Calculus + vector calculus¶

Weeks¶

Week 4¶

Spine: OpenStax Calculus Vol. 1

Reading: Ch. 2 Limits, esp. 2.2-2.5

Know: Interpret limits as local behavior; reason about continuity and approximation rather than symbolic manipulation only.

Reconstruct: Derive the epsilon-style intuition for continuity; regenerate standard limit laws from algebraic decomposition.

Do: Model sensor saturation with a piecewise transfer function and analyze where continuity/differentiability fails.

Defend: What physical conclusion can and cannot be drawn from a discontinuity in a mathematical model?

Gate: Closed-book: compute and explain 8 limits, including one non-existent limit and one asymptotic limit.

Source: source

Week 5¶

Spine: OpenStax Calculus Vol. 1

Reading: Ch. 3 Derivatives, esp. 3.1-3.6

Know: Treat derivatives as local linear maps/rates; connect geometry, units, sensitivity and dynamics.

Reconstruct: Derive product, quotient and chain rules from the derivative definition; derive velocity/acceleration relations.

Do: Build a numerical derivative estimator, inject measurement noise, and show the bias/noise tradeoff as step size changes.

Defend: Why is differentiation numerically ill-conditioned relative to integration?

Gate: derive chain rule and explain it geometrically and dimensionally without notation prompts.

Source: source

Week 6¶

Spine: OpenStax Calculus Vol. 1

Reading: Ch. 4 Applications: 4.2 linearization; 4.7 optimization; 4.9 Newton's method

Know: Linearize nonlinear systems, formulate local sensitivity, and use derivatives for optimization/root finding.

Reconstruct: Derive the tangent-line linearization and Newton iteration from it.

Do: Estimate the operating point of a nonlinear actuator using Newton's method; identify initial conditions that cause failure.

Defend: When does a local approximation become operationally dangerous?

Gate: Unseen nonlinear function: derive a local model, bound likely error empirically, and defend the operating region.

Source: source

Week 7¶

Spine: OpenStax Calculus Vol. 1

Reading: Ch. 5 Integration, esp. 5.2-5.4 Fundamental Theorem

Know: Treat integration as accumulation/conservation; connect rates to stocks and local to global quantities.

Reconstruct: Derive the accumulation-function form of the Fundamental Theorem of Calculus and dimensional-check it.

Do: From a noisy power-vs-time trace, estimate total energy and compare integration rules.

Defend: Why can integration suppress some measurement noise while differentiation amplifies it?

Gate: Closed-book derivation + numerical integration of an unseen signal with uncertainty statement.

Source: source

Week 8¶

Spine: OpenStax Calculus Vol. 3

Reading: Ch. 2 Vectors in Space; Ch. 3 Vector-Valued Functions

Know: Represent geometry, motion, forces and trajectories in 3D; manipulate dot/cross products and vector kinematics.

Reconstruct: Derive projection from the dot product; derive centripetal acceleration for uniform circular motion.

Do: Simulate a thrust-limited spacecraft trajectory in 3D with piecewise acceleration commands.

Defend: What information is coordinate-dependent and what geometric relation is invariant?

Gate: reconstruct position/velocity/acceleration vectors and projection formulas from first principles.

Source: source

Week 9¶

Spine: OpenStax Calculus Vol. 3

Reading: Ch. 4 Multivariable differentiation; Ch. 5 Multiple integration; Ch. 6 Vector calculus

Know: Own gradient, directional derivative, Jacobian, multiple integrals, flux/circulation and divergence/curl intuition.

Reconstruct: Derive directional derivative = grad f | u; derive Jacobian local-linearization interpretation.

Do: Given a synthetic temperature field, compute gradient flow, flux through a surface, and identify heat-source regions from divergence.

Defend: What does a Jacobian know that a scalar derivative cannot?

Gate: Oral: explain gradient/divergence/curl to both a physicist and a robot-control engineer, preserving mathematical meaning.

Source: source

Exit gate¶

Closed-book: 120 min: derive chain rule, Newton step, FTC accumulation relation, gradient directional derivative, Jacobian linearization.

Novel problem: Model an unfamiliar physical rate/field problem from units and geometry; derive a local approximation and one integral balance.

Artifact: Numerically differentiate/integrate noisy data; analyze error vs resolution.

Defend: Explain why local linearization works, when it fails, and what vector calculus quantities mean physically.

Pass criterion: Pass if derivations are dimensionally consistent and predictions match numerical checks within stated error.

Transfer problems¶

Try these before consulting solutions or asking for the complete answer.

1. Limits: Analyze a sensor transfer function with a dead zone and saturation; identify every continuity/differentiability boundary.

2. Derivative: Derive acceleration from a noisy position model and explain why numerical differentiation is fragile.

3. Chain rule: For y=f(g(h(x))), derive dy/dx and map each factor to a physical subsystem sensitivity.

4. Linearization: Linearize a nonlinear drag force around an operating velocity and estimate the range where error stays below 5%.

5. Optimization: Minimize material for a cylindrical pressure vessel under a fixed-volume toy constraint.

6. Newton: Construct a function/initial guess where Newton's method fails or converges to an unintended root.

7. Integration: Compute energy from a piecewise power profile analytically and numerically; compare errors.

8. Vector geometry: Given force and motion vectors, decompose work-producing and orthogonal components.

9. Gradient/Jacobian: Compute a Jacobian for a two-output sensor model and interpret each entry's units.

10. Flux/divergence: Given a 3D vector field, determine whether a region behaves like a source/sink and validate numerically.

Textbooks¶

See the five-book resource page.