How to read this course
Quadrature teaches physics in two registers at once — algebra-based and calculus-based — and lets a computer-algebra system prove, for every scenario, that the two agree. You are never told they agree; the proof is shown. This page is the short version of how the pieces fit, and how to practice.
The one idea
Stack a position–time graph over a velocity–time graph. The slope of the upper curve is the value of the lower one; the area under the lower is thechange in the upper. Slope is the derivative; area is the integral. That single pivot — slope↔value, area↔change — is the bridge between the two registers, and it recurs in every domain: position and velocity, charge and current , flux and induced EMF , pressure and work .
Three regimes — when algebra is enough, and when you need calculus
The relationship between the two registers has three honest regimes. Each lesson is tagged with its regime, so you always know which kind of problem you are looking at.
Algebra is calculus, evaluated
When the acceleration (or force) is constant, the calculus integrals can be done once and frozen into formulas. The constant-acceleration kinematics rules are exactly with the integrand held constant — a quadrature, an integral already evaluated. Here algebra and calculus give the same answer, and the lesson proves it.
Example: A ball thrown straight up →Calculus does what algebra cannot
When the force varies — drag, a spring, gravity over distance, a charging capacitor, a ringing circuit — there is no constant to freeze. You must set up a derivative or an integral or a differential equation and solve it. The algebra course hands you the answer as a rule; the calculus shows where it comes from. This is the half where calculus stops being optional.
Example: Charging a capacitor (RC) →Algebra-only — with a calculus underpinning where it is honest
Some topics (fluids, thermodynamics, optics, waves) have no calculus-based counterpart at this level. We do not force one. But where a clean calculus idea sits underneath — work as the area under a pressure–volume curve, — we surface it on the same area instrument, and where it does not, we say so.
Example: Work by an expanding gas →How to read a lesson
Each lesson is one verified scenario with several reconciled views:
- Scenario — the question, in words.
- Algebra and Calculus tabs — the same problem stepped through in each register. Watch the memorized algebra formula emerge from the calculus.
- Graph — an interactive instrument; drag the sliders and the curves, areas, and results move together, evaluated from a closed form in your browser (no server).
- SymPy proof — the shown, not asserted, machine check that the registers agree (or that the closed form solves the governing equation). Every tick is a verified identity.
- Common misconception — a documented wrong belief, refuted by the math and the graph rather than by assertion.
- Modeling assumptions — the author-asserted simplifications (e.g. g = −10, no air resistance), disclosed, never hidden inside the derivation.
How the practice works — solve it three ways
Where a lesson has a Practice tab, each question can be answered three ways — the same problem, your choice of register:
- Just the answer — multiple choice. Pick a value; the correct one is revealed. The wrong options are not random numbers — each is the answer you would get from a specific, named mistake (a dropped factor of ½, a sign error, the wrong formula), and choosing it tells you which.
- Algebra step-through — the memorized-formula route, one step at a time.
- Calculus step-through — the derivative/integral route, one step at a time.
This is practice as understanding, not a graded quiz: nothing is scored, nothing is stored, no sign-in. And like everything else here, every answer and every wrong option is computed and checked by SymPy when the site is built — a wrong option is even proven to differ from the right one.
The breadth of formulas lives in the reference and its concept graph; how the whole thing is verified is on the verification page.