100 written chapters · 0 published teaching videos

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Explore selected university topics across proof, algebra, geometry, analysis, and probability. Each lesson offers a focused entry point with an explicit problem and a worked argument you can reconstruct.

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100 of 100 chapters · in study order

Chapter 1 · Algebra
Why the order of quantifiers changes a claim
Distinguish a uniform choice from a choice that depends on the input.
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Chapter 2 · Algebra
Build a strong-induction postage argument
Use several base cases to support a recurrence that moves by four.
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Chapter 3 · Algebra
Compress integers into residue classes
Verify an equivalence relation and perform well-defined arithmetic on classes.
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Chapter 4 · Algebra
Recover a greatest common divisor as a linear combination
Run Euclid's algorithm and reverse it to obtain Bézout coefficients.
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Chapter 5 · Algebra
Undo multiplication modulo a prime
Find a modular inverse through a Bézout identity rather than trial division.
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Chapter 6 · Algebra
Combine two independent remainder schedules
Solve simultaneous congruences and identify the repetition period.
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Chapter 7 · Algebra
Reuse elimination with an LU factorization
Separate elimination from back substitution to solve repeated linear systems.
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Chapter 8 · Algebra
Find all invisible inputs of a linear map
Parameterize a nullspace and connect free variables to lost information.
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Chapter 9 · Algebra
Describe the same vector in another coordinate system
Solve for coordinates relative to a nonstandard basis.
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Chapter 10 · Algebra
Turn differentiation into a matrix
Represent an operator using its action on basis vectors.
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Chapter 11 · Algebra
Separate a signal from its perpendicular residual
Compute a least-distance projection onto a line.
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Chapter 12 · Algebra
Create perpendicular directions without changing the span
Orthogonalize independent vectors by subtracting a projection.
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Chapter 13 · Algebra
Fit a constant when measurements disagree
Derive normal equations from an orthogonal residual condition.
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Chapter 14 · Algebra
Find directions a coupled system preserves
Compute eigenvalues and eigenvectors of a symmetric coupling matrix.
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Chapter 15 · Algebra
Compute a large matrix power through its eigenbasis
Use diagonalization to avoid repeated matrix multiplication.
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Chapter 16 · Algebra
Keep the dominant direction of a rectangular map
Read singular values from AᵀA and interpret a rank-one approximation.
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Chapter 17 · Algebra
Certify that a quadratic energy is strictly positive
Complete squares to test positive definiteness.
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Chapter 18 · Algebra
Separate a small residual from a trustworthy solution
Measure how unequal matrix scales amplify input errors.
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Chapter 19 · Algebra
Track composition order in permutations
Compute a permutation product and recognize noncommutativity.
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Chapter 20 · Algebra
Count subgroup cosets in a finite symmetry group
Use disjoint equal-size cosets to constrain subgroup orders.
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Chapter 21 · Algebra
Find the information lost by a group homomorphism
Compute an image and kernel for reduction modulo four.
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Chapter 22 · Algebra
Separate a polynomial quotient from its remainder
Use division to reduce a polynomial modulo a quadratic.
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Chapter 23 · Algebra
Calculate in a finite field with four elements
Construct an extension field using an irreducible quadratic.
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Chapter 24 · Algebra
Turn a network into an energy matrix
Build a graph Laplacian and connect its quadratic form to edge differences.
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Chapter 25 · Algebra
Locate an optimum among feasible polygon corners
Solve a small linear program by checking the geometry of its constraints.
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Chapter 26 · Geometry
Compare distance rules and their unit balls
Distinguish Euclidean and taxicab metrics through their geometric balls.
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Chapter 27 · Geometry
Separate translation from a linear transformation
Compose an affine map using a matrix and a displacement.
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Chapter 28 · Geometry
Locate a point through triangle weights
Use barycentric coordinates to recognize points inside a triangle.
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Chapter 29 · Geometry
Recover area and orientation from a cross product
Use an oriented area vector to distinguish magnitude from direction.
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Chapter 30 · Geometry
Measure perpendicular distance to a plane
Project a displacement onto the unit normal of a plane.
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Chapter 31 · Geometry
Decide whether spatial lines intersect
Distinguish parallel, intersecting, and skew lines using coordinate equations.
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Chapter 32 · Geometry
Remove a cross term by rotating coordinates
Diagonalize a quadratic curve using principal directions.
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Chapter 33 · Geometry
Calculate length along a parameterized curve
Integrate speed rather than endpoint displacement.
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Chapter 34 · Geometry
Measure how quickly a curve changes direction
Compute curvature of a graph at a specified point.
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Chapter 35 · Geometry
Separate spatial twisting from bending
Compute curvature and torsion for a circular helix.
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Chapter 36 · Geometry
Find shortest spherical travel through a central angle
Convert a dot product into great-circle distance.
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Chapter 37 · Geometry
Connect a spherical triangle's angles to its area
Apply spherical excess on a unit sphere.
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Chapter 38 · Geometry
Test openness relative to the surrounding space
Distinguish a subset's geometry from its ambient topology.
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Chapter 39 · Geometry
Know when continuity guarantees an extreme value
Use compactness to distinguish attained maxima from mere bounds.
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Chapter 40 · Geometry
Count a topological invariant of a polyhedral surface
Calculate Euler characteristic and distinguish it from a metric measurement.
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Chapter 41 · Trigonometry
Separate periodic signals by orthogonality
Use a full-period integral to distinguish different harmonic modes.
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Chapter 42 · Trigonometry
Resolve four samples into discrete frequency components
Compute a small discrete Fourier transform with an explicit sign convention.
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Chapter 43 · Trigonometry
See how different waves produce identical samples
Derive aliasing from sampled complex phases.
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Chapter 44 · Trigonometry
Convert a forced oscillation into a complex amplitude
Solve a linear steady sinusoidal response using phasors.
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Chapter 45 · Trigonometry
Generate polynomial identities from cosine addition
Derive Chebyshev polynomials from a three-term recurrence.
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Chapter 46 · Statistics
Correct double counting in event unions
Use inclusion–exclusion without assuming independence.
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Chapter 47 · Statistics
Update odds with a likelihood ratio
Separate prior odds from the evidential strength of an observation.
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Chapter 48 · Statistics
Model sampling without replacement
Count samples with a fixed number of selected target items.
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Chapter 49 · Statistics
Read probabilities from a generating polynomial
Use a probability generating function to combine independent Bernoulli trials.
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Chapter 50 · Statistics
Understand a waiting time that forgets failures
Derive memorylessness for independent repeated Bernoulli trials.
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Chapter 51 · Statistics
Combine independent streams of rare events
Add independent Poisson counts and their rates.
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Chapter 52 · Statistics
Find which of two independent clocks rings first
Derive the distribution of a minimum exponential waiting time.
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Chapter 53 · Statistics
Transform a continuous random variable carefully
Include a Jacobian when changing the variable of a density.
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Chapter 54 · Statistics
Normalize probability on a triangular region
Integrate a joint density over its actual support.
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Chapter 55 · Statistics
Average conditional means with the correct weights
Apply the law of total expectation to a mixed population.
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Chapter 56 · Statistics
Separate within-group noise from between-group differences
Use the law of total variance to expose two sources of variation.
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Chapter 57 · Statistics
Derive a triangular distribution from two uniforms
Compute a convolution using overlap length.
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Chapter 58 · Statistics
Put an average on the central-limit scale
Standardize a sample mean using its standard error.
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Chapter 59 · Statistics
Prove concentration of an average without a normal model
Use Chebyshev's inequality to obtain a distribution-free variance bound.
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Chapter 60 · Statistics
Estimate a rate by maximizing its likelihood
Differentiate a log-likelihood and check its maximum.
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Chapter 61 · Statistics
Estimate an endpoint by matching a theoretical moment
Construct a method-of-moments estimator and compare it with a support constraint.
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Chapter 62 · Statistics
Compare two parameter values using observed evidence
Compute a likelihood ratio while keeping hypotheses distinct from posterior probabilities.
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Chapter 63 · Statistics
Quantify local information about a Bernoulli probability
Compute score variance for a single Bernoulli observation.
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Chapter 64 · Statistics
Check a fitted line through orthogonal residuals
Verify least-squares normal equations in a small regression.
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Chapter 65 · Statistics
Verify stationarity through balanced transitions
Solve and check a stationary distribution of a two-state Markov chain.
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Chapter 66 · Calculus
Turn a limit claim into an explicit error guarantee
Choose a delta that forces a requested output tolerance.
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Chapter 67 · Calculus
Check convergence through distances between late terms
Verify the Cauchy property independently of naming a limit.
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Chapter 68 · Calculus
Use an invariant interval to prove a recurrence converges
Combine monotonicity and boundedness before solving a fixed-point equation.
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Chapter 69 · Calculus
Detect nonconvergence through incompatible subsequences
Use subsequential limits to disprove convergence.
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Chapter 70 · Calculus
Distinguish cancellation from absolute summability
Compare a convergent alternating series with the sum of its magnitudes.
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Chapter 71 · Calculus
Watch a sequence converge pointwise but not uniformly
Compare pointwise behavior with a worst-case error over a whole interval.
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Chapter 72 · Calculus
Find a continuous function with a corner
Compare one-sided difference quotients at a nondifferentiable point.
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Chapter 73 · Calculus
Construct an integral from finite measurement strips
Evaluate a right-endpoint Riemann-sum limit explicitly.
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Chapter 74 · Calculus
Differentiate accumulation with two moving boundaries
Apply the fundamental theorem and chain rule to both integral endpoints.
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Chapter 75 · Calculus
Track a field along a moving trajectory
Differentiate a composite scalar field through all moving coordinates.
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Chapter 76 · Calculus
Find the direction of fastest local increase
Maximize a directional derivative under a unit-length constraint.
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Chapter 77 · Calculus
Classify a critical point through second-order geometry
Use Hessian eigenvalues to distinguish a minimum from a saddle.
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Chapter 78 · Calculus
Optimize while staying on a constraint
Use parallel gradients to find constrained extrema.
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Chapter 79 · Calculus
Integrate a radial quantity over a disk
Include the polar area factor in a double integral.
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Chapter 80 · Calculus
Measure how a coordinate map changes area
Use an absolute determinant as an area scaling factor.
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Chapter 81 · Calculus
Replace a path integral with endpoint values
Find a potential and verify path independence.
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Chapter 82 · Calculus
Turn boundary circulation into interior curl
Use Green's theorem with a consistent orientation.
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Chapter 83 · Calculus
Compute outward flux by measuring sources inside
Apply the divergence theorem to a closed surface.
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Chapter 84 · Calculus
Match curl flux with circulation around an edge
Use Stokes' theorem with compatible boundary orientation.
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Chapter 85 · Calculus
Solve a linear ODE by building a product derivative
Choose an integrating factor that combines two terms into one derivative.
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Chapter 86 · Calculus
Distinguish exact equilibrium from growth toward one
Solve a logistic equation and interpret its carrying-capacity limit.
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Chapter 87 · Calculus
Use two initial conditions in a second-order motion model
Solve an undamped oscillator and verify conserved energy.
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Chapter 88 · Calculus
Check whether a numerical decay method actually decays
Derive the step-size stability condition for forward Euler.
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Chapter 89 · Calculus
Use a midpoint slope to improve one numerical step
Compare a second-order Runge–Kutta update with the local Taylor expansion.
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Chapter 90 · Calculus
Linearize a nonlinear root problem and inspect the update
Derive Newton's iteration and check a concrete root approximation.
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Chapter 91 · Calculus
Guarantee a root bracket after a fixed number of steps
Bound bisection error through interval contraction.
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Chapter 92 · Calculus
Balance truncation error against measurement noise
Derive the accuracy of a centered derivative approximation.
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Chapter 93 · Calculus
Test incompressibility without requiring zero velocity
Compute divergence to distinguish volume preservation from motionlessness.
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Chapter 94 · Calculus
Separate change at a point from change along a moving particle
Calculate a material derivative by adding local and advective change.
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Chapter 95 · Calculus
Connect a velocity gradient with Newtonian shear stress
Distinguish dynamic viscosity, kinematic viscosity, and shear rate.
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Chapter 96 · Calculus
Compare inertial and viscous scales without declaring a regime automatically
Form a dimensionless Reynolds number from characteristic scales.
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Chapter 97 · Calculus
Verify an exact flow between moving plates
Check velocity, boundary conditions, and every term of a simple steady Navier–Stokes solution.
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Chapter 98 · Calculus
Balance a pressure drop with viscous curvature
Derive a parabolic velocity profile between stationary plates.
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Chapter 99 · Calculus
Check viscous energy loss in an exact periodic flow
Relate a decaying Fourier shear mode to its kinetic-energy dissipation.
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Chapter 100 · Calculus
Build incompressibility into a two-dimensional velocity field
Generate divergence-free planar motion from a streamfunction.
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Explore the complete written study sequence

Chapters and video availability

Why the order of quantifiers changes a claim

Build a strong-induction postage argument

Compress integers into residue classes

Recover a greatest common divisor as a linear combination

Undo multiplication modulo a prime

Combine two independent remainder schedules

Reuse elimination with an LU factorization

Find all invisible inputs of a linear map

Describe the same vector in another coordinate system

Turn differentiation into a matrix

Separate a signal from its perpendicular residual

Create perpendicular directions without changing the span

Fit a constant when measurements disagree

Find directions a coupled system preserves

Compute a large matrix power through its eigenbasis

Keep the dominant direction of a rectangular map

Certify that a quadratic energy is strictly positive

Separate a small residual from a trustworthy solution

Track composition order in permutations

Count subgroup cosets in a finite symmetry group

Find the information lost by a group homomorphism

Separate a polynomial quotient from its remainder

Calculate in a finite field with four elements

Turn a network into an energy matrix

Locate an optimum among feasible polygon corners

Compare distance rules and their unit balls

Separate translation from a linear transformation

Locate a point through triangle weights

Recover area and orientation from a cross product

Measure perpendicular distance to a plane

Decide whether spatial lines intersect

Remove a cross term by rotating coordinates

Calculate length along a parameterized curve

Measure how quickly a curve changes direction

Separate spatial twisting from bending

Find shortest spherical travel through a central angle

Connect a spherical triangle's angles to its area

Test openness relative to the surrounding space

Know when continuity guarantees an extreme value

Count a topological invariant of a polyhedral surface

Separate periodic signals by orthogonality

Resolve four samples into discrete frequency components

See how different waves produce identical samples

Convert a forced oscillation into a complex amplitude

Generate polynomial identities from cosine addition

Correct double counting in event unions

Update odds with a likelihood ratio

Model sampling without replacement

Read probabilities from a generating polynomial

Understand a waiting time that forgets failures

Combine independent streams of rare events

Find which of two independent clocks rings first

Transform a continuous random variable carefully

Normalize probability on a triangular region

Average conditional means with the correct weights

Separate within-group noise from between-group differences

Derive a triangular distribution from two uniforms

Put an average on the central-limit scale

Prove concentration of an average without a normal model

Estimate a rate by maximizing its likelihood

Estimate an endpoint by matching a theoretical moment

Compare two parameter values using observed evidence

Quantify local information about a Bernoulli probability

Check a fitted line through orthogonal residuals

Verify stationarity through balanced transitions

Turn a limit claim into an explicit error guarantee

Check convergence through distances between late terms

Use an invariant interval to prove a recurrence converges

Detect nonconvergence through incompatible subsequences

Distinguish cancellation from absolute summability

Watch a sequence converge pointwise but not uniformly

Find a continuous function with a corner

Construct an integral from finite measurement strips

Differentiate accumulation with two moving boundaries

Track a field along a moving trajectory

Find the direction of fastest local increase

Classify a critical point through second-order geometry

Optimize while staying on a constraint