⚡ Cheatsheet: Engineering Mathematics

Discrete Mathematics

Topic	Formula
Logic	$p \to q \equiv \neg p \lor q \equiv \neg q \to \neg p$
Sets	$
Graphs	$\sum deg(v) = 2 \times
Planar	$V - E + R = 2$
Functions	Injectives: $P(n,m)$ ($n \ge m$). Surjectives: $n^m - \binom{n}{1}(n-1)^m...$
Poset	Lattices must have unique LUB and GLB for every pair.

Rank: Number of linearly independent rows/cols. $\rho(A) = \rho(A^T)$.
Determinant: Product of eigenvalues. Triangular matrix det = Product of diagonal.
Eigenvalues:
$\sum \lambda_i = Trace(A)$.
$\prod \lambda_i = det(A)$.
Eigenvalues of $A^k$ are $\lambda^k$.
Eigenvalues of $A^{-1}$ are $1/\lambda$.
Cayley-Hamilton: Every square matrix satisfies its own characteristic equation. $A^n + c_{n-1}A^{n-1} ... + c_0 I = 0$.

Limits: L'Hopital's Rule (if $0/0$ or $\infty/\infty$, differentiate Num/Denom).
Mean Value: $\exists c \in (a,b)$ s.t. $f'(c) = \frac{f(b)-f(a)}{b-a}$.
Maxima/Minima: $f'(x)=0$. If $f''(x) > 0$ (Min), if $f''(x) < 0$ (Max).
Taylor Series: $f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + ...$

Bayes: $P(A|B) = \frac{P(B|A)P(A)}{P(B)}$.
Binomial: $P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}$. (Mean $np$, Var $npq$).
Poisson: $P(X=k) = \frac{e^{-\lambda} \lambda^k}{k!}$. (Mean $\lambda$, Var $\lambda$).
Normal: PDF $\frac{1}{\sqrt{2\pi}\sigma} e^{-\frac{1}{2}(\frac{x-\mu}{\sigma})^2}$.
Exponential: $f(x) = \lambda e^{-\lambda x}$ ($x \ge 0$). (Mean $1/\lambda$).

Topic	Formula
Logic	\(p \to q \equiv \neg p \lor q \equiv \neg q \to \neg p\)
Sets	$
Graphs	$\sum deg(v) = 2 \times
Planar	\(V - E + R = 2\)
Functions	Injectives: \(P(n,m)\) (\(n \ge m\)). Surjectives: \(n^m - \binom{n}{1}(n-1)^m...\)
Poset	Lattices must have unique LUB and GLB for every pair.