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3.1. Calculus and Analysis (CAN)
This area covers the fundamental study of change, limits, and the rigorous properties of functions. It spans from basic differentiation and integration to advanced measure theory and functional spaces, providing the essential toolkit for all mathematical modeling.
| Knowledge Area (KA) | Core Tier1 | Core Tier2 |
 3.1.1 Limits and Continuity | 1 | 1 |
|
3.1.2 Differential Calculus | 1 | 1 |
|
3.1.3 Integral Calculus | 1 | 1 |
|
3.1.4 Improper Integrals | 1 | 1 |
|
3.1.5 Sequences, Series, and Power Series | 1 | 1 |
|
3.1.6 Applications of the Definite Integral | 1 | 1 |
|
3.1.7 Vectors and the Geometry of Space | 1 | 1 |
|
3.1.8 Quadric Surfaces | 1 | 1 |
|
3.1.9 Parametric Equations and Polar Coordinates | 1 | 1 |
|
3.1.10 Multivariable and Vector Calculus | 1 | 1 |
3.1.1. CAN/Limits and Continuity (Core Tier1: 1 hr, Core Tier2: 1 hr) ↑ Back to top
Limits and continuity of single-variable functions, the foundation for rigorously studying change in architectural systems.
Topics:
Core
- Limits and continuity of single-variable functions
Learning Outcomes:
Core:
- Interpret the concept of limit and continuity of a function and its relevance to modeling the behavior of architectural systems near a point [Familiarity]
3.1.2. CAN/Differential Calculus (Core Tier1: 1 hr, Core Tier2: 1 hr) ↑ Back to top
Derivatives and rules of differentiation, with applications to architectural design and optimization.
Topics:
Core
- Derivatives and rules of differentiation
Learning Outcomes:
Core:
- Apply differentiation rules to find rates of change and optimize architectural design parameters [Usage]
- Formulate and solve optimization problems for architectural systems using differential calculus [Assessment]
3.1.3. CAN/Integral Calculus (Core Tier1: 1 hr, Core Tier2: 1 hr) ↑ Back to top
The Fundamental Theorem of Calculus, integration techniques, and their applications to architectural problems.
Topics:
Core
- The Fundamental Theorem of Calculus
- Integration techniques including substitution and parts
- Architectural applications of integration: areas, volumes, centroids, and moments of inertia
Learning Outcomes:
Core:
- State and explain the Fundamental Theorem of Calculus [Familiarity]
- Calculate definite integrals arising from architectural problems such as work, fluid pressure, and heat transfer [Usage]
3.1.4. CAN/Improper Integrals (Core Tier1: 1 hr, Core Tier2: 1 hr) ↑ Back to top
Integrals over unbounded intervals or of unbounded functions, and the convergence criteria architects use to determine whether such quantities (e.g. total impulse, accumulated decay) are finite.
Topics:
Core
- Improper integrals over unbounded intervals
- Improper integrals of unbounded (discontinuous) integrands
- Convergence criteria and comparison tests for improper integrals
Learning Outcomes:
Core:
- Determine whether an improper integral arising from an architectural model converges, using comparison and other convergence tests [Assessment]
- Evaluate an improper integral by taking the appropriate limit, when it converges [Usage]
3.1.5. CAN/Sequences, Series, and Power Series (Core Tier1: 1 hr, Core Tier2: 1 hr) ↑ Back to top
Convergence of infinite sequences and series and the power-series (Taylor/Maclaurin) approximations architects use to model, linearize, and numerically evaluate functions.
Topics:
Core
- Sequences and series: convergence tests (ratio, root, comparison, integral)
- Power series, radius of convergence, and Taylor/Maclaurin series
- Architectural use of Taylor polynomials for linearization and numerical approximation
Learning Outcomes:
Core:
- Determine convergence of a power series and construct a Taylor series for a standard architectural function [Assessment]
- Apply a Taylor polynomial to approximate or linearize an architectural function [Usage]
3.1.6. CAN/Applications of the Definite Integral (Core Tier1: 1 hr, Core Tier2: 1 hr) ↑ Back to top
Geometric and physical applications of the definite integral: areas, volumes of revolution, arc length, and work, as architectural computations built on integral calculus.
Topics:
Core
- Area between curves by integrating with respect to \(x\) or \(y\)
- Volume of solids of revolution by the disk and washer method
- Volume of solids of revolution by the cylindrical shell method
- Work via integration (springs, pumping fluids, cables)
Elective
- Moments and centroid of a plane region or solid
Learning Outcomes:
Core:
- Calculate the volume of a solid of revolution using the disk, washer, or cylindrical shell method [Usage]
- Calculate the work done by a variable force using integration [Usage]
3.1.7. CAN/Vectors and the Geometry of Space (Core Tier1: 1 hr, Core Tier2: 1 hr) ↑ Back to top
Vectors in three-dimensional space, the dot and cross products, and their application to lines and planes, the basis of vector analysis in statics and dynamics.
Topics:
Core
- Dot and cross products: definition, properties, and geometric interpretation
- Equations of lines and planes in space
Learning Outcomes:
Core:
- Compute the dot and cross products of vectors to solve architectural force and moment problems [Usage]
- Determine equations of lines and planes in space from the geometric data of an architectural problem [Usage]
3.1.8. CAN/Quadric Surfaces (Core Tier1: 1 hr, Core Tier2: 1 hr) ↑ Back to top
Classification of quadric surfaces, used to describe component geometries and surfaces in architectural modeling.
Topics:
Core
- Classification of quadrics (ellipsoids, paraboloids, hyperboloids) from their general equation
Learning Outcomes:
Core:
- Classify a quadric surface and identify its geometry for architectural component modeling [Usage]
3.1.9. CAN/Parametric Equations and Polar Coordinates (Core Tier1: 1 hr, Core Tier2: 1 hr) ↑ Back to top
Parametric curves and polar coordinates, used to describe trajectories and geometries in kinematics and architectural design.
Topics:
Core
- Parametric curves and their calculus, applied to trajectories in kinematics
- Polar coordinates and their conversion to Cartesian coordinates
Learning Outcomes:
Core:
- Calculate the velocity and acceleration of a trajectory described parametrically [Usage]
- Convert between polar and Cartesian coordinates when modeling architectural geometries [Usage]
3.1.10. CAN/Multivariable and Vector Calculus (Core Tier1: 1 hr, Core Tier2: 1 hr) ↑ Back to top
Calculus of functions of several variables, focusing on partial derivatives, multiple integrals, and vector field theorems essential for field analysis in architecture.
Topics:
Core
- Partial derivatives and the gradient vector
- Multiple integrals in Cartesian, cylindrical, and spherical coordinate systems
- Vector fields and line and surface integrals
- Integral theorems: Green, Stokes, and the Divergence Theorem
- Applications to flux, circulation, and conservative fields in architectural contexts
Learning Outcomes:
Core:
- Compute gradients and directional derivatives for scalar fields describing physical quantities [Usage]
- Evaluate double and triple integrals for volumes, masses, and centroids of architectural objects [Usage]
- Apply Stokes' and Gauss's Divergence Theorems to relate field integrals in electromagnetism and fluid mechanics [Assessment]
- Determine whether a vector force field is conservative and compute the potential function [Assessment]