding out the integrating factors

By inspection; Rules of finding the integrating factors

Differential Equations of First Order & Higher Degree

• Introduction
• Equation solvable for p
• Equation solvable for y
• Lagrange’s equation
• Equation solvable for x
• Clairaut’s equation
• Singular solutions

Geometrical meaning of singular solutions; Method of finding singular solutions in general case

• Extraneous loci and its type
• Procedure of finding the singular solutions and extraneous loci

Geometrical Meaning of Differential Equation

• Introduction
• Geometrical meaning of differential equation
• Family of curves
• Equation of family of curve
• Trajectories

Orthogonal trajectory; Self orthogonal family of curves

• Equation of orthogonal trajectory of a family of curves
• Applications of orthogonal trajectories

Linear Differential Equations with Constant Coefficient

• Introduction
• General form of linear differential equation
• Differential operator
• Inverse operator
• Homogeneous and non-homogeneous linear differential equation
• Complementary function and particular integral
• Method of finding complementary function

When all roots of auxiliary equations are different; When roots of auxiliary equation are same; When the roots of auxiliary equation are complex; When the roots of auxiliary equation are complex and same; When the roots of auxiliary equation are surd roots; When the roots of auxiliary equation are surd and equal

• General method for finding particular integral
• Short method for finding particular integral
• Short method for finding P.I., when Q = e^ax
• Short method for finding P.I., when Q = sin ax or cos ax
• Short method for P.I., when Q = xⁿ
• Short method of P.I., when Q = e^ax.v, where v is any function of x
• Short method of P.I., when Q = x.v, where v is any function of x

Homogeneous Linear Differential Equation

• Introduction
• Method for solving homogeneous linear differential equation
• Methods for finding the complementary function
• Alternative method of finding homogeneous linear differential equations
• Differential equations reducible to homogeneous linear form