First-year engineering mathematics shared by most AICTE and VTU-pattern programmes: Laplace transforms, Fourier series, eigenvalues and eigenvectors, partial differentiation, and vector calculus (gradient, divergence and curl).
5 reference pages
Laplace transforms: the definition, a table of standard transforms, the shifting and differentiation properties, initial and final value theorems, and first and second-order ODEs solved with checks.
Fourier series coefficients for period 2π and 2L, odd and even shortcuts, worked series for x and x² (with the sum of 1/n²), behaviour at a jump, and common mistakes.
Eigenvalues and eigenvectors: the characteristic equation, two fully worked 2x2 examples, the trace and determinant checks, eigenvalues of A², A⁻¹ and A + kI, triangular matrices, and common mistakes.
Partial differentiation: first and mixed second partial derivatives, the gradient and directional derivative, and maxima and minima of functions of two variables, with worked examples and common mistakes.
Gradient, divergence and curl: what each operator takes and returns, worked examples on polynomial and rotating fields, the identity curl(grad f) = 0, and common mistakes.