Fundamentals
The core facts every aspirant should own — each a titled nugget with a real-world story, the concept in plain words, and a memory trick. Works even when the internet doesn't.
300 fundamentals
“Acceleration is the second derivative of position.”
The derivative of the derivative; it measures how the slope itself changes (concavity).
d²y/dx²
Memory trick: Rate of change of the slope.
“A ball leaving a curved ramp flies off along the tangent.”
The tangent has slope f'(x); the normal is perpendicular to it.
Memory trick: Normal slope = −1/f'.
“Engineers estimate small errors quickly using differentials.”
For a small change, Δy ≈ f'(x)·Δx.
Memory trick: Small change ≈ slope × step.
“Logs turn a tangled product into an easy sum before differentiating.”
Take logs first to differentiate products, quotients or variable powers like x^x.
Memory trick: Log it, then differentiate.
“It handles curves traced out over time, like a projectile's path.”
For x=f(t), y=g(t), dy/dx = (dy/dt)/(dx/dt).
Memory trick: Divide the t-derivatives.
“Integration was born from adding up infinitely many slivers.”
The definite integral is the limit of Riemann sums of thin rectangles.
Memory trick: Sum of infinite strips.
“Symmetry can kill an integral before you compute a thing.”
Over [−a,a], an odd function integrates to 0; an even one to twice the half-integral.
Memory trick: Odd over symmetric limits → 0.
“Break one hard fraction into a few easy ones.”
Split a rational function into simpler fractions to integrate it.
Memory trick: Decompose, then integrate.
“The integrating factor is the magic multiplier that makes it solvable.”
dy/dx + Py = Q is solved using an integrating factor e^∫P dx.
Memory trick: Multiply by e^∫P dx.
“Clean it up first, then read off the degree.”
The power of the highest-order derivative after removing radicals and fractions.
Memory trick: Clear radicals first.
“A zero box-volume means they lie flat in one plane.”
Three vectors are coplanar when their scalar triple product is zero.
Memory trick: [a b c] = 0 → coplanar.
“Two aircraft on skew paths miss by this closest gap.”
The length of the common perpendicular between two non-intersecting, non-parallel lines.
Memory trick: Along the common perpendicular.
“Give a wall one point and its perpendicular, and the wall is determined.”
A plane is fixed by a point on it and a normal vector.
Memory trick: Point + normal defines it.
“Sunlight hitting a slope makes this very angle.”
Found from the line's direction and the plane's normal.
Memory trick: Use direction vs normal.
“Rotations are orthogonal — they turn shapes without distorting them.”
A matrix whose transpose is its inverse (AAᵀ = I); it preserves lengths.
Memory trick: Aᵀ = A⁻¹.
“It collapses space, so it can't be undone.”
A square matrix with zero determinant and no inverse.
Memory trick: det = 0 → singular.
“A few row rules can simplify a determinant enormously.”
Swapping two rows changes the sign; a repeated row makes the determinant zero.
Memory trick: Repeated row → zero.
“Determinants reveal whether equations agree, clash, or overlap.”
A system is consistent if it has at least one solution, judged by determinants/rank.
Memory trick: Check via determinant/rank.
“Add up every path that leads to the outcome.”
The probability of an event is the sum over all ways it can occur.
Memory trick: Sum over all cases.
“The number rolled on a die is a random variable.”
A variable whose value depends on the outcome of a random experiment.
Memory trick: Maps outcomes to numbers.
“Two bets with equal expected value can differ wildly in risk (variance).”
Measures the spread of a random variable around its mean.
Memory trick: Spread around the mean.
“Sometimes 'at least one' is easiest as 1 minus 'none'.”
P(not A) = 1 − P(A).
P(A')=1−P(A)
Memory trick: Subtract from 1.
“It fairly compares consistency between things measured in different units.”
Standard deviation as a percentage of the mean, for comparing spread across data sets.
Memory trick: SD relative to mean.
“Height and weight are positively correlated — taller tends to be heavier.”
Measures how two variables move together, from −1 to +1.
Memory trick: +1 together, −1 opposite.