• Sequence and Series
• Multivariate Calculus

Each chapter follows a structured format:

• “Chapter at a Glance” with key definitions and explanations
• Multiple-choice questions
• Short and long answer questions, often sourced from past WBUT examinations

Calculus (Integration)

Improper Integrals

Improper integrals are classified into two types:

• Type I: Integrals taken over an unbounded interval (for example, limits involving infinity)
• Type II: Integrals where the integrand becomes unbounded or discontinuous within the interval

Methods for evaluation using limits are discussed, along with conditions for convergence and divergence.

Special Functions

Beta Function

• Also known as Euler’s integral of the first kind
• Defined for positive real values of its parameters
• It is a symmetric function, meaning the order of parameters does not affect its value

Gamma Function

• Also known as Euler’s integral of the second kind
• Defined for positive real values
• The integral representation is convergent for all positive arguments

Relation Between Beta and Gamma Functions

• The Beta function can be expressed in terms of the Gamma function
• This relationship is frequently used to simplify calculations involving special integrals

Applications of Integration

• Rectification: Finding the length of plane curves
• Area: Determining the area of a plane region
• Volume: Calculating the volume of solids of revolution
• Surface Area: Finding the surface area of solids generated by rotation about the X-axis or Y-axis

Calculus (Differentiation)

Mean Value Theorems

• Lagrange’s Mean Value Theorem establishes the existence of at least one point where the average rate of change equals the instantaneous rate of change
• Taylor’s Theorem generalizes this concept using higher-order derivatives

Taylor and Maclaurin Series

• Taylor Series represents a function expanded about any point
• Maclaurin Series is a special case of Taylor Series expanded about zero

Indeterminate Forms (L’Hospital’s Rule)

• Used to evaluate limits that result in indeterminate forms such as zero over zero or infinity over infinity
• Other indeterminate forms are first transformed into these basic types before applying the rule

Maxima and Minima

• A necessary condition for a function to have an extreme value is that its first derivative vanishes at that point
• Higher-order derivative tests are used to determine the nature of the extremum

Matrices

Definition and Determinants

• A matrix is defined by its number of rows and columns
• The determinant is defined only for square matrices and is used to determine important properties such as invertibility

Minors and Cofactors

• A minor is obtained by deleting a specific row and column from a determinant
• A cofactor is obtained by assigning a sign to the corresponding minor

Systems of Linear Equations (Cramer’s Rule)

• A system of linear equations has a unique solution if the determinant of the coefficient matrix is non-zero

Vector Spaces

Definition

• A vector space is a collection of elements that satisfies axioms related to vector addition and scalar multiplication
• A real vector space is defined over the field of real numbers

Subspaces

• A non-empty subset of a vector space is a subspace if it is closed under vector addition and scalar multiplication
• The intersection of two subspaces is also a subspace

Linear Independence and Dependence

• A set of vectors is linearly independent if none of the vectors can be expressed as a linear combination of the others
• A set is linearly dependent if at least one vector can be expressed as a linear combination of the others

Basis and Dimension

• A basis is a linearly independent set that spans the entire vector space
• The dimension of a vector space is the number of vectors in its basis

Linear Transformation

• A mapping between two vector spaces is linear if it preserves vector addition and scalar multiplication

Kernel and Rank (Sylvester’s Law)

• Kernel (Null Space) consists of all vectors that are mapped to the zero vector
• Image (Range) consists of all vectors obtained as outputs of the transformation
• Sylvester’s Law of Nullity states that the sum of the rank and nullity equals the dimension of the domain space

Eigen Values and Eigen Vectors

Definitions

• The characteristic matrix is obtained by subtracting a scalar multiple of the identity matrix from a given matrix
• The characteristic equation is obtained by setting the determinant of the characteristic matrix to zero
• Eigen values are the roots of the characteristic equation
• Eigen vectors are non-zero vectors associated with an eigen value

Theorems and Properties

• Cayley–Hamilton Theorem states that every square matrix satisfies its own characteristic equation
• An orthogonal matrix preserves length and angle
• A matrix is diagonalizable if it has a full set of linearly independent eigen vectors
• A matrix is orthogonally diagonalizable if it is real and symmetric

Sequence and Series

Sequences

• Types of sequences include bounded, monotonic, convergent, divergent, and oscillatory sequences

Series Convergence Tests

• p-Series Test determines convergence based on the value of the exponent
• Comparison Test (Limit Form) compares a given series with a known series
• D’Alembert’s Ratio Test uses the ratio of successive terms
• Cauchy’s Root Test uses the root of the general term

Alternating Series

• Leibnitz’s Test provides conditions for the convergence of alternating series
• Absolute Convergence occurs when the series of absolute values converges
• Conditional Convergence occurs when a series converges but not absolutely

Multivariate Calculus

Functions of Several Variables

• Functions involving more than one independent variable are introduced

Limits and Continuity

• Continuity is defined using limits as multiple variables approach a point
• The limit must be independent of the path of approach

Partial Derivatives

• Partial derivatives measure the rate of change of a function with respect to one variable while keeping others constant
• Higher-order and mixed partial derivatives are also discussed

Mixed Derivative Theorems

• Young’s Theorem and Schwarz’s Theorem state conditions under which mixed partial derivatives are equal

Homogeneous Functions and Euler’s Theorem

• A homogeneous function exhibits uniform scaling behavior
• Euler’s Theorem relates the function to its partial derivatives

Jacobian

• The Jacobian is used to study transformations between coordinate systems and functions of multiple variables

Maxima and Minima (Two Variables)

• Stationary points are identified using first-order partial derivatives
• Second-order derivative tests classify points as maximum, minimum, or saddle points

Lagrange’s Multipliers

• A technique used to find extreme values of a function subject to one or more constraints

Vector Calculus

Velocity and Acceleration

• Velocity is defined as the rate of change of position
• Acceleration is defined as the rate of change of velocity

Vector Differential Operator

• Introduces the differential operator used in vector calculus

Gradient, Divergence, and Curl

• Gradient represents the rate and direction of maximum increase of a scalar field
• Divergence measures the net outward flow of a vector field
• Curl measures the rotational tendency of a vector field
• A field may be solenoidal or irrotational based on these properties