---
title: "Calculus & Analysis"
description: "83 topics in Mathematics, in the order they build on each other."
canonical: https://lightmysky.com/learn/mathematics/areas/calculus-and-analysis
source: https://lightmysky.com/learn/mathematics/areas/calculus-and-analysis.md
retrieved: 2026-09-02
---

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# Calculus & Analysis

83 topics in Mathematics, in the order they build on each other.

Page: https://lightmysky.com/learn/mathematics/areas/calculus-and-analysis

- [The Gradient of a Curve as a Limit](https://lightmysky.com/learn/mathematics/the-gradient-of-a-curve-as-a-limit-mt_LAlhcwVjln): See that a curve has a different gradient at every point, and that the gradient of a chord settles on a single value as the second point slides in. That limiting value is the gradient of the tangent.
- [Differentiation from First Principles](https://lightmysky.com/learn/mathematics/differentiation-from-first-principles-mt_pdfPztEjw_): Write the chord gradient as [f(x + h) - f(x)] / h, simplify it algebraically, and let h approach zero to get the derivative. This is the definition every rule later rests on.
- [The Power Rule for Differentiating Polynomials](https://lightmysky.com/learn/mathematics/the-power-rule-for-differentiating-polynomials-mt_l7f4j44bR3): Differentiate any sum of powers of x with the rule that xⁿ becomes nxⁿ⁻¹, including negative and fractional powers once the term is rewritten as a power.
- [The Derivative as a Rate of Change](https://lightmysky.com/learn/mathematics/the-derivative-as-a-rate-of-change-mt_RqyLE3jrAW): Read dy/dx as how fast one quantity changes per unit of another, with the units to match, so a derivative answers questions about cost per item, litres per second or metres per second.
- [Tangents and Normals to a Curve](https://lightmysky.com/learn/mathematics/tangents-and-normals-to-a-curve-mt_BzMw7IT6kp): Find the equation of the tangent at a point on a curve from the derivative there, and the normal from the perpendicular gradient.
- [Increasing and Decreasing Functions](https://lightmysky.com/learn/mathematics/increasing-and-decreasing-functions-mt_KOrXxQvBrZ): Use the sign of the derivative to say where a curve rises and where it falls, which turns a question about shape into an inequality in x.
- [Stationary Points and the Second Derivative](https://lightmysky.com/learn/mathematics/stationary-points-and-the-second-derivative-mt_VbGJEFFgfs): Solve f'(x) = 0 to locate maximum, minimum and inflection points, then use the sign of f''(x) to decide which is which.
- [Optimisation with Calculus](https://lightmysky.com/learn/mathematics/optimisation-with-calculus-mt_9I0ql39Lzw): Turn a worded problem into one function of one variable, use a constraint to remove the second variable, then differentiate to find the best value and justify that it is a maximum or minimum.
- [Integration as the Reverse of Differentiation](https://lightmysky.com/learn/mathematics/integration-as-the-reverse-of-differentiation-mt_edtk3ArxRk): Recover a function from its derivative by raising the power and dividing, and carry the constant of integration that differentiation destroyed. Use a known point to pin that constant down.
- [The Definite Integral and the Area Under a Curve](https://lightmysky.com/learn/mathematics/the-definite-integral-and-the-area-under-a-curve-mt_S2IO2S7kgd): Estimate the area under a curve with rectangles, then evaluate it exactly by substituting the limits into the antiderivative. Areas below the axis come out negative and have to be handled separately.
- [Areas Between Two Curves](https://lightmysky.com/learn/mathematics/areas-between-two-curves-mt_lDBZmnLK1b): Find where two curves meet, then integrate the difference of the two functions between those limits to get the area enclosed.
- [One-Sided Limits and When a Limit Fails to Exist](https://lightmysky.com/learn/mathematics/one-sided-limits-and-when-a-limit-fails-to-exist-mt__L0B_WQq-P): Treat a limit as a two-sided agreement: what the outputs approach from the left has to match what they approach from the right. Where the two disagree, or the outputs grow without bound, no limit exists.
- [The Limit Laws and Indeterminate Forms](https://lightmysky.com/learn/mathematics/the-limit-laws-and-indeterminate-forms-mt_rAJs-V4Jlx): Combine limits with the rules for sums, products and quotients, and recognise the forms those rules leave undecided. A 0/0 quotient is a signal to factorise, rationalise or squeeze.
- [Locating Roots and Iterative Methods](https://lightmysky.com/learn/mathematics/locating-roots-and-iterative-methods-mt_2COO6TWerM): Trap a root between two values by a change of sign, then rearrange the equation into the form x = g(x) and iterate from a starting value until the digits stop moving.
- [The Newton-Raphson Method](https://lightmysky.com/learn/mathematics/the-newton-raphson-method-mt_jkjJre5ETd): Follow the tangent at the current guess down to the x-axis to get the next guess, which is x - f(x)/f'(x). Note when it fails: a nearly flat tangent throws the next guess far away.
- [The Trapezium Rule](https://lightmysky.com/learn/mathematics/the-trapezium-rule-mt_rAlJ1WIRmZ): Estimate a definite integral by slicing the region into trapezia of equal width, and use the curvature to say whether the estimate is above or below the true value.
- [Continuity and the Intermediate Value Theorem](https://lightmysky.com/learn/mathematics/continuity-and-the-intermediate-value-theorem-mt_kaQR6cWjbV): Define continuity at a point as the limit agreeing with the value there, classify the ways that can fail, and use the intermediate value theorem to guarantee a root inside an interval.
- [The Epsilon-Delta Definition of a Limit](https://lightmysky.com/learn/mathematics/the-epsilon-delta-definition-of-a-limit-mt_jFzRYQzCmk): State what a limit claims with no appeal to motion: for every tolerance around the limiting value there is a distance around the point that keeps outputs inside it. Prove a simple limit by producing that distance.
- [The Product and Quotient Rules](https://lightmysky.com/learn/mathematics/the-product-and-quotient-rules-mt_NJ7INJ0jfv): Differentiate a product or a quotient without multiplying it out, and see where each rule comes from in the difference quotient.
- [The Chain Rule](https://lightmysky.com/learn/mathematics/the-chain-rule-mt_kdhl4dmwJn): Differentiate a composition by multiplying the rate of the outer function by the rate of the inner one. The rule is what makes substitution, related rates and implicit differentiation possible.
- [Derivatives of Exponential, Logarithmic and Trigonometric Functions](https://lightmysky.com/learn/mathematics/derivatives-of-exponential-logarithmic-and-trigonometric-functions-mt_695WPZKADT): Establish the derivatives of the standard non-polynomial functions, including why e is the base that makes the exponential its own derivative.
- [Implicit Differentiation and Derivatives of Inverse Functions](https://lightmysky.com/learn/mathematics/implicit-differentiation-and-derivatives-of-inverse-functions-mt_P4ABBGsKFT): Differentiate a relation that is not solved for y, and use the same move to get the derivative of an inverse function from the derivative of the original.
- [Related Rates](https://lightmysky.com/learn/mathematics/related-rates-mt_3XTIPt8Vmf): Link two changing quantities by an equation, differentiate the equation with respect to time, and solve for the rate you cannot measure directly.
- [The Mean Value Theorem](https://lightmysky.com/learn/mathematics/the-mean-value-theorem-mt_09iApYVelx): Over an interval where a function is continuous and differentiable, some interior point has instantaneous rate equal to the average rate across the whole interval. Most facts linking a derivative to a function's behaviour rest on this.
- [L'Hopital's Rule and Comparing Growth Rates](https://lightmysky.com/learn/mathematics/lhopitals-rule-and-comparing-growth-rates-mt_Ai-A2sPIoE): Resolve 0/0 and infinity over infinity limits by differentiating numerator and denominator separately, and use the result to rank how fast logarithms, powers and exponentials grow.
- [Riemann Sums and the Definite Integral as a Limit](https://lightmysky.com/learn/mathematics/riemann-sums-and-the-definite-integral-as-a-limit-mt_pwa2SRU6P9): Build the definite integral as the limit of sums of rectangle areas, with sample points anywhere in each subinterval. This is the definition every later property of the integral is argued from.
- [The Fundamental Theorem of Calculus](https://lightmysky.com/learn/mathematics/the-fundamental-theorem-of-calculus-mt_EsS_ovdz2A): Both halves: differentiating an accumulation function returns the integrand, and a definite integral can be evaluated from any antiderivative. This is why the two calculus operations undo each other.
- [Integration by Substitution](https://lightmysky.com/learn/mathematics/integration-by-substitution-mt_jH-j_XsdC7): Reverse the chain rule by renaming an inner function, changing the differential and, for a definite integral, changing the limits with it.
- [Integration by Parts](https://lightmysky.com/learn/mathematics/integration-by-parts-mt_v6jZVs2_tf): Reverse the product rule to trade one integral for another that is easier, and choose the parts so the trade is an improvement.
- [Trigonometric Integrals and Trigonometric Substitution](https://lightmysky.com/learn/mathematics/trigonometric-integrals-and-trigonometric-substitution-mt_JqysS41psg): Handle powers of sine and cosine with identities, and clear expressions such as √(a² - x²) by substituting a trigonometric function for x.
- [Partial Fractions for Rational Integrands](https://lightmysky.com/learn/mathematics/partial-fractions-for-rational-integrands-mt_PA8qxs424c): Split a rational function into simpler fractions whose integrals are logarithms and arctangents, after checking the degree of the numerator first.
- [Improper Integrals and Their Convergence](https://lightmysky.com/learn/mathematics/improper-integrals-and-their-convergence-mt_gvtZPFRA6Z): Give meaning to an integral with an infinite limit or an unbounded integrand by evaluating a proper integral and taking a limit, then decide whether the result is finite.
- [Vector-Valued Functions and Motion Along a Curve](https://lightmysky.com/learn/mathematics/vector-valued-functions-and-motion-along-a-curve-mt_Xcjt6exMy3): Describe a path in space by a vector that depends on one parameter. Differentiating gives velocity and acceleration, and the arc length is an integral of speed.
- [Functions of Several Variables and Level Curves](https://lightmysky.com/learn/mathematics/functions-of-several-variables-and-level-curves-mt_u04fZ_XiJ8): Read a function of two inputs as a surface, and read the surface through its level curves. A contour map is the working picture for the rest of the spine.
- [Partial Derivatives](https://lightmysky.com/learn/mathematics/partial-derivatives-mt_tu11fd9Xk9): Differentiate with respect to one variable while holding the others fixed. Mixed second partials agree for the functions met in practice, which is a statement worth noticing rather than assuming.
- [Tangent Planes and Linear Approximation](https://lightmysky.com/learn/mathematics/tangent-planes-and-linear-approximation-mt_EYMcjChX_m): Replace a surface near a point by the plane its partials define, and use that plane to estimate values and propagate small errors.
- [The Multivariable Chain Rule](https://lightmysky.com/learn/mathematics/the-multivariable-chain-rule-mt_CHSHdNxPmJ): Differentiate a composition when several inputs each depend on other variables, by summing one contribution per path through the dependency diagram.
- [Directional Derivatives and the Gradient](https://lightmysky.com/learn/mathematics/directional-derivatives-and-the-gradient-mt_LW_KckY5Ad): The rate of change in an arbitrary direction is the dot product of the gradient with a unit vector, so the gradient points the steepest way uphill and sits perpendicular to level curves.
- [Critical Points and Optimisation in Two Variables](https://lightmysky.com/learn/mathematics/critical-points-and-optimisation-in-two-variables-mt_4qB5pAGeuR): Find points where the gradient vanishes and classify them with the second derivative test, including the saddle point, which has no one-variable analogue.
- [Lagrange Multipliers](https://lightmysky.com/learn/mathematics/lagrange-multipliers-mt_0RPOBWZ1gs): Optimise subject to a constraint by setting the gradient of the objective parallel to the gradient of the constraint, which is the moment the level curves touch.
- [Double Integrals over General Regions](https://lightmysky.com/learn/mathematics/double-integrals-over-general-regions-mt_UKS9_AhAWw): Integrate over a plane region by iterating two single integrals, with the inner limits describing the region. Choosing the order of integration is often the whole problem.
- [Volumes of Revolution: Disks, Washers and Shells](https://lightmysky.com/learn/mathematics/volumes-of-revolution-disks-washers-and-shells-mt_mzZ22MkWD4): Set up a volume as an integral of cross-sections, and choose between slicing perpendicular to the axis and using cylindrical shells based on which integral is easier.
- [Arc Length and the Area of a Surface of Revolution](https://lightmysky.com/learn/mathematics/arc-length-and-the-area-of-a-surface-of-revolution-mt_W4uujT9Z1t): Set up the integrals that measure the length of a plane curve and the area swept when it is spun about an axis.
- [Sequences and Their Limits](https://lightmysky.com/learn/mathematics/sequences-and-their-limits-mt_PQl3Q6n5dc): Treat a sequence as a function on the positive integers and ask what its terms approach. Monotone bounded sequences converge, which is the first place completeness is doing real work.
- [Infinite Series and the Geometric Series](https://lightmysky.com/learn/mathematics/infinite-series-and-the-geometric-series-mt_EPE-ERFFQt): Define the sum of a series as the limit of its partial sums, work out the geometric and telescoping cases exactly, and use the term test to rule out convergence.
- [Convergence Tests for Series of Positive Terms](https://lightmysky.com/learn/mathematics/convergence-tests-for-series-of-positive-terms-mt_8N49Q4BlO8): Decide convergence with the integral, comparison, limit comparison, ratio and root tests, and choose the test that matches the shape of the terms.
- [Alternating Series and Absolute Convergence](https://lightmysky.com/learn/mathematics/alternating-series-and-absolute-convergence-mt_sk6F4n1USH): Handle series whose terms change sign: the alternating test, the error bound from the first omitted term, and the difference between converging absolutely and converging only conditionally.
- [Power Series and the Radius of Convergence](https://lightmysky.com/learn/mathematics/power-series-and-the-radius-of-convergence-mt_RnYb0JLKbD): Read a power series as a function defined where it converges, find its radius with the ratio test, and check the endpoints separately.
- [Taylor and Maclaurin Series](https://lightmysky.com/learn/mathematics/taylor-and-maclaurin-series-mt_SgI9Pn9RIO): Build the power series of a function from its derivatives at a point, use the remainder to say how good a truncation is, and reuse the standard expansions instead of starting over.
- [Double Integrals in Polar Coordinates](https://lightmysky.com/learn/mathematics/double-integrals-in-polar-coordinates-mt_93e-KSzQMT): Integrate over circular regions by switching to polar coordinates, where the area element carries an extra factor of r.
- [Polynomial Interpolation and Its Error Term](https://lightmysky.com/learn/mathematics/polynomial-interpolation-and-its-error-term-mt_NSnzY9Xodd): Fit the unique polynomial through given points, build it in Lagrange and Newton form, and bound how far it strays between the points.
- [Numerical Quadrature: Newton-Cotes and Adaptive Rules](https://lightmysky.com/learn/mathematics/numerical-quadrature-newton-cotes-and-adaptive-rules-mt_fT5kRCUiD5): Derive quadrature rules by integrating an interpolating polynomial, compare their error orders, and refine only where the integrand misbehaves.
- [The Completeness Axiom: Suprema and Infima](https://lightmysky.com/learn/mathematics/the-completeness-axiom-suprema-and-infima-mt_xjI-pIfh95): The rationals have gaps and the reals do not. Completeness is stated as every bounded set having a least upper bound, and it is the axiom every later theorem in analysis is traced back to.
- [Convergence of Sequences, Rigorously](https://lightmysky.com/learn/mathematics/convergence-of-sequences-rigorously-mt_yB5M-DCH8V): Prove convergence from the definition: for every tolerance there is a point in the sequence past which every term stays inside it. Limit laws become theorems with proofs rather than rules to apply.
- [Cauchy Sequences and the Bolzano-Weierstrass Theorem](https://lightmysky.com/learn/mathematics/cauchy-sequences-and-the-bolzano-weierstrass-theorem-mt_nreDylVkSU): Terms that eventually crowd together converge, in the reals but not in the rationals. Every bounded sequence has a convergent subsequence, which is the workhorse behind existence proofs.
- [Continuity and Uniform Continuity](https://lightmysky.com/learn/mathematics/continuity-and-uniform-continuity-mt_plWc9raOwz): Continuity as a sequence or epsilon-delta condition, then the stronger version where one delta works across the whole domain. A continuous function on a closed bounded interval is uniformly continuous and attains its bounds.
- [Differentiability and the Theorems Behind the Rules](https://lightmysky.com/learn/mathematics/differentiability-and-the-theorems-behind-the-rules-mt_1xLEeoA69z): Define the derivative as a limit, prove differentiability implies continuity, and prove Rolle and the mean value theorem. The rules used all year get their justification here.
- [The Riemann Integral and When a Function Is Integrable](https://lightmysky.com/learn/mathematics/the-riemann-integral-and-when-a-function-is-integrable-mt_Ahl2MYIA4N): Define the integral by upper and lower sums and call a function integrable when the two meet. Continuous functions qualify; a function that is discontinuous everywhere need not.
- [Pointwise and Uniform Convergence](https://lightmysky.com/learn/mathematics/pointwise-and-uniform-convergence-mt_iAGk1ML5MA): A sequence of functions can converge at every point and still lose continuity in the limit. Uniform convergence is the stronger condition that preserves continuity and permits term-by-term integration.
- [Triple Integrals and Coordinates for Solids](https://lightmysky.com/learn/mathematics/triple-integrals-and-coordinates-for-solids-mt_RFeK8LD_jT): Integrate over a solid region, and choose cylindrical or spherical coordinates when the solid has an axis or a centre of symmetry.
- [Change of Variables and the Jacobian](https://lightmysky.com/learn/mathematics/change-of-variables-and-the-jacobian-mt_1_LRE2cJft): Change coordinates in a multiple integral, and see the Jacobian determinant as the local factor by which area or volume is stretched.
- [Vector Fields and Line Integrals](https://lightmysky.com/learn/mathematics/vector-fields-and-line-integrals-mt_VZxKEmsVgO): Attach a vector to every point of a region, then integrate along a curve to accumulate work done by the field or mass along a wire.
- [Conservative Fields and Path Independence](https://lightmysky.com/learn/mathematics/conservative-fields-and-path-independence-mt_5djKpI5F33): Some fields are gradients of a potential function. For those, the line integral depends only on the endpoints, and there is a test to recognise them.
- [Green's Theorem in the Plane](https://lightmysky.com/learn/mathematics/greens-theorem-in-the-plane-mt_UDgivmPVEJ): Trade a line integral around a closed plane curve for a double integral of a derivative expression over the region inside it. The first of three theorems with the same shape.
- [Divergence and Curl](https://lightmysky.com/learn/mathematics/divergence-and-curl-mt_JbjW8WKOPE): Two derivative operations on a field: divergence measures net outflow at a point, curl measures local rotation. Both are the language of the theorems that follow.
- [Surface Integrals and Flux](https://lightmysky.com/learn/mathematics/surface-integrals-and-flux-mt_b3FIxOVaTO): Parameterise a surface, build its area element from a cross product, and integrate a field through it to measure flux.
- [Stokes' Theorem and the Divergence Theorem](https://lightmysky.com/learn/mathematics/stokes-theorem-and-the-divergence-theorem-mt_sRojZcrw-q): Both theorems say that integrating a derivative over a region equals integrating the original object over its boundary. With Green's theorem and the fundamental theorem, they are one statement in four settings.
- [Sigma-Algebras and Measurable Sets](https://lightmysky.com/learn/mathematics/sigma-algebras-and-measurable-sets-mt_cpvegazufk): A size cannot be assigned to every subset of the line, so the first move is to fix the collection of sets the size will be defined on. A sigma-algebra is closed under complement and countable union, which is exactly what limit arguments later need.
- [Outer Measure and the Construction of Lebesgue Measure](https://lightmysky.com/learn/mathematics/outer-measure-and-the-construction-of-lebesgue-measure-mt_6u5_Um71wE): Cover a set by intervals, take the cheapest cover, and every subset gets an outer measure. The Caratheodory condition then selects the sets on which that outer measure is additive, and those sets are the measurable ones.
- [Measurable Functions and Approximation by Simple Functions](https://lightmysky.com/learn/mathematics/measurable-functions-and-approximation-by-simple-functions-mt_VwNRjXA7ro): A function is measurable when the preimage of every interval is a measurable set. Every non-negative measurable function is an increasing limit of simple functions, and that staircase is what makes an integral definable in stages.
- [The Lebesgue Integral and What It Repairs](https://lightmysky.com/learn/mathematics/the-lebesgue-integral-and-what-it-repairs-mt_KO92nLc5YU): Define the integral first for simple functions, then as a supremum over the simple functions underneath. Functions the Riemann theory cannot touch become integrable, and the value agrees with the Riemann integral wherever that one exists.
- [Monotone Convergence, Fatou and Dominated Convergence](https://lightmysky.com/learn/mathematics/monotone-convergence-fatou-and-dominated-convergence-mt_7n6ZDkHeBe): Three theorems that say when a limit may be moved inside an integral, each with its own price. They are the reason this integral, and not the earlier one, is the one used in probability and in partial differential equations.
- [Product Measure and Fubini's Theorem](https://lightmysky.com/learn/mathematics/product-measure-and-fubinis-theorem-mt_1VzzVmMjTB): Two measure spaces combine into one, and an integral over the product can be done one variable at a time. The hypotheses of the theorem say exactly when the order of integration is free.
- [Lp Spaces and the Inequalities They Rest On](https://lightmysky.com/learn/mathematics/lp-spaces-and-the-inequalities-they-rest-on-mt_Qd8jU26qgE): Functions with integrable p-th power form a normed space once functions that agree almost everywhere are treated as equal. Holder's and Minkowski's inequalities are what make the norm behave like a length.
- [Normed Spaces, Completeness and Banach Spaces](https://lightmysky.com/learn/mathematics/normed-spaces-completeness-and-banach-spaces-mt_y_jueKnovX): A vector space with a length function, plus the question of whether every Cauchy sequence in it converges. Completeness is what lets an existence proof finish by taking a limit and knowing the limit is still in the space.
- [Bounded Linear Operators and the Operator Norm](https://lightmysky.com/learn/mathematics/bounded-linear-operators-and-the-operator-norm-mt_Qi_mbSNMMx): For a linear map between normed spaces, continuity and boundedness turn out to be the same condition. The operator norm records the worst stretching the map can perform on a unit vector.
- [Baire Category and the Uniform Boundedness Principle](https://lightmysky.com/learn/mathematics/baire-category-and-the-uniform-boundedness-principle-mt_11Rd73trAX): In a complete space, a countable union of nowhere dense sets cannot fill the space. That one fact upgrades pointwise bounds on a family of operators into a single uniform bound, and it is also what makes a surjective bounded operator an open map.
- [The Hahn-Banach Theorem and the Dual Space](https://lightmysky.com/learn/mathematics/the-hahn-banach-theorem-and-the-dual-space-mt_pJuHFxkvjS): A bounded linear functional defined on a subspace extends to the whole space without growing in norm. The dual space these functionals populate is what makes it possible to argue about a vector by testing it against everything.
- [The Fourier Transform on the Line](https://lightmysky.com/learn/mathematics/the-fourier-transform-on-the-line-mt_BMrt5FPe3U): Extend Fourier series from an interval to the whole line, where the sum over discrete frequencies becomes an integral over a continuum.
- [Hilbert Spaces and Orthogonal Projection in Infinite Dimensions](https://lightmysky.com/learn/mathematics/hilbert-spaces-and-orthogonal-projection-in-infinite-dimensions-mt_xmKq8SAgvy): A complete inner-product space keeps the geometry of angles and projections when the dimension is infinite. Closest points to a closed subspace still exist and are unique, and every bounded functional is an inner product with a single fixed vector.
- [Compact Self-Adjoint Operators and the Spectral Theorem](https://lightmysky.com/learn/mathematics/compact-self-adjoint-operators-and-the-spectral-theorem-mt_P_Jr9LWFQj): In infinite dimensions the spectrum is larger than the set of eigenvalues, so the matrix picture cannot be copied over. Compact self-adjoint operators are the case where it survives: an orthonormal basis of eigenvectors, with eigenvalues shrinking to zero.
- [Convolution, Plancherel and Transforming a Derivative](https://lightmysky.com/learn/mathematics/convolution-plancherel-and-transforming-a-derivative-mt_zSwMJrQIMt): Establish the three properties that make the transform useful: convolution becomes multiplication, energy is preserved, and differentiation becomes multiplication by the frequency.
- [Conditioning, Stability and Floating-Point Error](https://lightmysky.com/learn/mathematics/conditioning-stability-and-floating-point-error-mt_q8IoCElXIV): Conditioning says how much the answer moves when the data moves, and stability says how much an algorithm adds on top of that. Rounding error is tiny, and those two amplifiers are what make it visible.