Simulation · Calculus × Computing

01
Derivative
$f'(a)$ is the instantaneous rate of change of $f$ at $a$ : the slope of the tangent line to the graph, defined as the limit of slopes of secant lines.
FundamentalDerivatives
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02
Definite integral
$\int_a^b f(x)\,\dd x$ is the limit of Riemann sums: the signed area between the graph and the axis, or, more generally, the total accumulated by a rate $f$ over $[a, b]$ .
FundamentalIntegrals
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03
Ordinary differential equations
An equation relating an unknown function to its derivatives, $y' = f(t, y)$ . Its solutions are trajectories: given where you start, the equation says where you go next — the mathematical model of anything that evolves in time.
UniversityDifferential equations
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04
Euler's method
The simplest numerical ODE solver: follow the tangent for a small time step, $y_{k+1} = y_k + h\,f(t_k, y_k)$ . First order (error $O(h)$ ), easy to destabilize — and, in its semi-implicit form, the default integrator of game physics.
UniversityDifferential equations
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05
Runge–Kutta methods
Sample the slope at several points inside the step and combine them to cancel error terms. The classic RK4 has error $O(h^4)$ : halve the step, divide the error by 16. Adaptive pairs (Dormand–Prince, ode45, solve_ivp) adjust $h$ automatically.
AdvancedDifferential equations
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06
Systems of ODEs
Several coupled unknowns, $\mathbf x' = F(\mathbf x)$ . For linear systems $\mathbf x' = A\mathbf x$ the solution is $e^{At}\mathbf x_0$ and the eigenvalues of $A$ decide growth, decay and oscillation.
UniversityDifferential equations
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07
Dynamical systems
A state and a rule that moves it forward: continuous ( $\dot x = f(x)$ ) or discrete ( $x_{k+1} = g(x_k)$ ). The questions change from "find the formula" to "what happens in the long run?" — and every iterative algorithm, from gradient descent to a recurrent network, is a discrete dynamical system.
AdvancedDynamical systems and chaos
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08
Equilibria and stability
An equilibrium $f(x^\ast) = 0$ is stable if nearby states stay nearby (and asymptotically stable if they return). Linearize: if every eigenvalue of the Jacobian $J_f(x^\ast)$ has negative real part, it is asymptotically stable.
AdvancedDynamical systems and chaos
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09
Stiffness and implicit methods
Systems with very different time scales (a stiff spring on a slow body) force explicit methods to take tiny steps just to stay stable. Implicit methods, $y_{k+1} = y_k + h f(y_{k+1})$ , are stable for any step at the price of solving an equation each step.
AdvancedDifferential equations
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10
Chaos and sensitivity to initial conditions
Deterministic systems whose nearby trajectories separate exponentially, $|\delta(t)| \approx |\delta_0|e^{\lambda t}$ with $\lambda > 0$ . Prediction is possible only up to a horizon of about $\frac1\lambda\ln\frac{\text{tolerance}}{|\delta_0|}$ : better measurements buy time only logarithmically.
SpecializationDynamical systems and chaos
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11
Physics engines
Software that advances a world of bodies in small time steps: accumulate forces, integrate velocities and positions (semi-implicit Euler or Verlet), detect collisions, resolve contacts. A numerical ODE solver tuned for speed and stability rather than accuracy.
UniversityPhysics and simulation
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