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Math,maxed.

Duolingo-grade practice for the university math you actually have to pass — calculus, linear algebra, differential equations, probability. Every answer is verified by a computer algebra system, never guessed.

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  • 1,538 verified problems
  • 7 courses
  • Free to start · no account · works offline
Del, the Nabla mascot
1666

Chapter I · Woolsthorpe, 1666

An apple falls.

Everyone had watched apples fall. Newton asked a different question — not where the apple is, but how fast it is moving at this exact instant. The answer needed a kind of math that did not exist yet.

t0.00 s
fallen0.00 m
speed0.0 m/s

The question

Not where.
How fast.

Every tenth of a second the apple covers more ground than the tenth before. Its position is easy to write down. Its speed at a single instant is the whole problem.

s(t)=12 g t2s(t) = \tfrac{1}{2}\,g\,t^{2}position after
t seconds

Chapter II · The derivative

Speed is a slope.

Take two moments on the apple’s height curve and draw the line through them. Its slope is the average speed between those moments. Now slide the second moment toward the first.

The limit

Δt shrinks.
The line settles.

The gap never quite reaches zero, but the slope stops changing. That settled number is the speed at one instant. Newton called it a fluxion. You call it a derivative.

v(t)=dsdt=lim⁡Δt→0s(t+Δt)−s(t)Δtv(t)=\frac{ds}{dt}=\lim_{\Delta t\to 0}\frac{s(t+\Delta t)-s(t)}{\Delta t}

A function of its own

Do it at every instant.

Walk the tangent along the curve and record its slope as you go. What you trace is a new function — velocity — and for a falling apple it is a straight line.

v(t)=− g t=−9.8 tv(t) = -\,g\,t = -9.8\,t
t0.40 s
Δt0.50 s
slope-6.37 m/s

Chapter III · The integral

Add up the
infinitely thin.

Flip the question. You know the speed at every instant — how far did the apple fall? Cut time into strips, treat the speed as constant on each, and stack their areas.

Refine

Thinner strips.
Less error.

Two strips overshoot badly. Two hundred and fifty-six barely miss. Push the count to infinity and the sum stops being an estimate and becomes the integral.

∑i=1nv ⁣(in)1n  →  n→∞    ∫019.8 t dt\sum_{i=1}^{n} v\!\left(\tfrac{i}{n}\right)\tfrac{1}{n}\;\xrightarrow{\;n\to\infty\;}\;\int_{0}^{1} 9.8\,t\,dt

The fundamental theorem

4.9 metres.
Same number.

The area under the speed curve is exactly the height we dropped from. Slicing and slope-taking undo each other — that is why derivatives and integrals are one course, not two.

∫019.8 t dt=[ 4.9 t2 ]01=4.9\int_{0}^{1} 9.8\,t\,dt = \Big[\,4.9\,t^{2}\,\Big]_{0}^{1} = 4.9
strips2
sum7.350 m
integral4.900 m
Del planting a nabla flag on the Moon

Chapter IV · Newton’s cannon

Throw it harder.

Newton’s thought experiment: fire the apple from a mountaintop. Faster, and it lands farther. Fast enough, and the ground curves away exactly as quickly as the apple falls toward it. It never lands. It is in orbit.

launch speed2.8 km/s
resultlands

Pull back

The Moon is
falling, too.

The pull that dropped the apple bends the Moon’s path around the Earth — every second, forever. One law. Written with the derivative you just learned.

F=G m1 m2r2F = G\,\frac{m_{1}\,m_{2}}{r^{2}}

Moon size to scale. Distance not — it is 60 Earth radii and the screen is not that wide.

1687 · Principia

Same fall.
Same law.

From an apple to the Moon in twenty years of mathematics. The calculus you take in first year is the tool Newton had to invent to write this down.

Your turn

You just felt
Calculus I.

Limits, derivatives, integrals, and the theorem that ties them — in four scrolls. Nabla is where you learn to do it, one verified problem at a time.

Try a lesson right here

Lunar maps · NASA/GSFC Scientific Visualization Studio

Inside the app

Tap around.
It’s the real thing.

These are the actual screens from the current build, and the lesson is live — three problems from the Calculus I bank, answers verified by SymPy. Get one wrong on purpose. See what Del does.

9:41

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Screens

Seven courses

The ones you
have to pass.

Not arithmetic dressed up. Real first- and second-year engineering math, with the opening unit of every course free — a place to start, not a demo.

0

verified problems

0

skills

0

formulas

Calculus I

Limits → derivatives → integrals → the Fundamental Theorem.

Unit 1 free
ddx sin⁡(3x)=3cos⁡(3x)\frac{d}{dx}\,\sin(3x) = 3\cos(3x)
Units
6 units
Skills
21 skills
Problems
361 problems

Calculus II

Parts, partial fractions, series, Taylor, polar.

Unit 1 free
∫u dv=uv−∫v du\int u\,dv = uv - \int v\,du
Units
4 units
Skills
13 skills
Problems
209 problems

Linear Algebra

Row reduction to eigenvalues.

Unit 1 free
Av=λ vA\mathbf{v} = \lambda\,\mathbf{v}
Units
4 units
Skills
12 skills
Problems
210 problems

Multivariable Calculus

Gradients, tangent planes, triple integrals, curl.

Unit 1 free
∇f=⟨fx, fy, fz⟩\nabla f = \left\langle f_x,\, f_y,\, f_z \right\rangle
Units
4 units
Skills
11 skills
Problems
190 problems

Differential Equations

Separable to Laplace, with stability.

Unit 1 free
y′′+ω2y=0y'' + \omega^{2} y = 0
Units
3 units
Skills
10 skills
Problems
172 problems

Probability & Statistics

Bayes to hypothesis tests and regression.

Unit 1 free
P(A∣B)=P(B∣A) P(A)P(B)P(A \mid B) = \frac{P(B \mid A)\,P(A)}{P(B)}
Units
3 units
Skills
10 skills
Problems
169 problems

Math Foundations

An opt-in refresher, never the default.

Unit 1 free
(a+b)2=a2+2ab+b2(a+b)^{2} = a^{2} + 2ab + b^{2}
Units
5 units
Skills
14 skills
Problems
227 problems

Correct, always

Verified,
not guessed.

One wrong answer and an engineering student uninstalls — rightly. So a computer algebra system (SymPy) is the source of truth for every problem and every worked solution. A language model never decides what is right; it only explains steps that were already proven.

  • Every answer re-derived a second way — finite differences for derivatives, re-differentiation for antiderivatives, numeric sampling for limits.
  • Distractors are real misconceptions — each wrong choice is a mistake students actually make, verified distinct from the answer.
  • Frozen, then shipped offline — the whole bank lives inside the app. No account, no network, no surprise.

scripts/content/generate_calc1.py → src/content/calc1.json

  1. 1Author a template

    differentiate a·sin(kx) — parameters sampled, degenerate cases rejected

  2. 2SymPy solves

    canonical answer + step-by-step derivation from a CAS, not a model

  3. 3Independent re-check

    simplify(candidate − canonical) = 0, then numeric spot checks

  4. 4Frozen to JSON

    answer, steps, difficulty, tags — deterministic, $0 at runtime

  5. 5Ships offline

    the whole bank inside the app; Del explains, never arbitrates

One frozen problem, as shippedsolving…
{
  "id": "lim-direct-1",
  "skillId": "limits-basic",
  "difficulty": "intro",
  "questionLatex": "\\lim_{x \\to 1} \\dfrac{x^{2} + 3x}{x + 2}",
  "choices": [
    { "latex": "- \\frac{4}{3}", "correct": false },
    { "latex": "4",              "correct": false },
    { "latex": "\\frac{4}{3}",   "correct": true  },
    { "latex": "3",              "correct": false }
  ],
  "solutionSteps": [
    "The denominator is nonzero here, so substitute directly.",
    "\\lim_{x \\to 1} \\frac{x^{2} + 3x}{x + 2} = \\frac{4}{3}"
  ]
}
DEL2 days away — easy fix. Start with one short set.

Click him. He follows your cursor.

Your spotter

Meet Del.

Named after ∇ — the del operator. He counts your reps, hands you the strategy before the steps, and reads your misses for the misconception behind them. Supportive, never condescending. The same character who floats on this page lives in the app, drawn from the same code.

He changes with your day — on the Home Screen widget

One clean set.
One clean set.Morning
3 ideas are fading.
3 ideas are fading.Review
Keep today alive.
Keep today alive.Late
Goal crushed.
Goal crushed.Done

Free to start · iOS and Android

Math,
maxed.

The opening unit of every course is free — a real place to start in all seven. No account. No ads. Works offline. Nabla Pro unlocks the rest when you decide it is worth it.

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