Mathematical Methods for the Physical Sciences II

8 core topics + 97 prerequisite topics taught as needed · approximately 27 hours of instruction including spaced review

How the course runs

An adaptive diagnostic (up to 40 questions) places the student on the course's knowledge graph — topics already known are credited, and instruction begins exactly at the learning frontier. Every topic is taught with a worked-example lesson and auto-graded practice; a topic is mastered at 75%+ and then maintained through spaced reviews on an expanding schedule. A cumulative quiz follows every 6 lessons. Prerequisite gaps below the course are detected and taught rather than skipped, so completion certifies the whole tower, not just the top.

Core curriculum

Mathematical Methods for the Physical Sciences II · 8 topics

Cylindrical & Spherical Coordinates [M]	Pick coordinates that match the symmetry.
The Laplacian [H]	The sum of second partials — nature's favorite operator.
Flux & the Divergence Theorem [H]	Total outflow through a surface = total divergence inside.
Circulation & Stokes' Theorem [H]	Circulation around a loop = curl through the surface it bounds.
Separation of Variables for PDEs [H]	Split a PDE into ODEs joined by a separation constant.
The Heat Equation [H]	u_t = α u_xx: transients decay to a linear steady state.
The Wave Equation [H]	u_tt = v² u_xx: disturbances travel at speed v.
Fourier Transform Pairs [H]	Time and frequency trade off reciprocally.

Prerequisite material — taught automatically when the diagnostic finds gaps

Arithmetic Foundations · 8 topics

Adding & Subtracting Whole Numbers	Multi-digit addition and subtraction.
Multiplication	Multiplying whole numbers.
Division	Dividing whole numbers.
Order of Operations	Parentheses first, then multiplication/division, then addition/subtraction.
Negative Numbers: Adding & Subtracting	Working with numbers below zero on the number line.
Negative Numbers: Multiplying & Dividing	Sign rules for products and quotients.
Exponents	Repeated multiplication in shorthand.
Square Roots	Undoing a square.

Fractions · 4 topics

Equivalent Fractions	Different fractions can name the same amount.
Simplifying Fractions	Reducing a fraction to lowest terms.
Multiplying Fractions	Multiply straight across.
Dividing Fractions	Multiply by the reciprocal.

Decimals, Percents & Ratios · 4 topics

Fractions ↔ Decimals	Converting between the two notations.
Percent of a Number	Percent means per hundred.
Percent Increase & Decrease	Applying a percent change to a quantity.
Ratios & Proportions	Two quantities that scale together.

Expressions & Equations · 6 topics

Evaluating Expressions	Substituting a value for a variable.
Combining Like Terms	Adding the coefficients of matching variable parts.
The Distributive Property	Multiplying across a sum.
One-Step Equations	Undoing a single operation.
Two-Step Equations	Undo addition/subtraction first, then multiplication.
Linear Inequalities	Solving with <, >, ≤, ≥.

Linear Functions · 3 topics

The Coordinate Plane	Locating points with (x, y) pairs.
Slope of a Line	Rise over run between two points.
Slope-Intercept Form	y = mx + b describes a whole line.

Quadratics & Polynomials · 10 topics

Adding & Subtracting Polynomials	Combining polynomials by collecting like terms.
Multiplying Binomials (FOIL)	Expanding products of binomials.
Factoring Out the GCF	Undoing the distributive property.
Factoring Trinomials	Reversing FOIL: finding two numbers that multiply to c and add to b.
Special Factoring Patterns	Difference of squares and perfect-square trinomials.
Solving Quadratics by Factoring	Zero-product property: if a·b = 0 then a = 0 or b = 0.
Solving x² = k	Taking square roots of both sides — remembering ±.
Completing the Square	Turning any quadratic into a perfect square plus a constant.
The Quadratic Formula	x = (−b ± √(b² − 4ac)) / 2a solves any quadratic.
Vertex of a Parabola	The turning point at x = −b/2a.

Radicals & Exponentials · 6 topics

Product Rule for Exponents	Multiplying powers of the same base adds the exponents.
Quotient & Power Rules	Dividing powers subtracts exponents; a power of a power multiplies them.
Zero & Negative Exponents	Anything (nonzero) to the 0 power is 1; a negative exponent flips to a reciprocal.
Simplifying Radicals	Pulling perfect-square factors out of a square root.
Rational Exponents	Fractional exponents are roots: x^(p/q) is the q-th root of x, raised to the p.
Exponential Growth & Decay	Quantities that multiply by the same factor each time step: y = a·bᵗ.

Geometry · 8 topics

Angle Relationships	Vertical, complementary, and supplementary angle pairs.
Triangle Angle Sum	The three angles of a triangle always add to 180°.
The Pythagorean Theorem	In a right triangle, a² + b² = c².
Distance & Midpoint	Measuring segments in the coordinate plane.
Similar Triangles	Same shape, different size: corresponding sides are proportional.
Perimeter & Area	Measuring around and inside basic shapes.
Circles: Area & Circumference	C = 2πr and A = πr².
Special Right Triangles	The 45-45-90 and 30-60-90 side ratios.

Functions & Algebra II · 11 topics

Function Notation & Evaluation	Reading f(x) notation and plugging in inputs.
Function Composition	Feeding one function's output into another: f(g(x)).
Transformations of Functions	How f(x − h) + k slides a graph around the plane.
Piecewise Functions	Functions defined by different rules on different intervals.
Complex Numbers	The imaginary unit i = √(−1) and numbers of the form a + bi.
Operations with Complex Numbers	Multiplying complex numbers with FOIL and i² = −1.
Simplifying Rational Expressions	Factor top and bottom, then cancel the common factor.
Logarithms	log_b(x) asks: to what power must b be raised to get x?
Properties of Logarithms	Logs turn products into sums, quotients into differences, powers into multiples.
Arithmetic Sequences	Sequences that grow by a constant difference each step.
Geometric Sequences	Sequences that grow by a constant ratio each step.

Trigonometry · 4 topics

Right-Triangle Trigonometry	SOH-CAH-TOA: the three trig ratios of an acute angle in a right triangle.
Degrees & Radians	Two ways to measure the same angle: 180° equals π radians.
The Unit Circle	Exact sine, cosine, and tangent values at the special angles.
Graphs of Sine & Cosine	Reading amplitude, period, and midline from y = a sin(bx) + c.

Precalculus · 5 topics

Vectors: Components & Magnitude	A vector is a displacement: components ⟨Δx, Δy⟩ and a length.
Vector Operations	Scaling, adding, and dotting vectors — all component by component.
Polar Coordinates	Locating points by distance from the origin and angle from the x-axis.
Sigma Notation & Series	Σ compresses a sum: read the limits, add up the terms.
Average Rate of Change	The slope of the secant line: (f(b) − f(a)) / (b − a).

Limits & Continuity · 2 topics

Limits: Graphical & Numerical	What value a function approaches — which need not be the value it takes.
Evaluating Limits Algebraically	Direct substitution — and the factor-and-cancel fix for 0/0.

Differentiation · 7 topics

The Limit Definition of the Derivative	The derivative is the limit of average rates of change.
The Power Rule	d/dx xⁿ = n·xⁿ⁻¹ for any real n.
Sum & Constant-Multiple Rules	Differentiate term by term.
The Product Rule	(fg)′ = f′g + fg′.
Derivatives of Exponentials & Logs	eˣ is its own derivative; (ln x)′ = 1/x.
The Chain Rule	d/dx f(g(x)) = f′(g(x)) · g′(x).
Implicit Differentiation	Differentiating equations that mix x and y.

Integration · 5 topics

Antiderivatives	Reversing differentiation: the power rule backwards.
Riemann Sums	Approximating area with rectangles.
Properties of Definite Integrals	Linearity, additivity, and orientation.
The Fundamental Theorem: Evaluating Integrals	∫ₐᵇ f = F(b) − F(a).
u-Substitution	The chain rule in reverse.

Differential Equations · 3 topics

Differential Equations: Verifying Solutions	A solution is a function that satisfies the equation.
Separation of Variables	Move all y's left, all x's right, integrate both sides.
Exponential Growth & Decay Models	dy/dt = ky means y = y₀e^{kt}.

Linear Algebra · 1 topics

Vectors in Rⁿ

Ordered lists of numbers, added and scaled componentwise.

Multivariable Calculus · 5 topics

Partial Derivatives	Differentiate in one variable while holding the others constant.
Second Partials & Clairaut	Differentiate twice; mixed partials agree.
Double Integrals	Volume under a surface, computed one variable at a time.
Vector Fields, Divergence & Curl	Divergence measures spreading; curl measures rotation.
Line Integrals & Work	Add up F · dr along a path.

Advanced Differential Equations · 2 topics

First-Order Linear ODEs	y′ + ay = b: solved with an integrating factor.
Second-Order Linear ODEs	Constant coefficients: guess e^{rt}, factor the characteristic equation.

Mathematical Methods for the Physical Sciences I · 3 topics

Complex Arithmetic	Multiply out, use i² = −1.
Polar Form & Euler's Formula	z = re^{iθ}: modulus and angle instead of real and imaginary parts.
Fourier Series Basics	Any periodic signal is a sum of sines and cosines.

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