What is it?
Connected pieces of the real line: , , … Closed and bounded intervals are where the big theorems of calculus (Bolzano, Weierstrass) hold.
Formulas
Where it shows up in computing
Interval arithmetic computes with enclosures to get guaranteed error bounds.
Where is it used?
Computing topics reachable from here, through the chain of ideas that leads to them:
λ Scientific computing and algorithms
- Domain and range→Inverse functions→Logarithmic functions→Algorithm analysis and complexity★★★★★
- Inequalities→Convergence of sequences→Order of convergence→Scientific computing★★★★★
- Inequalities→Limit of a function→Derivative→Differentiation rules→Symbolic computation (CAS)★★★★★
- Inequalities→Limit of a function→Riemann sums→Definite integral→Monte Carlo methods★★★★★
- Domain and range→Inverse functions→Logarithmic functions→Floating point (IEEE 754)★★★★★
ℒ AI and machine learning
- Inequalities→Convexity and concavity→Loss function★★★★★
- Inequalities→Convexity and concavity→Support vector machines★★★★★
- Inequalities→Limit of a function→Derivative→Gradient descent★★★★★
- Inequalities→Convexity and concavity→Loss function→Logistic regression★★★★★
- Inequalities→Limit of a function→Derivative→Automatic differentiation★★★★★
- Inequalities→Convexity and concavity→Loss function→Regularization★★★★★
- +12
3D Computer graphics
- Inequalities→Limit of a function→Riemann sums→Definite integral→The rendering equation★★★★★
- Inequalities→Limit of a function→Continuity→Bézier curves and splines★★★★★
- Inequalities→Limit of a function→Continuity→Interpolation→Lighting and shading★★★★★
- Bolzano's theorem→Bisection method→Ray tracing★★★★★
- Bolzano's theorem→Bisection method→Ray tracing→Signed distance fields and ray marching★★★★★
⚙ Robotics and control
- Inequalities→Limit of a function→Derivative→Kinematics: position, velocity, acceleration★★★★★
- Inequalities→Limit of a function→Derivative→Kinematics: position, velocity, acceleration→Robot Jacobian (velocity kinematics)★★★★★
- Inequalities→Limit of a function→Derivative→Kinematics: position, velocity, acceleration→Robot dynamics★★★★★
- Inequalities→Limit of a function→Derivative→Ordinary differential equations→Control theory★★★★★
- Inequalities→Limit of a function→Derivative→PID control★★★★★
- Inequalities→Convergence of sequences→Newton's method→Inverse kinematics★★★★★
- +1
⚛ Physics and simulation
- Inequalities→Limit of a function→Derivative→Physics engines★★★★★
- Inequalities→Limit of a function→Derivative→Ordinary differential equations→Classical mechanics★★★★★
- Inequalities→Limit of a function→Derivative→Ordinary differential equations→Population and epidemic models★★★★★
- Inequalities→Limit of a function→Derivative→Physics engines→N-body gravitational simulation★★★★★
- Inequalities→Limit of a function→Derivative→Numerical differentiation→Heat equation and diffusion★★★★★
- Inequalities→Limit of a function→Derivative→Physics engines→Fluid dynamics and CFD★★★★★
- +1
∿ Signals, media and vision
- Inequalities→Convergence of sequences→Numerical series→Fourier series→Signal processing★★★★★
- Inequalities→Limit of a function→Continuity→Interpolation→Image processing and computer vision★★★★★
- Inequalities→Convergence of sequences→Numerical series→Fourier series→Media compression (JPEG, MP3, video)★★★★★
What depends on it
This page has the essentials. A fuller treatment (intuition, formal definition, worked example) is on the way.