知识脉络

UBC Knowledge Graph

Editorial note: this vault should be maintained as a connected Obsidian knowledge graph, not a folder of isolated notes. After each new note, read the existing vault notes and add meaningful links/backlinks.

Main Branches / 主枝

Summer Preparatory Courses - University Writing

AP Calculus BC

AP Physics C - Electricity and Magnetism

3Blue1Brown - The Essence of Calculus

3Blue1Brown - Differential Equations

Concept Bridges / 概念桥

  • Academic Writing connects inquiry, evidence, reasoning, disciplinary conventions, and authorial contribution.
  • Thesis Statement turns a broad topic into an arguable, supportable, and appropriately qualified central claim.
  • Reader-Centered Writing connects context, sentence flow, paragraph logic, counterargument, and introduction structure.
  • Integrals connects visual accumulation and AP Integral Test.
  • Polar Area connects Chapter 9 polar graphs with the same accumulation idea used in Integrals and Area Under a Curve.
  • Area Under a Curve connects circle area, accumulated area functions, and improper integrals.
  • Infinite Sequence and Series connects discrete accumulation with convergence tests.
  • Derivatives connects area functions back to original height functions and connects Chapter 2 to tangent-line slope.
  • Limits explains how average rates of change become derivatives without plugging in zero.
  • Fundamental Theorem of Calculus explains why derivative and integral undo each other.
  • Unit Circle explains why the sine derivative is exactly cosine, not just a similar-looking wave.
  • Tiny Nudges connects Chapter 2 derivative meaning with Chapter 3 derivative formulas.
  • Power Rule connects symbolic derivative fluency with the geometric idea of first-order tiny changes.
  • Product Rule adds the two first-order contributions created when both factors change.
  • Function Composition records how one function’s output becomes the next function’s input.
  • Chain Rule multiplies the local change factors carried through successive function layers.
  • Exponential Functions turn additive input changes into multiplicative output changes and therefore have derivatives proportional to themselves.
  • Euler’s Number e is the exponential base whose graph height equals its tangent slope everywhere.
  • Natural Logarithm gives the proportionality constant in and converts into .
  • Exponential Growth connects the differential relation to population, compounding, and decay models.
  • Implicit Curve represents points satisfying a relation without requiring one global input-output function.
  • Implicit Differentiation finds the local changes that preserve a constraint to first order.
  • Related Rates gives multiple changing quantities a common time input and relates their rates through a constraint.
  • Epsilon-Delta Definition makes the intuitive word “approach” precise through arbitrary output accuracy.
  • Indeterminate Form marks a substitution result that does not yet determine a limit.
  • L’Hôpital’s Rule compares derivative ratios when theorem conditions permit an indeterminate quotient limit.
  • Riemann Sum connects finite rectangle approximations to exact continuous accumulation.
  • Antiderivative reverses differentiation and recovers an accumulation function up to a constant.
  • Signed Area distinguishes net accumulation from total geometric magnitude.
  • Average Value of a Function connects continuous averaging to signed area and the secant slope of an antiderivative.
  • Second Derivative connects changing slope, graph concavity, and acceleration.
  • Higher-Order Derivatives repeatedly apply differentiation and supply local shape data for Taylor Series.
  • Concavity translates the sign of the second derivative into the direction in which tangent slopes change.
  • Acceleration is the derivative of velocity and the second derivative of position.
  • Taylor Polynomial uses finitely many derivative matches to approximate a function near a chosen center.
  • Taylor Series extends Taylor polynomials through a partial-sum limit, subject to convergence.
  • Transformational View of Derivatives reframes graph slope as the local scaling and orientation action of a function on tiny input separations.
  • Fixed-Point Stability uses local derivative magnitude to decide whether repeated iteration contracts toward or expands away from a fixed point.
  • Differential Equations describe unknown functions through local laws involving their derivatives.
  • Initial Conditions choose one trajectory from the family allowed by a differential equation.
  • Phase Space turns all possible system states and trajectories into geometry.
  • Euler Method follows the local vector field in finite steps to approximate continuous evolution.
  • Partial Derivatives measure one input-direction change while holding the other inputs fixed.
  • Heat Equation makes local temporal change proportional to spatial curvature.
  • Second Difference connects discrete neighbor averaging to the continuous Second Derivative.
  • Laplacian measures multidimensional curvature relative to surrounding values.
  • Fourier Series decomposes complex functions into simple sinusoidal modes and prepares the heat-equation solution.
  • Vectors links AP Physics C E&M with the force-component and field-superposition methods used throughout the course.
  • Force on a Current-Carrying Wire is the macroscopic sum of Lorentz forces on the wire’s moving charge carriers.
  • Magnetic Torque on a Current Loop turns equal-and-opposite wire forces into rotation through .
  • Biot-Savart Law constructs magnetic fields by integrating vector contributions from current elements.
  • Ampere’s Law links magnetic circulation to signed enclosed current and becomes directly calculational when symmetry is sufficient.
  • Solenoids and Toroids use repeated current turns to concentrate magnetic field into controlled regions.
  • Electromagnetic Induction unifies moving-conductor and changing-field routes to induced EMF.
  • Motional EMF connects Lorentz-force charge separation to a circuit source.
  • Magnetic Flux measures magnetic field through an oriented surface and supplies the changing quantity in Faraday’s law.
  • Faraday’s Law turns flux change rate into EMF, with Lenz’s-law direction encoded by the negative sign.
  • Lenz’s Law turns a flux increase or decrease into the induced-field and current direction required by energy conservation.
  • Induced Electric Field explains how changing magnetic flux drives charge through a circulating non-conservative field even without a wire.
  • Inductors connect self-induced EMF, current continuity, and magnetic-field energy.
  • RL Circuits connect inductor current memory to exponential rise, decay, and the time constant .
  • Electromagnetic Damping connects induced current, opposing magnetic force, energy conversion, and terminal speed.
  • Electric Field turns force between charges into a field description of space; later Gauss’s Law will compute fields from symmetry.
  • Coulomb’s Law is an inverse-square bridge to Universal Gravitation and introduces vector superposition in electrostatics.
  • Electric Charge starts the AP Physics C E&M branch and supplies the source quantity for electric force and electric field.
  • Electric Field Superposition mirrors Coulomb-force superposition but removes dependence on the test charge.
  • Continuous Charge Distribution turns point-charge fields into integrals over .
  • Electric Flux counts electric field passing through a surface and sets up Gauss’s Law.
  • Electric Potential turns electric potential energy into a source-defined scalar background and allows algebraic superposition.
  • Test Charge connects the field definition to the requirement that measurement should not disturb the original field.
  • Interval of Convergence connects power series back to convergence tests and endpoint checking.
  • Maclaurin Series is the centered-at-zero version of Taylor series and supplies standard expansions.
  • Taylor Series connects derivatives, power series, and function approximation.
  • Ratio Test completes the AP Calculus BC convergence-test toolkit and drives interval-of-convergence problems.

Concept Cards / 知识卡片

Current Vault Notes / 当前库内笔记

Maintenance Rule / 维护规则

When adding a note:

  1. Read current vault notes.
  2. Identify overlapping concepts.
  3. Add direct links in the new note.
  4. Add backlinks or graph connections in older related notes.
  5. Update this MOC when the note creates a new branch.

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