COURSE OBJECTIVES:
- To develop the use of matrix algebra techniques that are needed by engineers for practical
applications. - To familiarize the students with differential calculus.
- To familiarize the student with functions of several variables. This is needed in many
branches of engineering. - To make the students understand various techniques of integration.
- To acquaint the student with mathematical tools needed in evaluating multiple integrals and
their applications.
UNIT I MATRICES
Eigenvalues and Eigenvectors of a real matrix – Characteristic equation – Properties of Eigenvalues
and Eigenvectors – Cayley – Hamilton theorem – Diagonalization of matrices by orthogonal
transformation – Reduction of a quadratic form to canonical form by orthogonal transformation –
Nature of quadratic forms – Applications : Stretching of an elastic membrane.
UNIT II DIFFERENTIAL CALCULUS
Representation of functions – Limit of a function – Continuity – Derivatives – Differentiation rules (sum,
product, quotient, chain rules) – Implicit differentiation – Logarithmic differentiation – Applications :
Maxima and Minima of functions of one variable.
UNIT III FUNCTIONS OF SEVERAL VARIABLES
Partial differentiation – Homogeneous functions and Euler’s theorem – Total derivative – Change of
variables – Jacobians – Partial differentiation of implicit functions – Taylor’s series for functions of
two variables – Applications : Maxima and minima of functions of two variables and Lagrange’s
method of undetermined multipliers.
UNIT IV INTEGRAL CALCULUS
Definite and Indefinite integrals – Substitution rule – Techniques of Integration : Integration by parts,
Trigonometric integrals, Trigonometric substitutions, Integration of rational functions by partial
fraction, Integration of irrational functions – Improper integrals – Applications : Hydrostatic force and
pressure, moments and centres of mass.
UNIT V MULTIPLE INTEGRALS
Double integrals – Change of order of integration – Double integrals in polar coordinates – Area
enclosed by plane curves – Triple integrals – Volume of solids – Change of variables in double and
triple integrals – Applications : Moments and centres of mass, moment of inertia.
MA3151 MATRICES AND CALCULUS IMPORTANT QUESTIONS
PART A
UNIT 1
State Cayley-Hamilton Theorem
Rank,index,signature
Properties of Eigen values(Proof and problems)
Quadratic form to matrix form
UNIT 2
Domain,Range
Limits
Rolle’s theorem
Mean Value theorem
Differentiation (Implicit,Chain rule)
UNIT 3
Jacobians
Euler’s theorem
Implicit Differentiation
Parametric equation (u=xy,x=at²,y=2at find dy/dx type)
UNIT 4
Convergence and divergence
What is wrong with the equation
Substitution method
Simple integration
UNIT 5
Change of integration
Derive/determine the region for triple integration
Simple double/Triple integration
PART B
UNIT 1
Canonical Form(With index,rank and signature)
Cayley Hamilton Theorem(Verification,Inverse,Long equation)
Eigen values and Eigen vectors
Diagonalisation
UNIT 2
Maxima and Minima(Local and Absolute Maxima and Minima, Increasing and Decreasing, Points of deflection,concavity)
Continuous functions
Tangent line and Normal line
Normal Differentiation
UNIT 3
Taylor’s Series
Lagrangian Multiplier
Maxima Minima two variables
Euler’s theorem(Verification,Inverse)
UNIT 4
Integration by parts
Reduction formula
Partial fraction
Convergence and divergence
UNIT 5
Triple integration (Volume integral)
Change of order of Integration
Area problems(Cartesian and polar)
Change of variables from cartesian to polar