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3.1. Calculus and Analysis (CAN)

This area covers the fundamental study of change, limits, and integration. It spans from single-variable differential and integral calculus to multivariable analysis and harmonic methods, providing the essential analytical toolkit for algorithm analysis, signal processing, and machine learning.

Knowledge Area (KA)	CS Core	KA Core
3.1.1 Line Integrals and Vector Fields	Elective
3.1.2 Multiple Integrals	Elective
3.1.3 Multivariable Transformations	Elective
3.1.4 Surface Integrals and Fundamental Theorems	Elective
3.1.5 Vector Functions of a Real Variable	Elective
3.1.6 Limits and Continuity	Elective
3.1.7 Differential Calculus	Elective
3.1.8 Integral Calculus and Series	Elective
3.1.9 Multivariable Differential Calculus	Elective
3.1.10 Harmonic Analysis and Fourier Theory	Elective

3.1.1. CAN/Line Integrals and Vector Fields ↑ 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:

1. Compute scalar and vector line integrals over oriented curves [Usage]
2. Determine whether a vector field is conservative and construct its potential function [Usage]
3. Apply Green's Theorem to evaluate line integrals via double integrals [Assessment]

3.1.2. CAN/Multiple Integrals ↑ 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:

1. Evaluate double and triple integrals over general regions using iterated integrals [Usage]
2. Apply coordinate changes in polar, cylindrical, and spherical systems to simplify multiple integrals [Usage]
3. Compute areas, volumes, and centers of mass using multiple integrals [Assessment]

3.1.3. CAN/Multivariable Transformations ↑ 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:

1. Describe transformations between coordinate systems using the Jacobian matrix [Familiarity]
2. Compute the Jacobian and apply change of variables to evaluate multiple integrals [Usage]
3. Apply the inverse and implicit function theorems to analyze local invertibility [Assessment]

3.1.4. CAN/Surface Integrals and Fundamental Theorems ↑ 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:

1. Parametrize surfaces and evaluate scalar and flux surface integrals [Usage]
2. Apply Stokes' and Divergence Theorems to reduce complex integrals [Assessment]
3. Relate local differential properties (curl, divergence) to global behavior via integral theorems [Assessment]

3.1.5. CAN/Vector Functions of a Real Variable ↑ 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:

1. Identify limits, continuity, and derivatives of vector functions of a real variable [Familiarity]
2. Compute tangent, normal, and binormal vectors and reparametrize a curve by arc length [Usage]
3. Calculate curvature and torsion using the Frenet-Serret formulas [Assessment]

3.1.6. CAN/Limits and Continuity ↑ Back to top

Rigorous study of limits and continuity of single-variable functions, including the epsilon-delta definition, limit laws, and fundamental theorems.
Topics:
Core

• Epsilon-delta definition of a limit and formal proof techniques
• Limit laws, one-sided limits, and limits at infinity
• Continuity: types, properties, and the Intermediate Value Theorem
• L'Hôpital's rule and indeterminate forms
• Limits of sequences: convergence, Squeeze Theorem, and Bolzano-Weierstrass

Learning Outcomes:
Core:

1. State the epsilon-delta definition of a limit and explain its geometric meaning [Familiarity]
2. Evaluate limits using limit laws, L'Hôpital's rule, and asymptotic analysis [Usage]
3. Prove continuity or discontinuity of a function and apply the Intermediate Value Theorem [Assessment]

3.1.7. CAN/Differential Calculus ↑ Back to top

Derivatives of single-variable functions: definition, rules, higher-order derivatives, and applications to optimization and algorithm analysis.
Topics:
Core

• Derivative as a limit; geometric and physical interpretations; differentiability
• Product, quotient, and chain rules; implicit differentiation; related rates
• Rolle's theorem, Mean Value Theorem, and monotonicity criteria
• Higher-order derivatives, Taylor polynomials, and remainder estimation
• Extrema: first and second derivative tests, curve sketching, and applied optimization

Learning Outcomes:
Core:

1. Interpret the derivative as a rate of change and as the slope of the tangent line [Familiarity]
2. Apply differentiation rules and implicit differentiation to compute derivatives of composite and implicit functions [Usage]
3. Solve optimization problems and sketch curves using the Mean Value Theorem and derivative tests [Assessment]

3.1.8. CAN/Integral Calculus and Series ↑ Back to top

Definite and indefinite integrals, the Fundamental Theorem of Calculus, integration techniques, and convergence of sequences and series.
Topics:
Core

• The Fundamental Theorem of Calculus and the antiderivative
• Integration techniques: substitution, parts, partial fractions, and trigonometric substitution
• Improper integrals: convergence criteria and comparison tests
• Sequences and series: convergence tests (ratio, root, comparison, integral)
• Power series, radius of convergence, Taylor and Maclaurin series

Learning Outcomes:
Core:

1. State the Fundamental Theorem of Calculus and explain the relationship between differentiation and integration [Familiarity]
2. Compute definite and indefinite integrals using standard techniques including substitution and integration by parts [Usage]
3. Determine convergence of improper integrals and power series and construct Taylor series for standard functions [Assessment]

3.1.9. CAN/Multivariable Differential Calculus ↑ Back to top

Differential calculus of scalar functions of several variables: limits, partial derivatives, gradient, tangent plane, optimization, and Taylor's theorem.
Topics:
Core

• Limits and continuity of functions of several variables in \(\mathbb{R}^n\)
• Partial derivatives, directional derivatives, and the gradient vector
• Tangent plane, total differential, and linear approximation
• Extrema of multivariable functions and Lagrange multipliers
• Taylor's theorem for functions of several variables

Learning Outcomes:
Core:

1. Evaluate limits and determine continuity of functions of several variables [Familiarity]
2. Compute gradients, directional derivatives, and tangent planes for multivariable functions [Usage]
3. Apply Lagrange multipliers to solve constrained optimization problems in computing and data science [Usage]
4. Expand functions of several variables using Taylor's theorem and estimate error bounds [Assessment]

3.1.10. CAN/Harmonic Analysis and Fourier Theory ↑ Back to top

Fourier series and transforms with applications to signal processing, compression, and spectral analysis in computing.
Topics:
Core

• Fourier series: convergence, Parseval's theorem, and \(L^2\) completeness
• The Fourier transform: properties, inversion, and convolution
• Plancherel theorem and tempered distributions (Schwartz space)
• Wavelet theory: multiresolution analysis and the discrete wavelet transform
• Applications to signal processing, PDE solving, and data compression

Learning Outcomes:
Core:

1. Decompose periodic functions into Fourier series and assess pointwise and \(L^2\) convergence [Familiarity]
2. Apply the Fourier transform to solve linear systems and compute convolutions in digital signal processing [Usage]
3. Analyze the decay rate of Fourier coefficients in relation to the smoothness of a function [Assessment]