Math

List of topics for reference

Basics


Engineering

Prerequisites

0.1
  • exponential, log
  • sin, cos, identities
  • even, odd functions
  • tan
  • cosh, sinh
0.2
  • First order Partial Derivatives of f(x,y)
  • 2nd order partial derivatives
  • f(x,y,z) 1st and 2nd partial derivatives
0.3

  • System of equations
  • Determinants of second order
  • elements, row, columns
  • cramer's rule
  • homogeneous/non-homogeneous, trivial solutions
  • determinants of third order
  • existence of solutions, no.of solutions
  • minor, co-factor
  • properties of determinants
0.4
  • complex numbers, real/imaginary parts
  • cartesian coordinate system
  • complex plane/argand diagram
  • equality / addition / subtraction / multiply / division
  • conjugate
  • commutative, assosiative, etc laws
0.5
  • polar form of complex numbers/ trignometric form
  • absolute value, argument
  • usefulness in multiplication, division
  • properties in multiplication, division
0.6
  • numerical computation
  • absolute error
  • relative error
  • rounding error, guarding figures
0.7
  • solutions of equations f(x) = 0
  • quadratic equations
  • newton's method
  • method of false position (regula falsi)
  • iteration method
0.8
  • approximate integration
  • rectangular rule
  • trapezoidal rule (linear)
  • simpson's rule

ODE First Order

1.1
  • ode vs. partial differential equations
  • order of an ODE
  • solution of first-order ODE
  • implicit, explicit solution 
    • y = g(x), G(x,y) = 0
  • general, particular solution
  • singular solutions
    • not important for engineering
  • solutions which are linear
1.2
  • implicit form of ODE
    • F(x,y,y') = 0
  • explicit form of ODE
    • y' = f(x,y)
  • isoclines
    • curves of constant slope
    • f(x,y) = constant
  • direction field
    • perpendicular to isoclines
1.3
  • Seperable equations
    • g(y)y' = f(x)
    • g(y)dy = f(x)dx
    • integrate both sides
  • initial value problems, initial conditions
1.4
  • Reducable to seperable equations
  • change of variables
    • y' = g(y/x)
    • y/x = u, y' = u+u'x
    • u + u'x = g(u)
    • du/(g(u)-u) = dx/x
    • integrate both sides
1.5
  • exact differential equations
  • definition of exact
    • M(x,y)dx + N(x,y)dy = 0
    • du = pdu/pdx*du + pdu/pdy*dy
    • du = 0
    • integrate both sides
1.6
  • Integrating factors
    • P(x,y)dx + Q(x,y)dy = 0
    • make exact by multiplying with F(x)
1.7
  • Linear ODE
  • Linear form
    • y' + f(x)y = r(x)
  • homogeneous/non-homogeneous
  • homogeneous form solution
  • non-homogeneous solution
  • input, output/response of system
    • (image: block diagram of system with input and output)
1.8
  • variation of parameters
    • y' + f(x)y = r(x)
    • y(x) = u(x)r(x)
1.9
  • voltage-resistor circuit
  • RL, RC circuit
1.10
  • one-parameter family of curves
  • F(x,y,c) = 0 for fixed value of c
    • represents a curve in x-y plane
  • parameter of family
  • representation using differential equation y' = f(x,y)
  • orthogonal trajectories
1.11
  • Picard's iteration method
    • approximate solution of IVP
1.12
  • Existence, uniqueness of solutions
  • existence theorem (solutions of IVP)
  • uniqueness theorem (unique solution)
1.13
  • Numerical methods
    • Euler-cauchy method

Laplace Transform

useful for solving linear differential equations

4.1
  • 3 steps for solving equations with laplace transform
    • subsidiary equations
  • advantages
  • F(s) = L{f} = integral(0->inf)e^(-st)f(t)dt
  • f(t) = L-1{f}
  • linearity property
    • af(t) + bg(t) --> aF(s) + bG(s)
  • LT of some elementary functions
  • first shifting theorem
    • e^at*f(t) --> F(s-a)
  • existence theorem
    • sufficient conditions for existence of laplace transform of function
    • 1. piecewise continuous
    • 2. increase as t approaches infinity
  • uniqueness of function and its LT
4.2
  • diff/integration -> multiplication/division
  • differentiation theorem
    • f' --> sF(s) - f(0)
    • conditions
  • derivative of order n
    • f'(n) --> s^nF(s) - s^(n-1)f(0) - s^(n-2)f'(o) - ... - f'(n-1)(0)
    • existence theorem
  • integration of f(t)
    • int(0->t) of f(t) dt --> F(s)/s
4.3