Engineering maths notes

  • Matrices
  • Inverse of a matrix

inverse of a matrix

Where Adj(A) = adjoint A which is transpose matrix of cofactors of the original matrix.

adjoint of a matrix

Inverse of a diagonal matrix

inverse of diagonal matrix

  • Adjoint of a matrix

Let A be a nxn matrix, then

adjoint of matrix

  • Trace of a matrix

Let M be a n x n matrix, then

trace of a matrix

Where λ1, λ2,…,λn are eigen values of matrix M

  • Matrices of co-factors

matrices of cofactors

B is a matrix of cofactors of A; then

matrices of cofactors

  • Area of triangle determinant format

Three points A (a1, b1, c1), B (a2, b2, c2), C (a3, b3, c3). Area of triangle ABC

area of a triangle by determinant

  • Differential equation
  • Integrating factor solution for differential equation

integrating factor solution for differential equation

  • For roots ∝± iβ of differential equation, solution

complex root solution for differential equation

  • Odd function

For t < 0, e-|t| is an odd function

  • Calculating curl of a vector

Curl of a vector is defined as follows

curl of a vector

  • Taylor series

Taylor series: an infinite sum giving the value of a function f(z) in the neighbourhood of a point “a” in terms of the derivatives of the function evaluated at “a”.

curl of a vector

  • Ratio test

Ratio test for finding out whether a given series is convergent or divergent

ratio test for convergence of a series

  • Some hyperbolic functions

Following hyperbolic functions are defined

sinhx, coshx, limit of tanx, limit of tanhx

hyperbolic functions

  • Eigen values

Characteristic equation for Eigen value is expanded form of (A-λI) = 0. For finding Eigen values

eigen values of a matrix

If a matrix has eigen vector X then AX = KX; eigen vector should be a non-zero vector.

  • Average value of f(x)

average value of a function

  • Finding saddle point

saddle point of a function

Where fx is derivative of function f w.r.t. x, fxx is double derivative of function f w.r.t. x. Same is for y.

Function is harmonic if Fxx + Fyy = 0

  • Summation of tan-1A and tan-1B

tan-1A + tan-1B

tan inverse function

  • Calculating gradient of a function

To find unit vector normal to surface calculate gradient (grad ɸ)

gradient of a function

  • Divergence, curl and gradient of a vector

Divergence of curl of vector a

divergence curl and gradient of a vector

  • Cauchy Riemann equation

The Cauchy–Riemann equations on a pair of real-valued functions of two real variables u(x, y) and v(x, y) are the two equations. Then f = u + iv is complex-differentiable at that point if and only if the partial derivatives of u and v satisfy the Cauchy–Riemann equations

cauchy riemann equation

  • Laplace-Transformation
  • Laplace transformation of function f(t)

laplace transformation of a function

  • Periodic function

Laplace transfer function of periodic function with period T

laplace transformation of a periodic function

  • Improper integral by laplace transformation

Finding the value of improper integral by Laplace transformation

improper integral by laplace transformation

  • Complex equation
  • Polar form of complex equation

Polar form of complex equation x + iy

polar form of complex equation

  • Definite integral
  • For even function

definite integral for even function

  • Integral of inverse of 1+x2 

integral of inverse of 1+x

  • Breaking limits of definite integral

breaking limits of definite integral

  • Green’s Theorem

greens theoram

  • Fourier series
  • Infinite fourier series

For fourier seriesthe solution is infinite Fourier series

infinite fourier series

  • Probability density function

probability density function is defined as probability density function

Mean and variance of probability density function is defined as

mean and variance of probability density function

  • Random variable probability function (P(X))

random variable probability density function
random variable probability density function

Where: X = random variable.

In continuous random variable total area under curve = 1.

  • Normal distribution

Normal distribution is symmetrical about mean (µ)

normal distribution

  • Cauchy residual theorem/ Cauchy integral formula

cauchy integral formula

  • Residue of a function f(z)

residue of a function

  • Newton Raphson method

newton raphson method

Newton – Raphson method will converge in one step if the function is linear

  • Double integral

double integral

  • Euler’s method

euler method

Step size h is mentioned in the question.

  • Numerical integration
  • Trapezoidal rule

numerical integration trapezoidal rule

  • Simpson’s 1/3th rule

 numerical integration simpson's one third rule

  • Simpson’s 3/8th rule

numerical integration simpson three eight's rule

  • Poisson’s distribution

poisson distribution

  • i raise to power i

i raise to the power i

  • Probability
  • Summation n

probability venn diagram summation n

  • Probability A union B

probability A union B

  • Probability A intersection B

probability A intersection B

  • Del of f del of a function

del of a function

  • probability A given B conditional probability A given B

conditional probability A given B

  • For infinitely many solutions of set of equations

For infinitely many solutions of set of equations, rank of A, R(A) should be such that

condition for infinitely many solutions of set of equations

  • Summation involving both AP and GP

summation involving both AP and GP

  • Taylor series expansion of ez

taylor series expansion of e

  • Linearly independent set of equations

For a set of functions f(x), g(x), h(x) to be linearly independent w≠0

linearly independent set of equations

  • Residual of a function at x = a

To find residual of a function at a particular point

residual of a function

  • Determinant

For matrix A

determinant and skew matrix

If A=AT=>A is a skew matrix

  • Cumulative distribution function

Cumulative distribution function of random variable X

cumilative distribution function

  • Idempotent matrix

idempotent matrix

If AB is idempotent matrix then

idempotent matrix

  • Regula-Falsi method

Regula – Falsi method for convergence and divergence

regula falsi method

  • Covariance

covariance

  • Ramp function

ramp function

  • analytic function

Let f(z) be defined as

analytic function

For f(z) to be analytic

conditions for function to be analytic

If f(z) is analytic

analytic function

  • Types of errors
  • Integral square error (ISE)

integral square error

  • Integral absolute error (IAE)

integral square error

  • Integral time weighted absolute error (ITAE)

integral time weighted absolute error

  • Directional derivative

The directional derivative ∇uf(xo, yo, zo) is the rate at which the function f(x, y, z) changes at point (xo, yo, zo)in the direction u. Directional derivative along the path of maximum value is called gradient.

directional derivative

  • Exact differential

Mdx + Ndy is exact differential when

exact differential solution for differential equation

  • Initial value problem

The differential equationinitial value problem, with conditions y(0) = 0, and y(1) = 1 is called initial value problem.

  • Linear differential equation

A first order differential equation is said to be linear if it can be written as

linear differential equation

  • Sum of squares of errors

Sum of squares of error is defined as

sum of squares of error

Where yi is the data value that would be calculated from known correlations and Yi ist the data point that would be mentioned. It is used to find a value of a variable from an equation on basis of various data points.

  • Few random mathematical formulas
    1. cos(A-B)
    2. limit n tending to 0 1/n
    3. differential of a to the power g(x)
    4. integral e to the power -u square
    5. complex form of Z square
    6. sin 90+theta
    7. mean and standard deviation of distributed set
    8. i to the power i
  • Mathematics
  • Eigen values of symmetric matrix are real.

  • Transpose of a square matrix A have same eigen values as that of A.

  • Gradient of a scalar quantity is always vector.

  • If A and B are 3×3 matrix and AB = 0 then rank of matrix B is zero.

  • Three sets of equation will be independent and have a unique solution only if they have a non-vanishing determinant.

  • For solution of differential equation to be y = (C1 + C2x)emx, roots of characteristics equation needs to be equal.

  • A set of equations will have non-trivial solutions if det[A] = 0.

  • A is mxn matrix with rank n; B is nxp matrix with rank p. Assuming m ≥ n ≥ p, rank (AB) ≤ min (n, p) = p; Rank of a matrix is equal to number of linear independent rows.

  • Trapezoidal rule will give exact integral when f(x) is linear.

  • For vectors to be coplanar, determinant should be zero