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Inequalities

Level FundamentalDifficulty ★★★★★Concept⌖ Open in the map

What is it?

Calculus is mostly the art of bounding: proving that an error is smaller than $\varepsilon$ , that a term is negligible, that an algorithm takes at most so many steps. The tools are the triangle inequality, AM–GM, Cauchy–Schwarz and Jensen.

Formulas

$\frac{a + b}{2} \ge \sqrt{ab} \quad (a, b \ge 0)$
AM–GM: $|x \cdot y| \le \norm{x}\,\norm{y}$
Cauchy–Schwarz

Where it shows up in computing

Algorithm analysis and complexity★★★★★frequentScientific computing and algorithms
Upper and lower bounds on running time are proved by chains of inequalities.

Where is it used?

Computing topics reachable from here, through the chain of ideas that leads to them:

λ Scientific computing and algorithms

Limit of a function→Limits at infinity→Algorithm analysis and complexity★★★★★
Convergence of sequences→Order of convergence→Scientific computing★★★★★
Limit of a function→Derivative→Differentiation rules→Symbolic computation (CAS)★★★★★
Limit of a function→Riemann sums→Definite integral→Monte Carlo methods★★★★★
Convergence of sequences→Newton's method→Floating point (IEEE 754)★★★★★

ℒ AI and machine learning

Convexity and concavity→Loss function★★★★★
Convexity and concavity→Support vector machines★★★★★
Limit of a function→Derivative→Gradient descent★★★★★
Convexity and concavity→Loss function→Logistic regression★★★★★
Limit of a function→Derivative→Automatic differentiation★★★★★
Convexity and concavity→Loss function→Regularization★★★★★
+15

3D Computer graphics

Limit of a function→Riemann sums→Definite integral→The rendering equation★★★★★
Limit of a function→Derivative→Partial derivatives→Gradient→Surface normals★★★★★
Limit of a function→Derivative→Partial derivatives→Gradient→Signed distance fields and ray marching★★★★★
Limit of a function→Derivative→Higher-order derivatives→Curvature (basic differential geometry)→Mesh processing (discrete differential geometry)★★★★★
Limit of a function→Continuity→Bézier curves and splines★★★★★
Limit of a function→Continuity→Interpolation→Lighting and shading★★★★★
+1

⚙ Robotics and control

Limit of a function→Derivative→Kinematics: position, velocity, acceleration★★★★★
Limit of a function→Derivative→Kinematics: position, velocity, acceleration→Robot Jacobian (velocity kinematics)★★★★★
Limit of a function→Derivative→Kinematics: position, velocity, acceleration→Robot dynamics★★★★★
Limit of a function→Derivative→Ordinary differential equations→Control theory★★★★★
Limit of a function→Derivative→Kinematics: position, velocity, acceleration→Robot Jacobian (velocity kinematics)→Inverse kinematics★★★★★
Limit of a function→Derivative→Kinematics: position, velocity, acceleration→Robot dynamics→Trajectory optimization and MPC★★★★★
+2

⚛ Physics and simulation

Limit of a function→Derivative→Physics engines★★★★★
Limit of a function→Derivative→Ordinary differential equations→Classical mechanics★★★★★
Limit of a function→Derivative→Ordinary differential equations→Population and epidemic models★★★★★
Limit of a function→Derivative→Physics engines→N-body gravitational simulation★★★★★
Limit of a function→Derivative→Numerical differentiation→Heat equation and diffusion★★★★★
Limit of a function→Derivative→Physics engines→Fluid dynamics and CFD★★★★★
+4

∿ Signals, media and vision

Convergence of sequences→Numerical series→Fourier series→Signal processing★★★★★
Limit of a function→Continuity→Interpolation→Image processing and computer vision★★★★★
Convergence of sequences→Numerical series→Fourier series→Fourier transform→Sampling theorem (Nyquist–Shannon)★★★★★
Convergence of sequences→Numerical series→Fourier series→Fourier transform→Fast Fourier transform (FFT)★★★★★
Limit of a function→Riemann sums→Definite integral→Convolution→Digital filters★★★★★
Convergence of sequences→Numerical series→Fourier series→Signal processing→Media compression (JPEG, MP3, video)★★★★★
+1

⇄ Optimization and systems

Convexity and concavity→Operations research and logistics★★★★★
Limit of a function→Derivative→Gradient descent→Recommender systems★★★★★
Convergence of sequences→Numerical series→Geometric series→Queueing theory and performance★★★★★

⊕ Cryptography and security

Convergence of sequences→Numerical series→Fourier series→Signal processing→Side-channel analysis★★★★★
Limit of a function→Derivative→Chain rule→Implicit differentiation→Elliptic curve cryptography★★★★★

ψ Quantum computing and physics

Convergence of sequences→Numerical series→Fourier series→Fourier transform→Uncertainty principle★★★★★
Convergence of sequences→Numerical series→Power series→Taylor and Maclaurin series→Quantum computing★★★★★

What depends on it

This page has the essentials. A fuller treatment (intuition, formal definition, worked example) is on the way.