Intellect PlusCalculus and analysis
215 cards. Each one is a 40-second story with a five-minute read behind it. Page 4 of 4.
Taylor series rebuild a function from derivatives at one pointBrook Taylor introduced these expansions in 1715: an infinite sum whose coefficients are derivatives at a single center. Near that point many familiar functions match their series…
Limits tell calculus what a function is heading towardIn mathematics a limit is the value a function or sequence approaches as its input or index nears some point. That idea underwrites continuity, derivatives, and integrals across c…
Bolzano and Cauchy turned fuzzy approach into formal limitsThe limit of a function describes outputs near an input that may itself lie outside the domain. Informal talk says f(x) nears L as x nears a; early-nineteenth-century definitions …
Sequence limits ask what later terms are crowding towardThe limit of a sequence is the value its terms tend to, written with arrows or lim notation in analysis texts. Convergence means that beyond some index, every term stays arbitrari…
Analysis grew from calculus into the study of convergence itselfMathematical analysis studies functions, spaces, and operators through approximation and convergence. It grew out of calculus’s derivatives and integrals, formally blooming in sev…
Multiple integrals stack one integral sign per variableWhen a function depends on several real inputs—say f(x,y) or f(x,y,z)—its definite integral over a region is called a multiple integral, the higher-dimensional cousin of area-unde…
Multivariable calculus extends slope and area into higher dimensionsMultivariable—or multivariate—calculus extends one-variable differentiation and integration to functions of several inputs. Gradients, partials, multiple integrals, and vector fie…
Numerical analysis builds algorithms when exact answers won’t comeNumerical analysis studies algorithms for continuous-mathematics problems—real or complex variables—using approximation alongside symbolic manipulation. When closed forms fail, st…
ODEs tie an unknown function to derivatives in one variableAn ordinary differential equation links an unknown function of a single independent variable to that function’s derivatives. Newton’s second law written as mass times second deriv…
Partial derivatives freeze every input but oneA partial derivative differentiates a multivariable function with respect to one variable while holding the others fixed—unlike a total derivative that lets all inputs move. The i…- PDEs couple a field to how it bends in several directionsA partial differential equation involves a multivariable unknown and one or more of its partial derivatives. Laplace’s equation—sum of second pure partials in x, y, z equals zero—…
Fourier’s transform trades time localization for frequency spreadThe Fourier transform takes a function and returns another that measures how strongly each frequency is present, much as a chord can be split into the loudness of its separate pit…
Functional analysis studies infinite-dimensional linear worldsFunctional analysis studies vector spaces equipped with notions of limit, such as norms, inner products, and topologies, along with the linear maps that respect them. Unlike ordin…
One theorem proved differentiation and integration are inversesThe fundamental theorem of calculus links finding slopes with finding areas, showing that differentiation and integration roughly undo each other. Greek geometers could already wo…- Harmonic analysis breaks functions along their hidden symmetriesHarmonic analysis grew out of studying harmonic functions and how they behave at a boundary. It splits functions and measures into pieces by symmetry, scale, or frequency, then pr…
Hilbert spaces stretch Euclid’s geometry into infinite dimensionsA Hilbert space generalizes the Euclidean plane and three-space so calculus and geometry still work in any finite or infinite dimension. David Hilbert, Erhard Schmidt, and Frigyes…
One complex derivative forces infinite smoothnessA holomorphic function takes complex values and has a complex derivative throughout a neighbourhood of every point of its domain. That condition is so strong that the function is …
Modern infinity comes in many unequal sizesInfinity names whatever is boundless or endless, written with the symbol ∞ that John Wallis introduced in 1655. Greeks argued over its nature for centuries, and at the end of the …
Integrals turn continuous summing into area and volumeAn integral is the continuous counterpart of a sum, used to find areas, volumes, and their generalizations, and integration is one of calculus’s two basic operations alongside dif…
Laplace’s equation governs potentials with no local sourcesLaplace’s equation says that the Laplacian of a function, the divergence of its gradient, equals zero. Pierre-Simon Laplace first studied its properties in 1786, and its solutions…
Laplace’s transform turns calculus into algebra in s-spaceNamed for Pierre-Simon Laplace, the Laplace transform converts a function of a real variable, usually time, into a function of a complex variable s in the so-called s-domain. Diff…
Continuity bans jumps: tiny input shifts, tiny outputsA continuous function changes its value only a little when its input shifts a little—no sudden jumps allowed. Bernard Bolzano sketched an epsilon–delta style definition in 1817; C…
Nabla packs gradient, divergence, and curl into one symbolDel, written as the nabla symbol, is the vector differential operator of vector calculus. Applied in one dimension it behaves like an ordinary derivative; in higher dimensions the…
A derivative measures how fast a function's output reactsThe derivative of a single-variable function at a point is the limit of difference quotients as the step shrinks—an instantaneous rate of change. Leibniz wrote it as a ratio of di…- Differential calculus studies rates; integrals study totalsDifferential calculus focuses on derivatives, differentials, and how fast quantities change. Paired with integral calculus through the fundamental theorem, it turns secant slopes …
Newton already sorted differential equations into three kindsA differential equation links unknown functions to their derivatives—positions to velocities, heats to fluxes. Newton's 1671 work on fluxions listed three types; Bernoulli's 1695 …
Euler linked cosines and sines to the complex exponentialEuler's formula says the exponential of i times x equals cosine x plus i sine x, welding trigonometry to growth. Feynman called it mathematics' jewel; set x to π and it becomes th…
Euler's identity ties e, i, and π in one equationEuler's identity equates e raised to i times π, plus one, with zero—or writes the same relation as e to the i-π equals minus one. Named for Leonhard Euler, it is the π case of his…- Fibonacci numbers were Sanskrit prosody tools before FibonacciEach Fibonacci number is the sum of the previous two, giving 0, 1, 1, 2, 3, 5, 8, 13, and so on. Indian scholars such as Pingala described them by about 200 BC while counting poet…
Fourier analysis splits signals into oscillating partsFourier analysis studies how functions on lines, circles, or groups break into waves. The Fourier transform performs the split; the fast Fourier transform computes discrete cases …- Fourier series rebuild periodic functions from sines and cosinesA Fourier series expands a periodic function as a trigonometric sum. Named for Joseph Fourier, who lived from 1768 to 1830 and attacked the heat equation, the idea had earlier roo…
Best polynomials equioscillate their worst errorsApproximation theory asks how to replace hard functions with simpler ones and how to measure the damage. Computer libraries need polynomials using only arithmetic ops that nearly …- Newton's 1685 resistance problem birthed variational calculusCalculus of variations finds functions that maximize or minimize functionals—maps from functions to numbers, often integrals of a curve and its derivatives. Straight lines win sho…
One contour integral knows every disk valueCauchy's integral formula says a holomorphic function on a disk is fixed by its values on the boundary circle: f(a) equals a contour integral of f(z)/(z−a) against dz over 2πi. Th…- Complex differentiability secretly forces infinite smoothnessComplex analysis studies functions of complex variables, especially holomorphic maps that are complex-differentiable. Unlike real differentiability, holomorphicity implies infinit…