Roadmap

Adjoining a number i whose square is −1 extends the real numbers to the complex numbers a + bi, in which every polynomial equation has a solution and arithmetic still works exactly as before.

Writing a complex number by its modulus and argument instead of its real and imaginary parts turns multiplication into multiplying lengths and adding angles, and Euler's formula compresses the whole idea into re^(iθ).

A function assigns exactly one output to each input; it can be undone by an inverse function only when it is one-to-one, which is why domains often have to be restricted before an inverse exists.

Probability assigns each outcome a number between 0 and 1; the rules for combining events depend on whether they are mutually exclusive and whether they are independent, which are different questions with different tests.

A random variable attaches a number to each outcome, and its distribution says how that number is spread; expectation and variance summarise the distribution, and the normal distribution arises whenever many small effects add together.

A handful of named distributions — binomial for counting successes in a fixed number of independent trials, Poisson for counting rare events over a continuum, normal for continuous measurement error, and Student's t for small-sample estimation — cover most of the random variables statistics actually deals with, and each is fixed by only one or two parameters.

A vector in space carries both a magnitude and a direction, is written in coordinates as an ordered triple, and adds component by component — so geometry in three dimensions becomes arithmetic.

The dot product multiplies two vectors into a single number that measures how much they point the same way — it vanishes exactly when they are perpendicular, which makes it the standard test for a right angle.

The cross product of two vectors in space is a third vector perpendicular to both, whose length equals the area of the parallelogram they span and whose direction is fixed by the right-hand rule.

A line in space is fixed by a point and a direction vector, a plane by a point and a normal vector — and once both are written that way, questions about intersection, distance and angle become dot and cross products.

The limit of a function at a point is the value its outputs approach as the input closes in, whether or not the function is even defined there — which is what makes it the right tool for describing instantaneous rates.

The derivative of a function at a point is the limit of its difference quotient — the instantaneous rate of change, equal to the slope of the tangent line there.

The first derivative's sign locates where a function rises or falls and where it has local extrema; the second derivative's sign locates where the graph is convex or concave and where it has inflection points — together they turn an algebraic expression into an accurate sketch without plotting points.

Antidifferentiation reverses the derivative, recovering a family of functions that differ only by a constant — which is why every indefinite integral carries a +C.

A definite integral is the limit of a sum of thin slices, giving an accumulated total such as area or work; the fundamental theorem of calculus evaluates it using any antiderivative.

An improper integral extends the definite integral to an infinite limit of integration or an integrand that is unbounded within the interval; it has a value only when the corresponding limit of ordinary integrals actually converges, not merely because the integrand shrinks.

A matrix is a rectangular array of numbers that encodes a linear transformation; matrix multiplication composes those transformations, which is why it is not commutative.

The determinant of a square matrix is a single number giving the factor by which the transformation scales volume — zero exactly when the transformation collapses space and cannot be inverted.

A system of linear equations has exactly one solution, none, or infinitely many — never any other count — and Gaussian elimination decides which case holds while producing the solution.

A vector space is any set whose elements can be added and scaled under the usual rules; a set of vectors is linearly independent when none is a combination of the others, and an independent spanning set is a basis whose size is the dimension.

An infinite series is the limit of its partial sums; it converges only when those sums settle on a value, and terms shrinking to zero is necessary for that but nowhere near sufficient.

A partial derivative measures how a function of several variables changes when one variable moves and the rest are held fixed, so a surface has a different slope in every direction and needs several derivatives to describe it.

A double or triple integral accumulates a quantity over a region of the plane or of space by integrating one variable at a time, with the limits of the inner integral describing the shape of the region.

A differential equation relates a function to its own derivative; separable and linear first-order equations are the two types that can be solved by direct integration, giving a family of solutions narrowed by an initial condition.

A linear second-order differential equation with constant coefficients is solved by substituting a trial exponential solution, turning the differential equation into an algebraic characteristic equation whose roots determine whether the general solution is a sum of real exponentials, a single repeated exponential times a polynomial factor, or an oscillating exponential — exactly the mathematics behind damped and driven oscillators in physics.

A Fourier series represents a periodic function as an infinite sum of sines and cosines, built from the fact that these functions are mutually orthogonal over one period — the same orthogonality that gives each coefficient a closed-form integral, independent of every other term.

A point estimate is a single number computed from a sample to stand in for an unknown population parameter, but it is almost never exactly right; an interval estimate reports a range around it — built from the sample's own variability — that would contain the true parameter in a stated proportion of repeated samples, which is the honest way to communicate how much a point estimate can be trusted.

Hypothesis testing starts by assuming the null hypothesis is true, computes a test statistic measuring how far the sample departs from that assumption, and rejects the null only when that departure would be too unlikely to be explained by chance alone — it can disprove a null hypothesis but never proves one true, only fails to reject it.

A function of a complex variable is differentiable in an unusually strong sense — holomorphic — exactly when its real and imaginary parts satisfy the Cauchy–Riemann equations, and that rigidity is what makes complex analysis far more powerful than its real counterpart.

A conformal map is a function of a complex variable that preserves angles between intersecting curves at every point where its derivative is nonzero — a consequence of holomorphy, not an extra assumption — which makes such maps the tool of choice for warping an awkward geometric region into a simple one (a disk, a half-plane) while keeping the local angular structure that a physical field's boundary conditions depend on.

A Laurent series generalises a Taylor series by allowing negative powers of (z − z₀), which is exactly what's needed to represent a function around a point where it isn't holomorphic; the pattern of those negative-power terms classifies the singularity as removable (none), a pole of finite order (finitely many), or essential (infinitely many) — and only the essential case defeats every algebraic trick for evaluating a limit there.

The residue theorem says the integral of a function around a closed contour equals 2πi times the sum of its residues — a single number attached to each isolated singularity enclosed by the path — turning integrals that are hard or impossible by real-variable methods into an algebra problem of finding a handful of residues.

An unconstrained local extremum of f(x, y) occurs only where every partial derivative vanishes, and the second-derivative (Hessian) test then tells a maximum from a minimum from a saddle point; when the search is restricted to a constraint g(x, y) = c, Lagrange's method replaces that condition with ∇f = λ∇g — the geometric fact that at a constrained extremum the level curve of f is tangent to the constraint curve.

A vector field assigns a vector to every point in space; the gradient of a scalar field is the vector field pointing in the direction of steepest increase, and it is always perpendicular to the field's level surfaces.

Divergence measures how much a vector field spreads outward from a point, and curl measures how much it circulates around one — a fluid's velocity field has positive divergence at a source and non-zero curl wherever there is local rotation.

A tensor generalises vectors and matrices to an object with any number of indices that transforms in a fixed, predictable way under a change of coordinates; curvilinear systems like polar or spherical coordinates need this machinery because their basis vectors change direction from point to point, so gradient, divergence and curl each pick up extra terms that vanish in flat Cartesian coordinates.

The divergence theorem converts a volume integral of a field's divergence into a flux through the enclosing surface, and Stokes' theorem converts a surface integral of a field's curl into a circulation around the boundary curve — both are the fundamental theorem of calculus generalised to fields.

An eigenvector of a matrix is a direction the transformation only stretches, never rotates, and the stretch factor is its eigenvalue — finding them reveals the axes along which a complicated transformation acts as simply as multiplication.

Separating variables in Laplace's or the wave equation on a curved domain — a cylinder, a sphere, a quantum well — leaves behind a second-order ODE that isn't solved by sines, cosines or exponentials; each named family (Bessel for cylindrical symmetry, Legendre for spherical, Hermite for the quantum harmonic oscillator, Laguerre for the hydrogen atom's radial part) is the specific solution to one such equation, and together they form a complete orthogonal basis for expanding functions on that geometry, the same role sines and cosines play in ordinary Fourier series.

A general second-order linear PDE Au_xx + Bu_xy + Cu_yy + ⋯ = 0 is classified by the discriminant B² − 4AC in exact analogy with a conic section: negative means elliptic (like Laplace's equation, describing steady states), zero means parabolic (like the heat equation, describing diffusion), and positive means hyperbolic (like the wave equation, describing propagation) — and each class genuinely needs a different kind of boundary or initial condition to have a well-posed solution.

A Green's function G(x, ξ) is the response of a linear operator to a single point impulse at ξ, built once from the boundary conditions alone — once it exists, the solution to that same operator acting on any source term f(x) is just the integral ∫G(x, ξ)f(ξ)dξ, turning every future problem with those boundary conditions into a lookup and an integral instead of a fresh derivation.

A partial differential equation relates a function of several variables to its own partial derivatives, so a solution is a function rather than a number; the heat, wave and Laplace equations are the three archetypes worth recognising on sight, because each behaves qualitatively differently and models a different kind of physical process.

The Laplace transform rewrites a function of time as a function of a complex frequency variable s, turning derivatives into multiplication by s — so a linear differential equation with given initial conditions becomes an algebraic equation, solvable by ordinary algebra and then transformed back.

Separation of variables guesses a solution to a partial differential equation as a product of single-variable functions, which turns the PDE into several ordinary differential equations linked by a shared constant; the resulting boundary-value ODE is a Sturm-Liouville problem, whose eigenvalues give the allowed separation constants and whose eigenfunctions form a complete, orthogonal basis for building the general solution as a series.

A system of first-order differential equations can be pictured as a flow on a plane, where every point is pushed in a direction and speed set by the equations; a fixed point is where that flow stops, and linearising the system there — via the eigenvalues of the Jacobian — classifies it as a stable node, unstable node, saddle, or spiral without ever solving the full nonlinear system.