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Advanced Mathematics for Computer Science - Complete Reference

Advanced Mathematics for Computer Science - Complete Reference












Calculus
Multivariable Calculus
Differential Equations
Linear Algebra
Discrete Mathematics
Probability & Statistics
Numerical Analysis
Complex Analysis
Graph Theory
Number Theory
Algorithm Analysis
Cryptography
Optimization





Advanced Mathematics for Computer Science

Complete Reference: Calculus • Multivariable Calculus • Linear Algebra • Discrete Math • Graph Theory • Algorithms • Cryptography





Calculus

Multivariable Calculus

Differential Equations

Linear Algebra

Discrete Mathematics

Probability & Statistics

Numerical Analysis

Complex Analysis

Graph Theory

Number Theory

Algorithm Analysis

Cryptography

Optimization





Calculus



Limits



Definition of a Limit


$$\lim_{x \to a} f(x) = L$$
means: For every $\varepsilon > 0$, there exists $\delta > 0$ such that
$$0 < |x - a| < \delta \implies |f(x) - L| < \varepsilon$$






Important Limits


$$\lim_{x \to 0} \frac{\sin x}{x} = 1$$
$$\lim_{x \to 0} \frac{1 - \cos x}{x} = 0$$
$$\lim_{x \to 0} \frac{e^x - 1}{x} = 1$$
$$\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x = e$$






L'Hôpital's Rule


If $\lim_{x \to a} f(x) = \lim_{x \to a} g(x) = 0$ or $\pm\infty$, then:
$$\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$$
(provided the right side exists)










Derivatives



Definition


$$f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$$
Alternative notation: $\frac{df}{dx}$, $\frac{d}{dx}f(x)$, $D_x f$






Derivative Rules


Power Rule: $\frac{d}{dx}(x^n) = nx^{n-1}$

Product Rule: $(fg)' = f'g + fg'$

Quotient Rule: $\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}$

Chain Rule: $\frac{d}{dx}f(g(x)) = f'(g(x)) \cdot g'(x)$






Common Derivatives


$$\frac{d}{dx}(\sin x) = \cos x \qquad \frac{d}{dx}(\cos x) = -\sin x$$
$$\frac{d}{dx}(e^x) = e^x \qquad \frac{d}{dx}(\ln x) = \frac{1}{x}$$
$$\frac{d}{dx}(\tan x) = \sec^2 x \qquad \frac{d}{dx}(\arctan x) = \frac{1}{1+x^2}$$








Mean Value Theorem


If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then there exists $c \in (a,b)$ such that:
$$f'(c) = \frac{f(b) - f(a)}{b - a}$$








Integrals



Definite Integral


$$\int_a^b f(x)\,dx = \lim_{n \to \infty} \sum_{i=1}^n f(x_i^*) \Delt…