#4 MATRIX | Applied Mathematics -III UP Polytechnic Third Semester | UPBTE 3rd Semester As Technic

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🔴🔴Applied Mathematics🙏 (अनुप्रयुक्त गणित –3rd) by AS TECHNIC🔴🔴
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Syllabus -
1. Matrices
1.1 Algebra of Matrices, Inverse
Addition, Multiplication of matrices, Null matrix and a unit matrix, Square matrix,
Symmetric, Skew symmetric, Hermitian, Skew hermition, Orthagonal, Unitary,
diagonal and Triangular matrix, Determinant of a matrix.
Definition and Computation of inverse of a matrix.
1.2 Elementry Row/Column Transformation
Meaning and use in computing inverse and rank of a matrix.
1.3 Linear Dependence, Rank of a Matrix
Linear dependence/independence of vectors, Definition and computation of rank
of matrix. Computing rank through determinants, Elementary row transformation
and through the concept of a set of independent vectors, Consistency of equations.
1.4 Eigen Pairs, Cayley-Hamilton Theorem
Definition and evaluation of eign values and eign vectors of a matrix of order two
and three, Cayley-Hamilton theorem (without Proof)and its verification, Use in
finding inverse and powers of a matrix.

2. Differential Calculus
2.1 Function of two variables, identification of surfaces in space, conicoids
2.2 Partial Differentiation
Directional derivative, Gradient, Use of gradient f, Partial derivatives, Chain
rule, Higher order derivatives, Euler’s theorem for homogeneous functions,
Jacobians.
2.3 Vector Calculus
Vector function, Introduction todouble and triple integral,differentiation and
integration of vector functions, gradient, divergence and curl, differential
derivatives.

3. Differential Equation
3.1 Formation, Order, Degree, Types, Solution
Formation of differential equations through physical, geometrical,
mechanical and electrical considerations, Order, Degree of a differential equation,
Linear, nonlinear equation.
3.2 First Order Equations
Variable seperable, equations reducible to seperable forms, Homogeneous
equtions, equations reducible to homogeneous forms, Linear and Bernoulli
form exact equation and their solutions.
3.3 Higher Order Linear Equation :
Property of solution, Linear differential equation with constant coefficients
(PI for X= eax , Sinax, Cosax, Xn , axV, XV
3.4 Simple Applications
LCR circuit, Motion under gravity, Newton's law of cooling, radioactive
decay, Population growth, Force vibration of a mass point attached to spring
with and without damping effect. Equivalence of electrical and mechanical
system

4. Integral Calculus-II
4.1 Beta and Gamma Functions
Definition, Use, Relation between the two, their use in evaluating integrals.
4.2 Fourier Series
Fourier series, Odd and even function,Half range series.
4.3 Laplace Transform
Definition, Basic theorem and properties, Unit step and Periodic functions,
inverse laplace transform, Solution of ordinary differential equations

5. Probability and Statistics
5.1 Probability
Introduction, Addition and Multiplication theorem and simple problem.
5.2 Distribution
Discrete and continuous distribution, Bionimal Distribution, Poisson
distribution, Normal Distribution.

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