Explainer study library

Who it is for

This library is for readers who can follow a first calculus course and still feel unsure what a definition is asking. That includes undergraduates early in a degree, and self learners who finished AP Calculus AB, or a course like it, and want the next ideas without treating one page as a syllabus.

If algebra and graphs are the actual gap, start on calculus foundations or the glossary. Stokes and Lagrange are the wrong first door. The prose here is original to this site, and it is educational only. It is not a substitute for coursework, a textbook, or a tutor. See the disclaimer. When a page stops on purpose, the sources list points at open books and course notes.

How to use an explainer

Use each page in three passes. First, read for the picture. Leave the calculator closed until you can say, in one sentence, what its output should mean. Second, write the checkpoint on paper, with a small example you invent if you can. Third, open the linked calculator or a glossary anchor only to check the sentence you already wrote.

After the idea holds, put it back on a topic map so you can see what comes before and after. If you want sittings with checkpoints you produce, switch to a pathway. The FAQ on how to use the site describes the same loop: topic, explainer, glossary, calculator, then a pathway.

A useful sitting is short. Stop at the checkpoint. Nodding along is not the same as holding the idea, and a row of open tabs is not a study order. The clusters below are that order. They use only pages that already live here. Linear algebra sits before several variables because a Jacobian is a matrix. Probability and differential equations are separate doors. If you are on the AP Calculus AB bridge, stay in the calculus cluster, then read the separable page, and treat matrices as a different route.

Cluster

Calculus foundations

This is the usual first door when the work in front of you is change, accumulation, or the meaning under rules you already compute. The map is calculus foundations. The sittings, if you want them, are the AP Calculus AB bridge. Pair the derivative page with the numerical derivative calculator, epsilon delta with the numerical limit estimator, integrals with the Riemann sum calculator and the partial fraction decomposition calculator, and series with the sequence calculator and the Taylor polynomial calculator. That sequence tool adds finitely many terms. It does not decide whether an infinite sum converges. The Taylor tool builds the degree-n polynomial and a remainder bound. It is not a convergence proof either.

1. What a derivative actually measures. Start here when the rules are memorized and the meaning is thin. Units, the best local line, and the graphs where a slope fails.
2. Implicit differentiation intuition. The curve is an equation, not a formula that already solves for y. The chain rule finds the slope the equation was hiding.
3. Epsilon delta intuition without terror. The limit language under those slopes. Skip it if you only need computational calculus this month. It is central if the next course asks for arguments. The intro to proofs pathway and the real analysis topic both use this page as writing practice.
4. Sequences, series, and when infinite sums make sense. Partial sums first. The word convergence only after you can say what is being limited.
5. Convergence tests for series. Comparison, ratio, and root tests, plus the cases where a test stays silent. Read this after the sequences page, not instead of it.

Checkpoint. Explain an average rate and an instantaneous rate with units. Say what epsilon and delta are negotiating. Name one series test, and one case where that test stays silent.

Cluster

Linear algebra

One explainer, on purpose. The surrounding order lives on the linear algebra topic and the linear algebra first course. You can start this cluster without calculus. Read eigenvalues in plain language after you can multiply a matrix by a vector and picture a stretch along a line. The eigenvalue is the stretch factor on a direction the map refuses to tip onto a different line.

Then use the 2 by 2 eigenvalue calculator. Check that the eigenvalues sum to the trace and multiply to the determinant before you trust the screen. The 2 by 2 matrix calculator is for the determinant, the inverse, and a product. The vector operations calculator is for the geometry of a single vector. When the matrix is 3 by 3, the 3 by 3 matrix calculator expands a determinant and an inverse, the reduced row echelon form calculator writes elimination, and the 3x3 eigenvalue calculator expands the characteristic polynomial. Orthogonal bases are the Gram Schmidt calculator, and elimination factors are the LU decomposition calculator. Glossary anchors: eigenvalue, eigenvector, matrix, vector.

Checkpoint. On a 2 by 2 example you compute by hand, name an eigenvalue and a matching eigenvector, and pass both the trace check and the determinant check.

Cluster

Multivariable and vector calculus

Several variables, after single variable calculus is steady and after you can read a matrix. There is no separate multivariable pathway. The preparation is the linear algebra cluster above, through eigenvalues. The map is multivariable and vector calculus. Practice tools are the matrix calculator and the vector operations calculator.

1. Jacobian matrices in plain language. The best linear map near a point when several inputs move together. Compute the partials on the Jacobian matrix calculator. If matrix language is the gap, read eigenvalues first. If the phrase "best linear map" feels empty, repair it on what a derivative measures, then return.
2. Lagrange multipliers intuition. After you trust gradients. The multiplier measures how the extreme value responds when the constraint level changes a little. It is not a spare symbol with no job. Glossary: gradient, Lagrange multiplier, constraint.
3. Stokes theorem intuition. Circulation around a rim equals the curl through a surface, for a first undergraduate look. Use the vector calculator for a dot product or a cross product while you read. Do not start the cluster on this page. Glossary: Stokes theorem, vector field.

Checkpoint. Say what a Jacobian approximates, what a Lagrange multiplier measures, and what Stokes equates, in words, before you reach for a symbol.

Cluster

Probability

Also one explainer. Calculus is not a gate. The map is probability and statistics, and that pillar is the order, because there is no separate probability pathway. Read Bayes theorem without the fog from a table of counts you can draw, before any formula. The two directions of a conditional probability are not the same sentence.

After the model has a name, the Bayes theorem calculator turns a prior and two likelihoods into a posterior. The binomial probability calculator or the normal CDF calculator can evaluate a named count or a continuous tail. The combinatorics calculator only counts. It does not update a belief. Glossary: Bayes theorem, conditional probability, probability.

Checkpoint. From a small table of counts, compute both directions of the conditional probability and say which one the question asked for.

Cluster

Differential equations

Read separable differential equations when the unknown is a function and the equation relates that function to its derivative. The map is differential equations first look. On the AP Calculus AB bridge, this page comes after the derivative and integral steps, not before them.

Separate, integrate, and list constant solutions you might lose by dividing. Then differentiate your answer and recover the original equation. The numerical derivative calculator can audit the slope of an explicit solution you already found. The Riemann sum calculator practices the integral that appears after you separate. A constant-coefficient equation is a different family. The second order differential equation calculator writes the characteristic equation, the Wronskian calculator tests linear independence of the solutions, and the Laplace transform calculator applies a short table and inverts a proper rational transform. Glossary: differential equation, initial value problem, slope field.

Checkpoint. For one equation, show a constant solution that division would hide, and show the derivative check that recovers the original right hand side.

When two clusters both fit

If you can differentiate and then freeze on the sentence "for every epsilon there exists a delta," finish what a derivative measures, then epsilon delta intuition, then the intro to proofs pathway. The calculus bridge gets you the meaning. The proofs pathway gets you the writing.

If the next course is multivariable calculus, do eigenvalues before the Jacobian, Lagrange after gradients, and Stokes last. Starting at Stokes makes the earlier pages look like a vocabulary list.

If the interesting object is a rate that depends on the current amount, read the derivative page only until slope language feels solid, then the separable explainer. Series can wait unless the course lists them. If the interesting object is a conditional probability, stay on Bayes. A derivative will not sort out which conditional you were asked for.

How to know you understood

A checkpoint is something you produce, not a mood. The cluster notes above are the specific tests. These four work on any page in the library.

• Teach the idea with a small example you invent. If you can only retell the example printed on the page, read it again.
• Say what the idea is not. The derivative page and the Bayes page both spend space on the wrong reading, because the wrong reading is the usual one.
• Name the calculator output in a sentence before you look. Units, or a comparison you expect, count as a sentence. The FAQ on calculators is the same warning in shorter form.
• Point at the topic map and say which page should come next. If you cannot, you are still inside the article. That is a fine place to be. It is not a reason to hop clusters.

Common pitfalls when reading math explainers

These pages fail a reader in predictable ways, even when the paragraph is clear.

• The picture stands in for the definition. The opening image gets you in the door. The definition is the sentence you can check against an example, including an example where the claim fails.
• Units get skipped. A derivative without units is a slope with no story. A probability without a count is easy to reverse. If you cannot name the unit, you do not yet know what the number measures.
• The calculator opens first. A numerical derivative with a large step is an average rate. Calling that number the derivative undoes the page you came to read. A central difference at a corner can even print a calm value that is not a derivative at all. The derivative explainer shows that trap with absolute value at zero.
• The advanced title wins. Stokes is not preparation for the derivative. It assumes the derivative, and it assumes more than that. Pick the cluster that matches the course, not the cluster that sounds further along.
• Recognition gets mistaken for production. Following a paragraph is not writing the checkpoint. Close the page and try the checkpoint from a blank sheet before you call the sitting finished.
• One explainer gets treated as the course. Each page stops on purpose. When you need every theorem, use the sources list, or the pathway that orders several pages into sittings.

Topic maps, pathways, and tools

The clusters point at maps that already exist. Use them when you want the neighborhood, not another long read.

• Topic maps: calculus foundations, linear algebra, multivariable and vector calculus, probability and statistics, differential equations, and real analysis and proof habits. The topics hub says how a pillar differs from a pathway.
• Pathways: the AP Calculus AB bridge, the linear algebra first course, and intro to proofs. Compare audience and time shape on the pathways hub.
• Calculators, after the sentence is written: the calculator index.
• Short definitions: the glossary.

Quick shelf

All eleven pages

Same library, listed in the cluster order above, if you already know which door you want.