Where each idea is used on this site
Volume II is the on-ramp the rest of the site quietly assumes: the vector calculus, matrix derivatives and differential equations that the AI, vision and robotics guides use without stopping to derive. Each row below names one idea, links to the part that builds it, and points at the concrete page whose argument leans on it.
How to use this card
Filter, then follow the link
This is a routing table, not a lesson — nothing here is proved. The first table takes an idea from this volume and names the page elsewhere that consumes it. The second collects the operators the volume introduces, so a symbol you meet in a solver or a transformer paper resolves to the part that derived it. Type into the box to filter every row of both tables at once, case-insensitively; clear it to see everything again.
No row matches that — try a looser word, or clear the box.
Where each idea is used
From the part that derives it to the page that consumes it
| Idea | Volume II part | Where it is used |
|---|---|---|
| Vector fields & streamlines | Part 1 · Vector fields | Slope fields for the ODEs in Differential equations, and the velocity fields a controller follows. |
| Line integrals & work | Part 2 · Line integrals & conservative fields | The effort a path costs a robot — joint torques integrated along a trajectory. |
| Divergence & curl | Part 3 · Divergence and curl | Incompressible and potential flow, and the rotation/curvature diagnostics that shape a loss landscape in Nonlinear Optimization. |
| Green, Stokes & Divergence | Part 4 · Green, Stokes, Divergence: one theorem | The change-of-variables argument in Part 5 and the swept region integrals of Part 6. |
| Jacobian determinant | Part 5 · The Jacobian determinant | Reprojection and bundle adjustment in Multi-View Geometry, and the density scaling of Part 11. |
| Multiple integrals | Part 6 · Multiple integrals & coordinates | Normalising and marginalising probability densities over a region — the machinery behind Calculus of probability. |
| Lagrange multipliers & KKT | Part 7 · Lagrange multipliers & KKT | Every constrained fit — pose estimation with equality constraints, trust-region limits — in Nonlinear Optimization. |
| Matrix calculus | Part 8 · Matrix calculus | The $J^{\top}J$ normal equations and Gauss–Newton step in Nonlinear Optimization, and every weight gradient in LLM Training. |
| Backpropagation | Part 9 · Backpropagation is the chain rule | The training loop itself: reverse-mode differentiation through the network in LLM Training. |
| Curvature & convergence | Part 10 · Curvature and convergence | Why GD zig-zags, and where Newton, Gauss–Newton and Levenberg–Marquardt come from — Gradient Descent and Nonlinear Optimization. |
| Calculus of probability | Part 11 · Calculus of probability | The KL penalty and reparameterised sampling that steer RLHF and the rest of LLM Training. |
| Differential equations | Part 12 · Differential equations | Forward kinematics and the integrated pose updates of mobile robots. |
| Calculus on manifolds | Part 13 · Calculus on manifolds | Why rotations do not add, and the exponential-map update used throughout 3D Rotations: the math. |
| Calculus of variations & optimal control | Part 14 · Calculus of variations & optimal control | Minimum-jerk and minimum-time trajectories for robot navigation. |
The operators, collected
One symbol, one meaning, one part
| Operator | Meaning | Derived in |
|---|---|---|
∇f (grad) |
The gradient: the vector of partial derivatives, pointing uphill and normal to the level sets of $f$. | Part 1 |
∇·F (div) |
The divergence: net outward flux per unit volume — source strength, positive where a field spreads. | Part 3 |
∇×F (curl) |
The curl: the circulation per unit area — how hard a field would spin a tiny paddlewheel. | Part 3 |
J = ∂F/∂x (Jacobian) |
The first derivatives arranged as a matrix: the best linear map of a vector-valued function. | Part 8 |
H = ∂²f/∂x² (Hessian) |
The second derivatives arranged as a symmetric matrix: the local curvature of $f$. | Part 10 |
d/dt |
The derivative with respect to time — a rate. It is what turns a trajectory into a velocity. | Part 12 |
∫ |
The integral: signed accumulation, the limit of a sum of thin pieces. Along a path it is work. | Part 2 |
∂f/∂x |
A partial derivative: the slope in one coordinate direction with every other input held fixed. | Part 1 |
Where to start, given where you came from
Two common entry points
If you came here from Nonlinear Optimization, start with Part 10 and Part 8 — curvature explains the convergence races, and matrix calculus supplies the $J^{\top}J$ that Gauss–Newton is built on. If you came from LLM Training, start with Part 9: reverse-mode differentiation is the one idea that turns every other row of this card into a training step.