Advanced Calculus

Advanced Differential Calculus, Integral Calculus and Vector Calculus

Authors: Dr. Vimal Saraswat, Dr. Anil Kumar Menaria

This book includes the following topics

Continuity

• Introduction
• Limit
• Left and right limit
• To find the R.H.L. and L.H.L. of a function
• Existence of limit)/li>
• Distinction between the value and limit of a function
• Some theorems based on limits
• Methods of finding the limit of functions
• Some standard limits
• Cauchy’s definition of continuity
• Continuity from left and right
• Continuity of a function in an interval

Continuity in the open interval); Continuity in the closed interval

• Continuous function
• Heine’s definition of continuity or sequential continuity
• Discontinuity and its types

Removable discontinuity; Discontinuity of first kind or ordinary discontinuity; Discontinuity of the second kind; Mixed discontinuity

• Properties of continuous functions
• Uniform continuity
• Function of two variables
• Limit of function of two variables
• Continuity of function of two variables

Derivability

• Differentiable or derivable function
• Right hand and left hand derivative
• Differentiability of a function in an interval
• Some standard results on differentiability
• Necessary condition for the existence of a finite derivative
• Algebraic properties of derivatives
• Chain rule or derivative of function of a function
• Derivative of the inverse function
• Properties of derivative
• Darboux intermediate value theorem for derivatives
• Differentiability of functions of two variables
• Necessary and sufficient condition for differentiability
• Total derivative
• Algebraic property of differentiability of real valued functions of two variables
• Condition for differentiability in polar coordinates

Partial Differentiation

• Introduction
• Partial differential coefficients
• Partial derivatives of higher order
• Homogeneous functions
• Euler’s theorem for homogeneous functions
• Total differential coefficient
• Differentiation of implicit functions

Maxima and Minima

• Introduction
• Extreme values of functions of two variables
• Criteria for extreme value of f (x, y)
• Working method of finding extreme values
• Lagrange’s method for undetermined multipliers

Envelopes and Evolutes

• Family of curves
• Envelope
• Method of finding the envelope
• Working method for finding envelope
• Relation between envelope and each member of the family of curves
• To find the envelope, when two parameters are connected by a relation
• Evolute

Jacobians

• Introduction
• Definition
• Jacobians of functions of functions
• Jacobians of implicit functions
• Dependence of functions

Double Integrals

• Introduction
• Double integration
• Method of finding double integral
• Double integral in polar coordinates
• Change of double integral from cartesian to polar coordinates
• Change of order of integration
• Applications of double integrals

Area; Mass

Triple Integrals

• Introduction
• Evaluation of triple integral
• Dirichlet’s integral
• Liouville’s extension of Dirichlet’s integral
• Dirichlet’s general theorem
• Applications of triple integral

Volume and Surface of Solids of Revolution

• Introduction
• Volume of solids of revolution of cartesian curves
• Volume generated by revolution about any line
• Volume of solids of revolution of polar curves
• Prolate and oblate spheroid
• Surface area of solids of revolution of cartesian curves

Differential Operators

• Introduction
• Differential formulae for vectors
• Partial derivatives of vectors
• Vector differential operator ▽
• Scalar point function and scalar field
• Vector point function and vector field
• Gradient of scalar point function
• Theorem based on gradient
• a ⋅ ▽ operator
• Directional derivative
• Theorems based on directional derivatives
• Vector equation of tangent plane
• Vctor equation of the normal
• Divergence of a vector point function
• Solenoidal vector
• Theorem based on divergence
• Curl of a vector point function
• Irrotational vector
• Theorem based on curl
• Some important vector identities
• Second order differential function and its properties

Vector integration

• Introduction
• Vector constant of integration
• Some important integral result
• Linear integral or line integral
• Irrotational vector
• Work

Integral theorems

• Introduction
• Surface integral
• Volume integral
• Gauss’s divergence theorem
• Stoke’s theorem
• Green’s theorem
• Cartesian form of Green’s theorem