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4.1. 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.
4.1.1. CAN/Line Integrals and Vector Fields (Core Tier1: 1 hr, Core Tier2: 1 hr) ↑ Back to top
Scalar and vector line integrals, conservative fields, potential functions, path independence, and Green's Theorem.
Topics:
Core
- Vector fields: definition, visualization, divergence, and curl
- Line integrals of scalar functions: arc length and mass
- Line integrals of vector fields: work and circulation
- Conservative fields, potential functions, and path independence
- Green's Theorem and its applications in the plane
Learning Outcomes:
Core:
- Compute scalar and vector line integrals over oriented curves [Usage]
- Determine whether a vector field is conservative and construct its potential function [Usage]
- Apply Green's Theorem to evaluate line integrals via double integrals [Assessment]
4.1.2. CAN/Multiple Integrals (Core Tier1: 1 hr, Core Tier2: 1 hr) ↑ Back to top
Integration of functions of several variables: double and triple integrals in Cartesian, polar, cylindrical, and spherical coordinates, with applications.
Topics:
Core
- Double integrals over rectangles and general regions; iterated integrals (Fubini's theorem)
- Double integrals in polar coordinates and change-of-variables formula
- Triple integrals in Cartesian coordinates and applications to volume and mass
- Triple integrals in cylindrical and spherical coordinates
- Applications: area, volume, centers of mass, and moments of inertia
Learning Outcomes:
Core:
- Evaluate double and triple integrals over general regions using iterated integrals [Usage]
- Apply coordinate changes in polar, cylindrical, and spherical systems to simplify multiple integrals [Usage]
- Compute areas, volumes, and centers of mass using multiple integrals [Assessment]
4.1.3. CAN/Multivariable Transformations (Core Tier1: 1 hr, Core Tier2: 1 hr) ↑ Back to top
Vector functions of vector variables: differentiability, the Jacobian matrix, change of variables, and the inverse and implicit function theorems.
Topics:
Core
- Transformations from \(\mathbb{R}^n\) to \(\mathbb{R}^m\) and their differentiability
- Jacobian matrix and Jacobian determinant
- Change of variables in polar, cylindrical, and spherical coordinates
- Inverse function theorem
- Implicit function theorem in multivariable contexts
Learning Outcomes:
Core:
- Describe transformations between coordinate systems using the Jacobian matrix [Familiarity]
- Compute the Jacobian and apply change of variables to evaluate multiple integrals [Usage]
- Apply the inverse and implicit function theorems to analyze local invertibility [Assessment]
4.1.4. CAN/Surface Integrals and Fundamental Theorems (Core Tier1: 1 hr, Core Tier2: 1 hr) ↑ Back to top
Parametric surfaces, scalar and flux surface integrals, curl, divergence, and the fundamental theorems of Stokes and Gauss.
Topics:
Core
- Parametric surfaces, tangent plane, and surface area
- Surface integrals of scalar functions
- Flux integrals: surface integrals of vector fields
- Stokes' Theorem: relation between line integrals and surface integrals
- Divergence Theorem (Gauss): relation between flux and volume integral
Learning Outcomes:
Core:
- Parametrize surfaces and evaluate scalar and flux surface integrals [Usage]
- Apply Stokes' and Divergence Theorems to reduce complex integrals [Assessment]
- Relate local differential properties (curl, divergence) to global behavior via integral theorems [Assessment]
4.1.5. CAN/Vector Functions of a Real Variable (Core Tier1: 1 hr, Core Tier2: 1 hr) ↑ Back to top
Space curves as vector functions of a real variable, including differential geometry of curves: Frenet frame, curvature, and torsion.
Topics:
Core
- Definition, domain, and range of vector functions of a real variable
- Limits, continuity, and derivatives of vector functions
- Arc length and natural parametrization of a curve
- Tangent, normal, and binormal vectors; moving Frenet frame
- Curvature, torsion, and Frenet-Serret formulas
Learning Outcomes:
Core:
- Identify limits, continuity, and derivatives of vector functions of a real variable [Familiarity]
- Compute tangent, normal, and binormal vectors and reparametrize a curve by arc length [Usage]
- Calculate curvature and torsion using the Frenet-Serret formulas [Assessment]
4.1.6. 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 engineering 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 engineering systems near a point [Familiarity]
4.1.7. CAN/Differential Calculus (Core Tier1: 1 hr, Core Tier2: 1 hr) ↑ Back to top
Derivatives and rules of differentiation, with applications to engineering design and optimization.
Topics:
Core
- Derivatives and rules of differentiation
Learning Outcomes:
Core:
- Apply differentiation rules to find rates of change and optimize engineering design parameters [Usage]
- Formulate and solve optimization problems for engineering systems using differential calculus [Assessment]
4.1.8. 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 engineering problems.
Topics:
Core
- The Fundamental Theorem of Calculus
- Integration techniques including substitution and parts
- Engineering 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 engineering problems such as work, fluid pressure, and heat transfer [Usage]
4.1.9. 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 engineers 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 engineering model converges, using comparison and other convergence tests [Assessment]
- Evaluate an improper integral by taking the appropriate limit, when it converges [Usage]
4.1.10. 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 engineers 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
- Engineering 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 engineering function [Assessment]
- Apply a Taylor polynomial to approximate or linearize an engineering function [Usage]
4.1.11. 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 engineering 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]
4.1.12. 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 engineering force and moment problems [Usage]
- Determine equations of lines and planes in space from the geometric data of an engineering problem [Usage]
4.1.13. 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 engineering 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 engineering component modeling [Usage]
4.1.14. 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 engineering 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 engineering geometries [Usage]
4.1.15. 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 engineering.
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 engineering 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 engineering 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]
4.1.16. CAN/Complex Analysis and Conformal Mapping ↑ Back to top
Study of functions of a complex variable, analyticity, contour integration, and geometric transformations with applications to potential theory and signal analysis.
Topics:
Core
- Holomorphic functions and Cauchy-Riemann equations
- Contour integration and the Residue Theorem
- Conformal mapping and its applications in potential flow and heat conduction
- Connection between complex analysis and Laplace transforms in circuit analysis
Learning Outcomes:
Core:
- Evaluate real definite integrals using the Residue Theorem [Usage]
- Apply conformal mapping to transform boundary value problems in potential flow analysis [Assessment]
- Explain the analyticity conditions relevant to engineering potential theory [Familiarity]
- Determine if a complex function is analytic using the Cauchy-Riemann equations [Usage]
4.1.17. CAN/Harmonic Analysis and Fourier Series ↑ Back to top
Representation of periodic and aperiodic engineering signals as superpositions of sinusoidal components using Fourier series and transforms.
Topics:
Core
- Fourier series representation of periodic functions
- The Fourier transform and its properties in signal and system analysis
- Introduction to wavelet theory for non-stationary signal analysis
- Spectral analysis of engineering signals and filtering concepts
Learning Outcomes:
Core:
- Decompose periodic signals into their Fourier components and interpret the frequency spectrum [Usage]
- Apply the Fourier transform to analyze linear time-invariant engineering systems [Usage]
- Design frequency-domain filters using Fourier analysis to remove noise from engineering signals [Assessment]
- Explain the conditions for convergence of Fourier series and their practical implications [Familiarity]