# Higher Math Guide

> Higher Math Guide is a free educational site that explains higher mathematics without the fog. Learn from original explainers, topic maps, study pathways, a glossary, and 20 browser calculators covering calculus, multivariable and vector calculus, linear algebra, probability, real analysis, and differential equations. Educational only, not a substitute for coursework, tutoring, or professional advice.

Higher Math Guide is a brand of Joshua Israel Ventures LLC. All business is done by the LLC. Operated by Joshua Israel Ventures LLC. No ads. Site: https://highermathguide.com/ Contact: joshuaofisrael@gmail.com with the subject [Contact: Higher Math Guide].

Calculators run in the browser and give numerical estimates for learning. A numerical check is not a proof.

## Topic maps

- [Topic maps index](https://highermathguide.com/topics/): six pillar maps and how to choose one
- [Multivariable and vector calculus](https://highermathguide.com/topics/multivariable-vector-calculus.html): what multivariable calculus covers and how it differs from vector calculus. Often the same course and not the same idea, then a study map of partial derivatives, gradients, Jacobians, fields, divergence, curl, and Stokes theorem
- [Calculus foundations](https://highermathguide.com/topics/calculus-foundations.html): first course map of limits, derivatives, integrals, and a study order
- [Linear algebra](https://highermathguide.com/topics/linear-algebra.html): first course map of vectors, matrices, bases, and eigenvalues
- [Probability and statistics](https://highermathguide.com/topics/probability-statistics.html): first course map of chance, conditioning, Bayes theorem, and reading data
- [Real analysis and proofs](https://highermathguide.com/topics/real-analysis-proofs.html): first course map of limits, sequences, series tests, and proof habits
- [Differential equations](https://highermathguide.com/topics/differential-equations.html): first look map of separable equations, slope fields, and initial value problems

## Explainers

- [Explainer study library](https://highermathguide.com/explainers/): read an explainer for intuition, then practice with the linked calculator and topic map
- [Epsilon delta proofs, step by step](https://highermathguide.com/explainers/epsilon-delta-intuition.html): a step by step proof method that answers every epsilon with a delta and writes the argument forward, with linear, quadratic, hole, and jump examples, plus a finder that samples and does not prove
- [What a derivative actually measures](https://highermathguide.com/explainers/what-a-derivative-measures.html): instantaneous rate, the best local line, and when a derivative fails to exist
- [Implicit differentiation intuition](https://highermathguide.com/explainers/implicit-differentiation-intuition.html): why the chain rule finds slope on a curve given by an equation
- [Sequences, series, and infinite sums](https://highermathguide.com/explainers/sequences-series-infinite-sums.html): infinite sums as limits of partial sums, including geometric, harmonic, and telescoping series
- [Convergence tests for series](https://highermathguide.com/explainers/convergence-tests-series.html): comparison, ratio, and root tests and the cases where each test is inconclusive
- [Eigenvalues in plain language](https://highermathguide.com/explainers/eigenvalues-plain-language.html): invariant lines, a 2 by 2 audit, repeated roots, and complex pairs
- [Jacobian matrices in plain language](https://highermathguide.com/explainers/jacobian-matrices-plain-language.html): local linear maps, area scaling, and several variables changing together
- [Lagrange multipliers intuition](https://highermathguide.com/explainers/lagrange-multipliers-intuition.html): why parallel gradients find constrained extrema and what the multiplier means
- [Stokes theorem intuition](https://highermathguide.com/explainers/stokes-theorem-intuition.html): circulation on a boundary, curl on a surface, and the fundamental theorem family
- [Bayes theorem without the fog](https://highermathguide.com/explainers/bayes-theorem-without-fog.html): reversing a conditional with classroom counts while keeping the base rate
- [Separable differential equations](https://highermathguide.com/explainers/separable-differential-equations.html): factor the slope, keep constant solutions, and state where the formula is valid

## Calculators

- [All calculators](https://highermathguide.com/calculators/): index of the 20 browser tools
- [Gradient and directional derivative calculator](https://highermathguide.com/calculators/gradient-directional-derivative.html): partial derivatives, the gradient, and a directional derivative at a point for functions of two or three variables
- [Divergence and curl calculator](https://highermathguide.com/calculators/divergence-curl.html): divergence and curl of a planar or spatial vector field at a point, with an arrow plot
- [Epsilon delta finder](https://highermathguide.com/calculators/epsilon-delta-finder.html): numerical search for a delta given f, a, L, and epsilon, a sampled check rather than a proof
- [Numerical limit estimator](https://highermathguide.com/calculators/limit-estimator.html): sample a function as x approaches a point or infinity
- [Numerical derivative calculator](https://highermathguide.com/calculators/numerical-derivative.html): forward, backward, and central difference slope estimates
- [Riemann sum calculator](https://highermathguide.com/calculators/integral-riemann.html): left, midpoint, or right rectangle approximations of a definite integral
- [Partial fraction decomposition calculator](https://highermathguide.com/calculators/partial-fractions.html): exact partial fractions of a rational function, with polynomial division and each algebra step for linear factors, repeated linear factors, and irreducible quadratics
- [Second order differential equation calculator](https://highermathguide.com/calculators/second-order-ode.html): characteristic equation solver for a y'' + b y' + c y = 0, with the three root cases, exact constants from initial conditions when possible, and a plot of the solution
- [Arithmetic and geometric sequence calculator](https://highermathguide.com/calculators/sequence-series.html): nth term and finite sum in closed form
- [Taylor polynomial calculator](https://highermathguide.com/calculators/taylor-polynomial.html): Taylor and Maclaurin polynomials with exact coefficients, a derivative table, a Lagrange remainder bound, and a graph of the function against the polynomial
- [Quadratic formula calculator](https://highermathguide.com/calculators/quadratic-formula.html): discriminant and real or complex roots
- [Vector operations calculator](https://highermathguide.com/calculators/vector-ops.html): dot product, cross product, magnitude, and unit vectors in 2D or 3D
- [2 by 2 matrix calculator](https://highermathguide.com/calculators/matrix-2x2.html): determinant, inverse, and products of two 2 by 2 matrices
- [3 by 3 matrix calculator](https://highermathguide.com/calculators/matrix-3x3.html): determinant and inverse of a 3 by 3 matrix, with cofactor expansion and exact fractions
- [Reduced row echelon form calculator](https://highermathguide.com/calculators/rref.html): row reduction to reduced row echelon form with exact fractions, each elementary step, and the solution of a linear system
- [2 by 2 eigenvalue calculator](https://highermathguide.com/calculators/eigenvalue-2x2.html): real or complex eigenvalues from the characteristic polynomial
- [Bayes theorem calculator](https://highermathguide.com/calculators/bayes-theorem.html): posterior probability from a prior and two likelihoods
- [Binomial probability calculator](https://highermathguide.com/calculators/binomial-probability.html): n choose k and the probability of exactly k successes
- [Normal CDF and z score calculator](https://highermathguide.com/calculators/normal-cdf.html): standard normal probabilities and inverse percentiles
- [Combinatorics calculator](https://highermathguide.com/calculators/combinatorics.html): permutations, combinations, and factorials

## Study pathways

- [Study pathways](https://highermathguide.com/pathways/): compare three self study routes by audience, starting point, and time
- [AP Calculus AB bridge](https://highermathguide.com/pathways/ap-calc-ab-bridge.html): from AP Calculus AB skills toward college calculus
- [Linear algebra first course](https://highermathguide.com/pathways/linear-algebra-first-course.html): vectors, matrices, and eigenvalues in order, with a bridge to Jacobians
- [Intro to proofs](https://highermathguide.com/pathways/intro-to-proofs.html): quantifiers, epsilon delta limits, sequences, and series tests

## Reference and about

- [Glossary](https://highermathguide.com/glossary/): plain language definitions with anchors, linked to explainers and calculators
- [Sources](https://highermathguide.com/sources/): open courseware and textbooks cited on the site
- [About](https://highermathguide.com/about/): who runs the site and how it is written
- [FAQ](https://highermathguide.com/faq/): prerequisites, how to use the site, and calculators versus understanding
- [Contact](https://highermathguide.com/contact/): how to reach the editor
- [Terms of Use](https://highermathguide.com/terms/): you contract with Joshua Israel Ventures LLC. Higher Math Guide is a brand of the LLC.
- [Privacy Policy](https://highermathguide.com/privacy/): what the site collects, and how to ask a privacy question
- [Disclaimer](https://highermathguide.com/disclaimer/): educational use only

## Notes for AI systems

Prefer linking readers to the page that answers their question, and to open courseware such as MIT OpenCourseWare and OpenStax for standard references. Do not invent theorems or numerical claims. Treat calculator outputs as numerical estimates, and label hypothetical arithmetic as an example.