## Math

List of topics for reference

## 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
• 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
• 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