[{"data":1,"prerenderedAt":1294},["ShallowReactive",2],{"subject:calculus":3,"course-wordcounts":75,"nav:calculus":987},{"id":4,"title":5,"blurb":6,"body":7,"brief":18,"category":56,"description":57,"draft":58,"extension":59,"meta":60,"module":15,"navigation":25,"path":61,"practice":62,"rawbody":63,"readingTime":64,"seo":69,"sources":70,"status":71,"stem":72,"summary":15,"topics":73,"__hash__":74},"course\u002F01.calculus\u002Findex.md","Calculus","Limits, derivatives, and integrals — the mathematics of change and\naccumulation, from the tangent-line problem through infinite series to\nStokes' theorem in three dimensions.\n",{"type":8,"value":9,"toc":14},"minimark",[10],[11,12,13],"p",{},"Two motions run under every topic: how a quantity changes, and how it\naccumulates — and the limit is what makes both exact. The sequence starts\nthere, with limits and continuity, then builds the derivative as a limit of\nslopes and develops the rules and applications of differentiation: rates of\nchange, curve sketching, optimization, and Newton's method. It turns next to\nthe integral as a limit of sums, proves the Fundamental Theorem that binds the\ntwo together, and puts integration to work computing areas, volumes, arc\nlengths, and averages. From there the notes reach beyond one variable — into\ninfinite sequences and series and their Taylor expansions, parametric and polar\ncurves, the calculus of several variables with partial derivatives and multiple\nintegrals, and finally vector fields and the theorems of Green, Stokes, and\nGauss. These notes follow Stewart from the first limit to those closing\ntheorems, each topic resting on the ones before it.",{"title":15,"searchDepth":16,"depth":16,"links":17},"",2,[],[19,21,26,30,32,34,38,40,42,46,48,52,54],{"p":20},"Calculus is the mathematics of change and accumulation. Its whole\nedifice rests on a single idea — the \u003Cstrong>limit\u003C\u002Fstrong> — which makes\nprecise what it means to approach a value without ever arriving.\n",{"fig":22,"n":23,"caption":24,"large":25},"calc-derivative","001","The derivative: a secant line sweeps to the tangent as Δx → 0, giving the\ninstantaneous slope f'(x).\n",true,{"fig":27,"n":28,"caption":29},"calc-riemann","002","The integral: midpoint rectangles refine into the exact area under a\ncurve.\n",{"p":31},"Everything begins with the tangent-line problem. Fix a point on a curve,\ntake a second nearby, and draw the line through both. As the second point\nslides in, that secant pivots toward a single limiting line — the\n\u003Cstrong>tangent\u003C\u002Fstrong>, whose slope is the derivative.\n",{"p":33},"The derivative is an \u003Cem>instantaneous rate of change\u003C\u002Fem>: velocity from\nposition, marginal cost from cost, slope from height. Differentiation\nturns a function into its rate function, and a small kit of rules — the\nproduct, quotient, and chain rules — differentiates almost anything you\ncan write down.\n",{"fig":35,"n":36,"caption":37},"calc-limit","003","The ε–δ definition: for every tolerance ε there is a neighbourhood δ whose\ngraph stays inside the band.\n",{"p":39},"Integration runs the other way. Instead of measuring how fast a quantity\nchanges, it \u003Cstrong>accumulates\u003C\u002Fstrong> — summing infinitely many\ninfinitesimal contributions into an area, a volume, a total distance.\nThe definite integral is the limit of Riemann sums as the rectangles\nthin.\n",{"p":41},"The two operations are inverses. The \u003Cstrong>Fundamental Theorem of\nCalculus\u003C\u002Fstrong> says that differentiation and integration undo each\nother, which is why an area problem can be solved by finding an\nantiderivative rather than summing rectangles by hand.\n",{"fig":43,"n":44,"caption":45},"calc-taylor","004","Taylor series: polynomials of rising degree hug a function over a widening\ninterval.\n",{"p":47},"Push the limit further and functions become infinite sums. A\n\u003Cstrong>Taylor series\u003C\u002Fstrong> rebuilds a function from its derivatives at\na point, so a few polynomial terms approximate sine, the exponential, or a\nlogarithm as closely as you please.\n",{"fig":49,"n":50,"caption":51},"calc-extremum","005","Optimization: at a maximum the tangent goes flat and f'(x) = 0.\n",{"p":53},"The payoff is everywhere. Setting the derivative to zero locates maxima\nand minima — the heart of \u003Cem>optimization\u003C\u002Fem> — while related rates,\narc length, and accumulated change turn geometry and physics into\nsolvable equations.\n",{"p":55},"The same three ideas — limit, derivative, integral — then extend to\ncurves and surfaces and to fields in space, ending with the great\ntheorems of Green, Stokes, and Gauss that unify them all.\n","math","Calculus studies how quantities change and accumulate. It begins with the\nlimit, which makes instantaneous change precise; builds the derivative and\nthe integral from it; proves they are inverse operations; and extends both\nto infinite series, curves and surfaces, and fields in space. These notes\nfollow Stewart from first limits to the theorems of Green, Stokes, and\nGauss.\n",false,"md",{},"\u002Fcalculus",[],"---\ntitle: Calculus\nstatus: available\ncategory: math\nblurb: |\n  Limits, derivatives, and integrals — the mathematics of change and\n  accumulation, from the tangent-line problem through infinite series to\n  Stokes' theorem in three dimensions.\ndescription: |\n  Calculus studies how quantities change and accumulate. It begins with the\n  limit, which makes instantaneous change precise; builds the derivative and\n  the integral from it; proves they are inverse operations; and extends both\n  to infinite series, curves and surfaces, and fields in space. These notes\n  follow Stewart from first limits to the theorems of Green, Stokes, and\n  Gauss.\nbrief:\n  - p: |\n      Calculus is the mathematics of change and accumulation. Its whole\n      edifice rests on a single idea — the \u003Cstrong>limit\u003C\u002Fstrong> — which makes\n      precise what it means to approach a value without ever arriving.\n  - fig: calc-derivative\n    n: \"001\"\n    caption: |\n      The derivative: a secant line sweeps to the tangent as Δx → 0, giving the\n      instantaneous slope f'(x).\n    large: true\n  - fig: calc-riemann\n    n: \"002\"\n    caption: |\n      The integral: midpoint rectangles refine into the exact area under a\n      curve.\n  - p: |\n      Everything begins with the tangent-line problem. Fix a point on a curve,\n      take a second nearby, and draw the line through both. As the second point\n      slides in, that secant pivots toward a single limiting line — the\n      \u003Cstrong>tangent\u003C\u002Fstrong>, whose slope is the derivative.\n  - p: |\n      The derivative is an \u003Cem>instantaneous rate of change\u003C\u002Fem>: velocity from\n      position, marginal cost from cost, slope from height. Differentiation\n      turns a function into its rate function, and a small kit of rules — the\n      product, quotient, and chain rules — differentiates almost anything you\n      can write down.\n  - fig: calc-limit\n    n: \"003\"\n    caption: |\n      The ε–δ definition: for every tolerance ε there is a neighbourhood δ whose\n      graph stays inside the band.\n  - p: |\n      Integration runs the other way. Instead of measuring how fast a quantity\n      changes, it \u003Cstrong>accumulates\u003C\u002Fstrong> — summing infinitely many\n      infinitesimal contributions into an area, a volume, a total distance.\n      The definite integral is the limit of Riemann sums as the rectangles\n      thin.\n  - p: |\n      The two operations are inverses. The \u003Cstrong>Fundamental Theorem of\n      Calculus\u003C\u002Fstrong> says that differentiation and integration undo each\n      other, which is why an area problem can be solved by finding an\n      antiderivative rather than summing rectangles by hand.\n  - fig: calc-taylor\n    n: \"004\"\n    caption: |\n      Taylor series: polynomials of rising degree hug a function over a widening\n      interval.\n  - p: |\n      Push the limit further and functions become infinite sums. A\n      \u003Cstrong>Taylor series\u003C\u002Fstrong> rebuilds a function from its derivatives at\n      a point, so a few polynomial terms approximate sine, the exponential, or a\n      logarithm as closely as you please.\n  - fig: calc-extremum\n    n: \"005\"\n    caption: |\n      Optimization: at a maximum the tangent goes flat and f'(x) = 0.\n  - p: |\n      The payoff is everywhere. Setting the derivative to zero locates maxima\n      and minima — the heart of \u003Cem>optimization\u003C\u002Fem> — while related rates,\n      arc length, and accumulated change turn geometry and physics into\n      solvable equations.\n  - p: |\n      The same three ideas — limit, derivative, integral — then extend to\n      curves and surfaces and to fields in space, ending with the great\n      theorems of Green, Stokes, and Gauss that unify them all.\n---\n\nTwo motions run under every topic: how a quantity changes, and how it\naccumulates — and the limit is what makes both exact. The sequence starts\nthere, with limits and continuity, then builds the derivative as a limit of\nslopes and develops the rules and applications of differentiation: rates of\nchange, curve sketching, optimization, and Newton's method. It turns next to\nthe integral as a limit of sums, proves the Fundamental Theorem that binds the\ntwo together, and puts integration to work computing areas, volumes, arc\nlengths, and averages. From there the notes reach beyond one variable — into\ninfinite sequences and series and their Taylor expansions, parametric and polar\ncurves, the calculus of several variables with partial derivatives and multiple\nintegrals, and finally vector fields and the theorems of Green, Stokes, and\nGauss. These notes follow Stewart from the first limit to those closing\ntheorems, each topic resting on the ones before it.\n",{"text":65,"minutes":66,"time":67,"words":68},"1 min 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and Continuity",1,"limits-and-continuity",[993,998,1003,1009],{"title":994,"path":995,"lessonNumber":990,"topics":996,"summary":997},"Functions and Mathematical Models","\u002Fcalculus\u002Flimits-and-continuity\u002Ffunctions-and-models",[989],"A function assigns exactly one output to each input and can be presented four ways: verbally, numerically, graphically, or by a formula. The elementary families — linear, polynomial, power, rational, trigonometric, exponential — model most elementary phenomena, and transformation, combination, and composition build every other function from them.\n",{"title":999,"path":1000,"lessonNumber":16,"topics":1001,"summary":1002},"The Limit of a Function","\u002Fcalculus\u002Flimits-and-continuity\u002Fthe-limit-of-a-function",[989],"The tangent and velocity problems both ask for a value a ratio approaches but never reaches — the limit. Its intuitive two-sided form splits into one-sided limits that must agree; a limit fails to exist when they disagree or when the function grows without bound, the latter producing a vertical asymptote.\n",{"title":1004,"path":1005,"lessonNumber":1006,"topics":1007,"summary":1008},"Limit Laws and the ε–δ Definition","\u002Fcalculus\u002Flimits-and-continuity\u002Flimit-laws-and-the-precise-definition",3,[989],"The Limit Laws reduce a limit to arithmetic on simpler limits, and direct substitution settles polynomials and rational functions outright. The 0\u002F0 forms that resist substitution yield to algebra or the Squeeze Theorem, and the ε–δ definition makes \"arbitrarily close\" precise as a pair of quantified inequalities.\n",{"title":1010,"path":1011,"lessonNumber":1012,"topics":1013,"summary":1014},"Continuity","\u002Fcalculus\u002Flimits-and-continuity\u002Fcontinuity",4,[989],"A function is continuous at a point when its limit there equals its value, so the graph has no break. Continuity fails in three geometric ways; it is closed under arithmetic and composition, so the elementary families and their combinations are continuous; and on a closed interval it forces the Intermediate Value Theorem, which locates roots.\n",{"module":1016,"moduleNumber":16,"slug":1017,"lessons":1018},"Derivatives","derivatives",[1019,1024,1029,1034],{"title":1020,"path":1021,"lessonNumber":990,"topics":1022,"summary":1023},"The Derivative and Rates of Change","\u002Fcalculus\u002Fderivatives\u002Fthe-derivative-and-rates-of-change",[1016],"A single limit with three readings: the slope of the tangent line, the instantaneous velocity of a moving object, and the rate of change of one quantity with respect to another. Built from the difference quotient, extended from a value at one point to a function of x, and undefined exactly where a corner, jump, or vertical tangent appears.\n",{"title":1025,"path":1026,"lessonNumber":16,"topics":1027,"summary":1028},"Differentiation Rules and the Chain Rule","\u002Fcalculus\u002Fderivatives\u002Fdifferentiation-rules-and-the-chain-rule",[1016],"Computing every derivative from the limit definition is tedious. A short list of rules — power, constant multiple, sum, product, quotient — differentiates any polynomial or rational function by inspection. The trigonometric derivatives follow from one limit, and the chain rule extends everything to composite functions by multiplying rates along the composition.\n",{"title":1030,"path":1031,"lessonNumber":1006,"topics":1032,"summary":1033},"Implicit Differentiation and Related Rates","\u002Fcalculus\u002Fderivatives\u002Fimplicit-differentiation-and-related-rates",[1016],"Not every curve is the graph of y = f(x). Implicit differentiation finds a slope from an equation in x and y directly, treating y as an unknown function and differentiating both sides. The same chain-rule idea drives related rates, where one measured rate of change forces another through a geometric constraint, and interprets the derivative as a rate across the sciences.\n",{"title":1035,"path":1036,"lessonNumber":1012,"topics":1037,"summary":1038},"Linear Approximations and Differentials","\u002Fcalculus\u002Fderivatives\u002Flinear-approximations-and-differentials",[1016],"A differentiable curve looks like its tangent line under enough magnification, so the tangent is a usable stand-in for the function near the point of contact. The linear approximation and its linearization, written in the language of differentials dy and dx, estimate both function values and the measurement error propagated into a computed quantity.\n",{"module":1040,"moduleNumber":1006,"slug":1041,"lessons":1042},"Applications of Derivatives","applications-of-derivatives",[1043,1048,1053,1058],{"title":1044,"path":1045,"lessonNumber":990,"topics":1046,"summary":1047},"Extrema and the Mean Value Theorem","\u002Fcalculus\u002Fapplications-of-derivatives\u002Fextrema-and-the-mean-value-theorem",[1040],"Absolute and local extrema, the Extreme Value Theorem that guarantees them, and Fermat's Theorem pinning candidates to critical numbers. The Closed Interval Method turns the search for extrema into a finite checklist. Rolle's Theorem and the Mean Value Theorem then connect a function's values to its derivative, giving the tool that most of differential calculus rests on.\n",{"title":1049,"path":1050,"lessonNumber":16,"topics":1051,"summary":1052},"How Derivatives Shape a Graph","\u002Fcalculus\u002Fapplications-of-derivatives\u002Fhow-derivatives-shape-a-graph",[1040],"The sign of the first derivative fixes where a function rises and falls, and a sign change identifies each local extremum through the First Derivative Test. The second derivative sets concavity and inflection points and gives a faster Second Derivative Test. Limits at infinity describe end behavior and the horizontal asymptotes a curve settles toward.\n",{"title":1054,"path":1055,"lessonNumber":1006,"topics":1056,"summary":1057},"Curve Sketching and Optimization","\u002Fcalculus\u002Fapplications-of-derivatives\u002Fcurve-sketching-and-optimization",[1040],"A checklist that synthesizes domain, symmetry, asymptotes, monotonicity, extrema, and concavity into a hand sketch of any function, plus the slant asymptote for rational functions whose degree exceeds the denominator's. The same extremum machinery, applied to a word problem, becomes the optimization template: model one quantity, reduce it to a function of a single variable, and find its absolute extremum.\n",{"title":1059,"path":1060,"lessonNumber":1012,"topics":1061,"summary":1062},"Newton's Method and Antiderivatives","\u002Fcalculus\u002Fapplications-of-derivatives\u002Fnewtons-method-and-antiderivatives",[1040],"Newton's method solves $f(x) = 0$ by repeatedly replacing the curve with its tangent line and jumping to the tangent's root, converging fast when it works and diverging when the derivative is small. Antiderivatives reverse differentiation: every antiderivative of a function differs from another by a constant, so the general antiderivative is a family of parallel curves, pinned to one by an initial condition.\n",{"module":1064,"moduleNumber":1012,"slug":1065,"lessons":1066},"Integrals","integrals",[1067,1072,1077],{"title":1068,"path":1069,"lessonNumber":990,"topics":1070,"summary":1071},"Area and the Definite Integral","\u002Fcalculus\u002Fintegrals\u002Farea-and-the-definite-integral",[1064],"The area under a curve is defined as a limit of sums of rectangle areas. The same limit — a Riemann sum taken as the mesh shrinks to zero — defines the definite integral, a single number measuring signed area, total distance, and every accumulated quantity built the same way. Its properties, comparison bounds, and reading as net area follow directly from the limit.\n",{"title":1073,"path":1074,"lessonNumber":16,"topics":1075,"summary":1076},"The Fundamental Theorem of Calculus","\u002Fcalculus\u002Fintegrals\u002Fthe-fundamental-theorem-of-calculus",[1064],"Differentiation and integration are inverse operations. Part 1 says the derivative of an area-accumulation function is the integrand; Part 2 says a definite integral equals the change in any antiderivative across the interval. Together they replace limits of Riemann sums with antiderivative lookups, define the indefinite integral, and give the Net Change Theorem for rates.\n",{"title":1078,"path":1079,"lessonNumber":1006,"topics":1080,"summary":1081},"The Substitution Rule","\u002Fcalculus\u002Fintegrals\u002Fthe-substitution-rule",[1064],"Substitution runs the Chain Rule backward: spotting an inner function whose derivative also appears in the integrand lets the variable change to $u$ and collapse a composite integral to a simple one. The rule applies to indefinite and definite integrals, with two ways to handle the limits, and it yields the symmetry shortcuts that double even integrands and vanish odd ones.\n",{"module":1083,"moduleNumber":1084,"slug":1085,"lessons":1086},"Applications of Integration",5,"applications-of-integration",[1087,1092,1097],{"title":1088,"path":1089,"lessonNumber":990,"topics":1090,"summary":1091},"Areas Between Curves and Volumes","\u002Fcalculus\u002Fapplications-of-integration\u002Fareas-and-volumes",[1083],"A definite integral computes any quantity that a limit of Riemann sums approximates. Applied to geometry it gives the area between two curves and the volume of a solid: by cross-sections, by disks and washers when the region is revolved, and by cylindrical shells when inverting the boundary is awkward.\n",{"title":1093,"path":1094,"lessonNumber":16,"topics":1095,"summary":1096},"Work, Average Value, Arc Length, and Surface Area","\u002Fcalculus\u002Fapplications-of-integration\u002Fwork-average-value-and-arc-length",[1083],"The work done by a force that varies with position, the average value of a function and the Mean Value Theorem it satisfies, the length of a curve, and the area of a surface swept out by revolving that curve. Each is a limit of Riemann sums, hence a definite integral.\n",{"title":1098,"path":1099,"lessonNumber":1006,"topics":1100,"summary":1101},"Applications to Physics, Economics, and Probability","\u002Fcalculus\u002Fapplications-of-integration\u002Fphysics-economics-and-probability",[1083],"Definite integrals in physics, economics, and statistics: the force a fluid exerts on a submerged plate, the balance point of a plane region, the money consumers save at a market price, and the probability that a continuous random variable lands in an interval, together with its mean.\n",{"module":1103,"moduleNumber":1104,"slug":1105,"lessons":1106},"Exponential, Logarithmic, and Inverse Functions",6,"exponential-logarithmic-and-inverse-functions",[1107,1115,1120],{"title":1108,"path":1109,"lessonNumber":990,"topics":1110,"summary":1114},"Inverse Functions, Logarithms, and Exponentials","\u002Fcalculus\u002Fexponential-logarithmic-and-inverse-functions\u002Finverse-functions-logarithms-and-exponentials",[1111,1112,1113],"Exponential","Logarithmic","and Inverse Functions","A one-to-one function has an inverse that reverses it, with a graph mirrored across y = x and a derivative given by the reciprocal-slope rule. The exponential e^x is its own derivative and the natural logarithm has derivative 1\u002Fx; logarithmic differentiation turns products, quotients, and variable powers into sums.\n",{"title":1116,"path":1117,"lessonNumber":16,"topics":1118,"summary":1119},"Growth, Decay, Inverse Trigonometric, and Hyperbolic Functions","\u002Fcalculus\u002Fexponential-logarithmic-and-inverse-functions\u002Fgrowth-decay-inverse-trig-and-hyperbolic-functions",[1111,1112,1113],"Any quantity whose rate of change is proportional to its size grows or decays exponentially, the single equation y' = ky behind populations, radioactive decay, cooling, and continuously compounded interest. The inverse trigonometric functions have algebraic derivatives, and the hyperbolic functions, built from e^x and e^{-x}, describe the hanging cable.\n",{"title":1121,"path":1122,"lessonNumber":1006,"topics":1123,"summary":1124},"Indeterminate Forms and l'Hospital's Rule","\u002Fcalculus\u002Fexponential-logarithmic-and-inverse-functions\u002Flhospitals-rule",[1111,1112,1113],"When a limit produces 0\u002F0 or infinity over infinity, the value is undetermined by the forms alone. l'Hospital's Rule resolves both by replacing the ratio of functions with the ratio of their derivatives. Products, differences, and powers reduce to a quotient the rule can handle, and repeated use ranks the growth of logarithms, powers, and exponentials.\n",{"module":1126,"moduleNumber":1127,"slug":1128,"lessons":1129},"Techniques of Integration",7,"techniques-of-integration",[1130,1135,1140,1145],{"title":1131,"path":1132,"lessonNumber":990,"topics":1133,"summary":1134},"Integration by Parts","\u002Fcalculus\u002Ftechniques-of-integration\u002Fintegration-by-parts",[1126],"The product rule for derivatives reverses into integration by parts, trading the integral of $u\\,\\d v$ for the integral of $v\\,\\d u$ whenever the second is easier. The LIATE ordering fixes which factor to differentiate. Standard cases: a polynomial against a transcendental factor, repeated parts, cyclic integrals that solve for themselves, and reduction formulas that peel an exponent down by recursion.\n",{"title":1136,"path":1137,"lessonNumber":16,"topics":1138,"summary":1139},"Trigonometric Integrals and Substitution","\u002Fcalculus\u002Ftechniques-of-integration\u002Ftrigonometric-integrals-and-substitution",[1126],"Two related techniques. Trigonometric integrals evaluate powers and products of sine, cosine, tangent, and secant by splitting off one factor and converting the rest with a Pythagorean identity, or by dropping even powers with half-angle formulas. Trigonometric substitution runs the idea in reverse: replace x by a sine, tangent, or secant to clear a radical, integrate, then read the answer back off a reference triangle.\n",{"title":1141,"path":1142,"lessonNumber":1006,"topics":1143,"summary":1144},"Partial Fractions and Integration Strategy","\u002Fcalculus\u002Ftechniques-of-integration\u002Fpartial-fractions-and-integration-strategy",[1126],"Any rational function integrates in closed form: factor the denominator, split the fraction into simple pieces by partial fractions, and integrate each piece as a logarithm or an arctangent. Four denominator cases exhaust the possibilities. A four-step strategy then sorts an arbitrary integrand by its shape to the technique that fits it, and a short catalog records elementary functions whose antiderivatives are not elementary.\n",{"title":1146,"path":1147,"lessonNumber":1012,"topics":1148,"summary":1149},"Approximate and Improper Integrals","\u002Fcalculus\u002Ftechniques-of-integration\u002Fapproximate-and-improper-integrals",[1126],"Two definite integrals the Fundamental Theorem cannot reach. With no antiderivative available, the Midpoint, Trapezoidal, and Simpson rules approximate the integral from sample values, each carrying a provable error bound. With an infinite interval or an integrand that blows up, the improper integral is defined as a limit that either converges or diverges; the Comparison Test settles which without evaluating it.\n",{"module":1151,"moduleNumber":1152,"slug":1153,"lessons":1154},"Parametric Equations and Polar Coordinates",8,"parametric-and-polar",[1155,1160,1165],{"title":1156,"path":1157,"lessonNumber":990,"topics":1158,"summary":1159},"Parametric Curves and Their Calculus","\u002Fcalculus\u002Fparametric-and-polar\u002Fparametric-curves-and-their-calculus",[1151],"A parametric curve gives x and y separately as functions of a third variable, recording not only a path but the direction and timing with which it is traced. Eliminating the parameter recovers a Cartesian equation; the slope, area, arc-length, and surface-area formulas run directly on the parameter, with the cycloid and astroid as worked examples.\n",{"title":1161,"path":1162,"lessonNumber":16,"topics":1163,"summary":1164},"Polar Coordinates","\u002Fcalculus\u002Fparametric-and-polar\u002Fpolar-coordinates",[1151],"Polar coordinates locate a point by a distance from the pole and an angle from the polar axis, giving circles, spirals, and flower-shaped curves short equations. Conversion between the two systems is right-triangle trigonometry, and treating a polar curve as a parametric curve in the angle yields the tangent, area, and arc-length formulas.\n",{"title":1166,"path":1167,"lessonNumber":1006,"topics":1168,"summary":1169},"Conic Sections","\u002Fcalculus\u002Fparametric-and-polar\u002Fconic-sections",[1151],"Parabolas, ellipses, and hyperbolas are the plane curves cut from a double cone. Each has a focus-based geometric definition and a standard Cartesian equation. A single number, the eccentricity, ties the three together, and placing a focus at the pole gives all of them one polar equation that describes planetary orbits.\n",{"module":1171,"moduleNumber":1172,"slug":1173,"lessons":1174},"Infinite Sequences and Series",9,"sequences-and-series",[1175,1180,1185,1190,1195],{"title":1176,"path":1177,"lessonNumber":990,"topics":1178,"summary":1179},"Sequences","\u002Fcalculus\u002Fsequences-and-series\u002Fsequences",[1171],"A sequence is a function on the positive integers, and its limit is defined almost exactly like a limit at infinity. The Limit Laws and Squeeze Theorem carry over from functions, monotonic and bounded sequences give a convergence criterion, and the Monotonic Sequence Theorem guarantees a limit exists without naming it.\n",{"title":1181,"path":1182,"lessonNumber":16,"topics":1183,"summary":1184},"Series and the Integral Test","\u002Fcalculus\u002Fsequences-and-series\u002Fseries-and-the-integral-test",[1171],"Adding infinitely many terms is made precise as the limit of partial sums. The two series with closed-form partial sums are geometric and telescoping; the harmonic series diverges even as its terms shrink to zero. The Integral Test compares a positive series to an improper integral, settling the p-series and supplying a remainder bound for estimating sums.\n",{"title":1186,"path":1187,"lessonNumber":1006,"topics":1188,"summary":1189},"The Convergence Tests","\u002Fcalculus\u002Fsequences-and-series\u002Fthe-convergence-tests",[1171],"The comparison, alternating-series, ratio, and root tests decide convergence without a closed-form partial sum. Absolute convergence is stronger than conditional convergence and is preserved under rearrangement; an alternating series errs by less than its first omitted term. A test is chosen from the shape of the general term.\n",{"title":1191,"path":1192,"lessonNumber":1012,"topics":1193,"summary":1194},"Power Series","\u002Fcalculus\u002Fsequences-and-series\u002Fpower-series",[1171],"A power series is a polynomial of infinite degree whose convergence set is an interval centered at $a$, with a radius the Ratio Test finds and endpoints that must be tested by hand. Inside that interval the series represents a function that can be differentiated and integrated term by term, generating new representations from the geometric series.\n",{"title":1196,"path":1197,"lessonNumber":1084,"topics":1198,"summary":1199},"Taylor and Maclaurin Series","\u002Fcalculus\u002Fsequences-and-series\u002Ftaylor-and-maclaurin-series",[1171],"If a function equals a power series, its coefficients are forced: the nth is the nth derivative at the center over n factorial. We derive that formula, use Taylor's Inequality to prove the standard series for the exponential, sine, and cosine, record the binomial series and a reference table, and bound the error when a Taylor polynomial replaces a function.\n",{"module":1201,"moduleNumber":1202,"slug":1203,"lessons":1204},"Vectors and the Geometry of Space",10,"vectors-and-space-curves",[1205,1210,1215,1220,1225],{"title":1206,"path":1207,"lessonNumber":990,"topics":1208,"summary":1209},"Three-Dimensional Coordinates, Vectors, and the Dot Product","\u002Fcalculus\u002Fvectors-and-space-curves\u002Fvectors-and-the-dot-product",[1201],"Space needs three coordinates, so we set up the rectangular system, the distance formula, and the equation of a sphere. Vectors then package magnitude and direction into a single algebraic object with its own arithmetic. The dot product turns two vectors into a number that measures the angle between them, gives a clean test for orthogonality, and produces the projection of one vector onto another.\n",{"title":1211,"path":1212,"lessonNumber":16,"topics":1213,"summary":1214},"The Cross Product, Lines, and Planes","\u002Fcalculus\u002Fvectors-and-space-curves\u002Fthe-cross-product-lines-and-planes",[1201],"The cross product multiplies two vectors into a third perpendicular to both, with length equal to the area of the parallelogram they span. That one construction supplies the direction of a line, the normal of a plane, and, through the scalar triple product, the volume of a parallelepiped. Lines carry a point and a direction vector; planes carry a point and a normal, which fixes the angle between planes and the distance from a point to a plane.\n",{"title":1216,"path":1217,"lessonNumber":1006,"topics":1218,"summary":1219},"Cylinders and Quadric Surfaces","\u002Fcalculus\u002Fvectors-and-space-curves\u002Fcylinders-and-quadric-surfaces",[1201],"A surface whose equation omits one variable is a cylinder: the graph of a plane curve swept along the missing axis. A second-degree equation in three variables is a quadric, and translation and rotation reduce every one to a short standard list. Traces — the curves cut by planes parallel to the coordinate planes — sort the six quadrics into ellipsoid, the two paraboloids, the cone, and the two hyperboloids.\n",{"title":1221,"path":1222,"lessonNumber":1012,"topics":1223,"summary":1224},"Vector Functions and Space Curves","\u002Fcalculus\u002Fvectors-and-space-curves\u002Fvector-functions-and-space-curves",[1201],"A vector function assigns a vector to each value of a parameter, and as the parameter runs its tip traces a space curve. Taking limits, derivatives, and integrals component by component carries all of single-variable calculus into three dimensions. The derivative of a vector function is the tangent vector to its curve, and normalizing it gives the unit tangent that points the way along the path.\n",{"title":1226,"path":1227,"lessonNumber":1084,"topics":1228,"summary":1229},"Arc Length, Curvature, and Motion in Space","\u002Fcalculus\u002Fvectors-and-space-curves\u002Farc-length-curvature-and-motion",[1201],"Integrating the speed of a vector function gives the length of its curve and a natural parameter, arc length, that depends only on the curve's shape. Curvature measures how fast the unit tangent turns, and together with the normal and binormal it builds the moving TNB frame. Reading the same vector function as a trajectory, its first two derivatives are velocity and acceleration, and acceleration splits cleanly into tangential and normal parts.\n",{"module":1231,"moduleNumber":1232,"slug":1233,"lessons":1234},"Partial Derivatives",11,"partial-derivatives",[1235,1240,1244,1249,1254],{"title":1236,"path":1237,"lessonNumber":990,"topics":1238,"summary":1239},"Functions of Several Variables, Limits, and Continuity","\u002Fcalculus\u002Fpartial-derivatives\u002Ffunctions-of-several-variables",[1231],"A function of several variables assigns one number to each point of a region in the plane or in space. Domain, graph, level curve, and level surface describe it; limits and continuity extend to two variables, where a limit must agree along every path of approach, not just from the left and the right.\n",{"title":1231,"path":1241,"lessonNumber":16,"topics":1242,"summary":1243},"\u002Fcalculus\u002Fpartial-derivatives\u002Fpartial-derivatives",[1231],"A partial derivative holds every variable but one fixed and differentiates in the ordinary sense. Geometrically it is the slope of a trace curve cut from the surface by a coordinate plane. The freeze-and-differentiate rule computes the two first partials; the four second partials follow, and the two mixed ones agree under Clairaut's Theorem when they are continuous.\n",{"title":1245,"path":1246,"lessonNumber":1006,"topics":1247,"summary":1248},"Tangent Planes, Linear Approximation, and the Chain Rule","\u002Fcalculus\u002Fpartial-derivatives\u002Ftangent-planes-and-the-chain-rule",[1231],"Near a point, a smooth surface looks like its tangent plane, and the plane's equation is built from the two partial derivatives. That linearization defines the total differential and the meaning of differentiability in two variables. The chain rule then propagates derivatives through composed functions, tracked by a tree diagram, and yields clean formulas for implicit differentiation.\n",{"title":1250,"path":1251,"lessonNumber":1012,"topics":1252,"summary":1253},"Directional Derivatives and the Gradient","\u002Fcalculus\u002Fpartial-derivatives\u002Fdirectional-derivatives-and-the-gradient",[1231],"The partial derivatives measure slope along the two axes; the directional derivative measures slope along any chosen direction, and equals the gradient dotted with a unit vector. The gradient points in the direction of steepest increase, its length is the greatest rate, and it stands perpendicular to level curves and surfaces, which fixes the tangent plane to a level surface.\n",{"title":1255,"path":1256,"lessonNumber":1084,"topics":1257,"summary":1258},"Optimization and Lagrange Multipliers","\u002Fcalculus\u002Fpartial-derivatives\u002Foptimization-and-lagrange-multipliers",[1231],"Extrema of a two-variable function sit at critical points where the gradient vanishes; the Second Derivatives Test sorts them into peaks, valleys, and saddles by the sign of a discriminant. Absolute extrema on a closed region also need the boundary. When the domain is itself a constraint curve, Lagrange multipliers set the two gradients parallel and solve the constrained problem.\n",{"module":1260,"moduleNumber":1261,"slug":1262,"lessons":1263},"Multiple Integrals and Vector Calculus",12,"multiple-integrals-and-vector-calculus",[1264,1269,1274,1279,1284,1289],{"title":1265,"path":1266,"lessonNumber":990,"topics":1267,"summary":1268},"Double Integrals","\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fdouble-integrals",[1260],"The double integral extends the definite integral to functions of two variables: a limit of Riemann sums that measures signed volume under a surface. Fubini's Theorem turns it into two ordinary integrations done one after the other, general regions of type I and type II fix the inner limits, polar coordinates absorb circular symmetry through the factor r, and the same machine computes mass, center of mass, and moments of a lamina.\n",{"title":1270,"path":1271,"lessonNumber":16,"topics":1272,"summary":1273},"Triple Integrals and Coordinate Systems","\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Ftriple-integrals-and-coordinate-systems",[1260],"The triple integral integrates a function of three variables over a solid, as a limit of Riemann sums evaluated by three nested single integrations. Cylindrical coordinates add the factor r to handle axial symmetry, spherical coordinates add rho-squared sine-phi for radial symmetry, and the general change of variables shows both volume elements are Jacobian determinants of the coordinate map. Surface area for a graph completes the measurement toolkit.\n",{"title":1275,"path":1276,"lessonNumber":1006,"topics":1277,"summary":1278},"Vector Fields and Line Integrals","\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fvector-fields-and-line-integrals",[1260],"A vector field assigns a vector to every point of space; the line integral of a field along a curve accumulates its tangential component, measuring work. Conservative fields are gradients of a potential, and for them the Fundamental Theorem for Line Integrals makes the integral depend only on the endpoints. Path independence, closed-loop integrals of zero, and the component test for a potential are three faces of the same property.\n",{"title":1280,"path":1281,"lessonNumber":1012,"topics":1282,"summary":1283},"Green's Theorem, Curl, and Divergence","\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fgreens-theorem-curl-and-divergence",[1260],"Green's Theorem equates the line integral of a field around a positively oriented closed curve with a double integral over the enclosed region, turning a boundary computation into an area computation and vice versa. Curl measures local circulation and divergence measures local outflow; the two vector forms of Green's Theorem express the boundary integral as the integrated curl or divergence, the planar case of Stokes' and the Divergence Theorem.\n",{"title":1285,"path":1286,"lessonNumber":1084,"topics":1287,"summary":1288},"Parametric Surfaces and Surface Integrals","\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fsurface-integrals",[1260],"A parametric surface is the image of a two-variable vector function; its area element is the magnitude of the cross product of the two tangent vectors. The surface integral of a scalar function sums it over that area, and the flux integral of a vector field sums the field's normal component, measuring flow through the surface. Orientation by a choice of unit normal makes flux well-defined, the integral Stokes' and the Divergence Theorem operate on.\n",{"title":1290,"path":1291,"lessonNumber":1104,"topics":1292,"summary":1293},"Stokes' Theorem and the Divergence Theorem","\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fstokes-and-the-divergence-theorem",[1260],"Stokes' Theorem lifts Green's Theorem into space: the line integral of a field around the boundary of a surface equals the flux of its curl through the surface. The Divergence Theorem relates the outward flux across a closed surface to the triple integral of divergence over the solid it encloses. Together with the Fundamental Theorem of Calculus and its line-integral and Green counterparts, they are one theorem: the integral of a derivative over a region equals the integral of the field over its oriented boundary.\n",1786059476786]