Engineering Mathematics — Complete Revision Notes

Matrix Rank, Systems & Eigenvalues

System of Linear Equations $AX = B$:

• $\text{Rank}(A) = \text{Rank}(A|B) = n$ (number of variables) $\implies$ Unique Solution.
• $\text{Rank}(A) = \text{Rank}(A|B) = r < n \implies$ Infinitely Many Solutions with $(n - r)$ linearly independent free parameters.
• $\text{Rank}(A) \neq \text{Rank}(A|B) \implies$ No Solution (Inconsistent system).

Eigenvalues & Eigenvectors ($|A - \lambda I| = 0$):

• $\sum \lambda_i = \text{Trace}(A) = \sum a_{ii}$
• $\prod \lambda_i = \det(A)$
• Eigenvalues of Symmetric Matrix ($A = A^T$) are Always Real.
• Eigenvalues of Skew-Symmetric Matrix ($A = -A^T$) are Purely Imaginary or Zero.
• Eigenvalues of Orthogonal Matrix ($A^T A = I$) have unit absolute magnitude: $|\lambda_i| = 1$.

Limits, Maxima-Minima & Vector Theorems

• L'Hôpital's Rule: For $\frac{0}{0}$ or $\frac{\infty}{\infty}$, $\lim_{x\to a}\frac{f(x)}{g(x)} = \lim_{x\to a}\frac{f'(x)}{g'(x)}$.
• Taylor Series Expansion around $x = a$: $f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n$.
• Vector Theorems:
  • Gauss Divergence: $\iint_S \vec{F}\cdot d\vec{S} = \iiint_V (\nabla \cdot \vec{F}) dV$
  • Stokes' Theorem: $\oint_C \vec{F}\cdot d\vec{r} = \iint_S (\nabla \times \vec{F})\cdot d\vec{S}$
  • Green's Theorem in Plane: $\oint_C (M dx + N dy) = \iint_R \left(\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y}\right) dx dy$

Euler-Cauchy & 1st Order Linear ODEs

Euler-Cauchy Equation: $x^2 \frac{d^2 y}{dx^2} + p x \frac{dy}{dx} + q y = 0$. Substitute $x = e^z \iff z = \ln x$:

$$m(m-1) + p m + q = 0 \iff m^2 + (p-1)m + q = 0$$

• Real distinct roots $m_1, m_2 \implies y = C_1 x^{m_1} + C_2 x^{m_2}$
• Equal roots $m_1 = m_2 = m \implies y = (C_1 + C_2 \ln x) x^m$
• Complex roots $\alpha \pm j\beta \implies y = x^\alpha [C_1 \cos(\beta \ln x) + C_2 \sin(\beta \ln x)]$

Distributions, Correlation & Regression

• Normal Distribution: $P(\mu - \sigma \le X \le \mu + \sigma) \approx 68.27\%$, $P(\mu - 2\sigma \le X \le \mu + 2\sigma) \approx 95.45\%$.
• Poisson Distribution: $P(X=k) = \frac{e^{-\lambda}\lambda^k}{k!}$, $\text{Mean} = \text{Variance} = \lambda$.
• Exponential Distribution: $f(x) = \lambda e^{-\lambda x}$ ($x \ge 0$), Mean $= \frac{1}{\lambda}$, Variance $= \frac{1}{\lambda^2}$. Memoryless property: $P(X > s+t | X > s) = P(X > t)$.
• Correlation Coefficient: $r = \frac{\text{Cov}(X,Y)}{\sigma_X \sigma_Y} = \pm\sqrt{b_{yx} \cdot b_{xy}}$, where $b_{yx}$ is regression slope of $Y$ on $X$.