Course Syllabus
This syllabus outlines the modules of Mathematics III.
- Formation of PDEs: elimination of arbitrary constants and functions
- Classification of PDEs: order, degree, linearity; quasi-linear and nonlinear PDEs
- First-order PDEs: Lagrange's method; Charpit's method
- Second-order linear PDEs: classification into elliptic, parabolic, and hyperbolic types
- Canonical forms of second-order PDEs; method of separation of variables
- Wave equation: D'Alembert's solution; one-dimensional vibrating string
- Heat equation: solution by separation of variables; temperature distribution
- Laplace's equation: solution in Cartesian and polar coordinates
- Bessel's equation; Bessel functions of the first and second kind
- Legendre's equation; Legendre polynomials; orthogonality
- Sturm–Liouville problems; eigenfunction expansions
- Fourier integral and Fourier transform applications to PDEs
- Boundary-value problems in engineering: heat conduction, vibration
- Green's functions for PDEs (introductory treatment)
- Sample space, events, and axioms of probability; addition and multiplication rules
- Conditional probability; independence; Bayes' theorem and total probability
- Random variables (discrete and continuous); probability mass and density functions
- Cumulative distribution function; mathematical expectation and variance
- Moment-generating function; Chebyshev's inequality
- Discrete distributions: Bernoulli, Binomial, Poisson, Geometric, Negative Binomial
- Continuous distributions: Uniform, Exponential, Normal (Gaussian)
- Standard normal distribution; standard normal table; $z$-scores
- Bivariate distributions: joint, marginal, and conditional distributions
- Covariance and correlation coefficient; independence of random variables
- Functions of random variables; transformation of distributions
- Central Limit Theorem and its applications
- Descriptive statistics: measures of central tendency and dispersion
- Moments, skewness, and kurtosis; frequency distributions
- Curve fitting by the method of least squares: straight line, parabola, exponential
- Correlation: Karl Pearson's coefficient; rank correlation (Spearman)
- Linear regression: lines of regression; regression coefficients
- Sampling distributions: $t$, $\chi^2$, and $F$ distributions
- Hypothesis testing: null and alternative hypotheses; type I and type II errors
- Tests for population mean and proportion (large samples): $z$-test
- Tests for small samples: Student's $t$-test (one-sample and two-sample)
- Chi-square test: goodness of fit and test for independence of attributes
- $F$-test for equality of variances; one-way ANOVA (introduction)
- Confidence intervals for mean and proportion