Calculus, vectors, and differential equations
Build the mathematical language that turns physical stories into local laws and testable trajectories.
Connect derivatives, integrals, vector fields, linear algebra, Fourier modes, and differential equations to the site's mechanics, fields, relativity, and quantum pathways.
Before you begin
- • Level 2 mechanics
- • Algebra, trigonometry, and functions
- • Comfort reading graphs
By the end, you can
- • Interpret derivatives and integrals physically.
- • Compute with vectors, matrices, gradients, divergence, and curl.
- • Solve representative first- and second-order differential equations.
- • Use dimensional analysis, scaling, and numerical integration to test a model.
Interactive model
Explore before calculating
Live laboratory
Numerical oscillator bench
Integrate one oscillator with symplectic Euler. Change physical parameters and step size, then watch the trajectory and energy-error warning respond.
ω: 2.000 rad/s
Period: 3.142 s
Max energy drift: 5.26%
Level 3 · Undergraduate core teaching kit
Record the investigation. Teach the reasoning.
A learner-facing lab record and a course-specific instructor guide turn the live model into a repeatable classroom investigation.
Learner record
Oscillator convergence and energy-drift study
When does a numerical trajectory represent the differential equation rather than the stepping method's error?
Download learner recordInstructor guide
Teach for evidence, not button pushing
Learners separate physical parameters from numerical controls and justify convergence using trajectory and invariant diagnostics.
Download instructor guideLesson 1 of 3
Derivatives, integrals, and local change
How does an infinitesimal rate produce a finite measurable history?
A derivative is a local linear approximation: velocity is dx/dt, acceleration is d²x/dt², and a field gradient gives the steepest local change.
An integral accumulates local contributions. Displacement accumulates velocity; work accumulates force along a path; probability accumulates density over a region.
Worked example
A particle has v(t)=3t² m/s. Find displacement from t=0 to 2 s.
- 1. Integrate v over time.
- 2. ∫₀²3t²dt = [t³]₀².
- 3. Evaluate 8−0.
The displacement is 8 m.
Try it
Slope-to-area duality
Materials: Graph paper or a plotting notebook.
- 1. Plot x=t³.
- 2. Estimate tangent slopes.
- 3. Plot v=3t².
- 4. Estimate area under v and compare with changes in x.
Notice: Differentiation and integration connect one physical history in two complementary ways.
Check your understanding: What does ∫F·dr represent?
Answer: Work transferred by a force along a path.
The dot product keeps the component parallel to each displacement element.
Lesson 2 of 3
Vector fields, flux, and circulation
How do local arrows encode sources, sinks, and rotation?
The gradient maps scalar change, divergence measures local outward flux, and curl measures local circulation. These operators make Maxwell's equations compact.
Integral theorems connect local differential statements to measurable boundary fluxes and loops.
Worked example
For F=(x,y,z), compute ∇·F.
- 1. Differentiate Fx with respect to x: 1.
- 2. Differentiate Fy with respect to y: 1.
- 3. Differentiate Fz with respect to z: 1.
∇·F=3, a uniform positive source density in this mathematical field.
Try it
Field topology sketch
Materials: Paper and several vector-field formulas.
- 1. Sample arrows on a grid.
- 2. Mark regions of positive or negative divergence.
- 3. Trace a small loop to test circulation.
- 4. Compare local and boundary descriptions.
Notice: A picture can suggest topology, while derivatives quantify it.
Check your understanding: What does zero divergence guarantee?
Answer: No local net source or sink; it does not guarantee the field is zero or curl-free.
A divergence-free field can still circulate.
Lesson 3 of 3
Differential equations, modes, and computation
How do laws plus initial conditions generate trajectories and spectra?
A differential equation specifies how a state changes. Initial or boundary conditions select one solution from a family.
Linear systems decompose into modes; Fourier analysis represents complex signals as frequency components. Numerical stepping approximates systems without closed-form solutions.
Worked example
Solve dx/dt=−kx with x(0)=x₀.
- 1. Separate variables: dx/x=−kdt.
- 2. Integrate: ln x=−kt+C.
- 3. Apply x(0)=x₀.
x(t)=x₀e⁻ᵏᵗ, the universal exponential-relaxation form.
Try it
Euler-versus-exact decay
Materials: Spreadsheet or code notebook.
- 1. Choose k and x₀.
- 2. Step xₙ₊₁=xₙ−k xₙΔt.
- 3. Compare with x₀e⁻ᵏᵗ.
- 4. Reduce Δt and measure error.
Notice: Numerical answers converge only when the step is small enough and the scheme is stable.
Check your understanding: Why are boundary conditions essential for a cavity-mode problem?
Answer: They select which solutions of the field equation are allowed.
The differential equation alone permits a broader family than the physical geometry.
Formula-to-meaning deck
Read the equation in ordinary language.
v=dx/dt; a=d²x/dt²
Successive time derivatives turn position into velocity and acceleration.
∇f; ∇·F; ∇×F
Gradient, divergence, and curl quantify scalar change, flux sources, and circulation.
dy/dt=f(t,y)
A first-order differential equation generates state evolution from initial data.
Independent practice
Problem set
Work each problem before opening its hint and solution.
1. Differentiate x(t)=A cos(ωt) twice and identify the equation it satisfies.
Reveal hint
Each derivative brings a factor ω and alternates sine/cosine.
Reveal solution
x¨=−ω²x, so x¨+ω²x=0.
2. Compute the gradient of f=x²+y²+z².
Reveal hint
Differentiate once with respect to each coordinate.
Reveal solution
∇f=(2x,2y,2z).
3. Estimate one Euler step for dy/dt=−2y, y(0)=1, Δt=0.1.
Reveal hint
y₁=y₀+f(y₀)Δt.
Reveal solution
y₁=1−2(1)(0.1)=0.8.
Derivation studio
Build the result, line by line.
Keep the assumptions visible so the mathematics remains auditable.
Starting point
Simple harmonic motion from Hooke's law
F=−kx and F=ma
- 1. Set m x¨=−kx.
- 2. Divide by m: x¨+(k/m)x=0.
- 3. Test x=A cos(ωt+φ).
- 4. Match coefficients to obtain ω²=k/m.
x(t)=A cos(√(k/m)t+φ)
A local restoring force generates a global periodic trajectory; initial conditions set amplitude and phase.
Starting point
Wave equation from a stretched string
Transverse force balance on a short string element
- 1. Use small-slope tension components at both ends.
- 2. Net transverse force is T(∂²y/∂x²)Δx.
- 3. Element mass is μΔx.
- 4. Apply Newton's law and cancel Δx.
∂²y/∂t²=(T/μ)∂²y/∂x²
Wave speed c=√(T/μ) emerges from restoring tension and inertia per length.
Computational notebook
Turn the model into an experiment.
Numerical oscillator laboratory
When does Euler integration distort an oscillator's conserved energy?
Inputs
- • m, k, initial x and v
- • time step Δt
- • Euler and symplectic-Euler choices
Algorithm
- 1. Integrate position and velocity.
- 2. Compute K+U each step.
- 3. Repeat for several Δt.
- 4. Compare phase and energy drift.
Evidence to produce
- • Trajectory and phase-space plots
- • Relative energy error versus time
- • A justified stable-step recommendation
Continue into the evidence