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5.16. Advanced Differential and Integral Calculus (Mandatory)

5.16.1. Justification ↑ Back to top

Advanced differential and integral calculus provides the rigorous foundations for understanding multidimensional phenomena in engineering and computer science. This course covers vector functions, multivariable calculus, differentiable transformations, multiple integrals, and integral calculus on manifolds, including the fundamental theorems of Green, Stokes, and Gauss. These tools are essential for modeling complex systems, optimizing multivariable functions, and solving problems in computer graphics, computer vision, and physical simulations.

5.16.2. Generales Goals ↑ Back to top

1. Study vector functions of a real variable and the differential geometry of space curves.
2. Apply differential calculus to functions of several variables: partial derivatives, gradient, tangent plane, optimization, and Taylor's theorem.
3. Analyze differentiable transformations between vector spaces using the Jacobian matrix and the inverse and implicit function theorems.
4. Evaluate multiple integrals in different coordinate systems and compute areas, volumes, and centers of mass.
5. Compute line integrals and apply Green's Theorem and the concept of conservative fields.
6. Compute surface integrals and apply Stokes' Theorem and the Divergence Theorem.

5.16.3. Contribution to Outcomes ↑ Back to top

AG-C07) Computing Knowledge: Applies knowledge of mathematics, science, and computing. (Assessment)

5.16.4. Content ↑ Back to top

5.16.4.1. Vector Functions of a Real Variable (14 hours) [Skills AG-C07] ↑ Back to top

Bibliography: (Apostol, 1997; Marsden and Tromba, 2011)

Topics

1. Definition, domain, and range of vector functions of a real variable
2. Limits, continuity, and derivatives of vector functions
3. Arc length and natural parametrization of a curve
4. Tangent, normal, and binormal vectors; moving Frenet frame
5. Curvature, torsion, and Frenet-Serret formulas

Learning Outcomes

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]

5.16.4.2. Multivariable Differential Calculus (18 hours) [Skills AG-C07] ↑ Back to top

Bibliography: (Apostol, 1997; Marsden and Tromba, 2011)

Topics

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

Learning Outcomes

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]

5.16.4.3. Multivariable Transformations (12 hours) [Skills AG-C07] ↑ Back to top

Bibliography: (Apostol, 1997; Marsden and Tromba, 2011)

Topics

1. Transformations from \(\mathbb{R}^n\) to \(\mathbb{R}^m\) and their differentiability
2. Jacobian matrix and Jacobian determinant
3. Change of variables in polar, cylindrical, and spherical coordinates
4. Inverse function theorem
5. Implicit function theorem in multivariable contexts

Learning Outcomes

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]

5.16.4.4. Multiple Integrals (18 hours) [Skills AG-C07] ↑ Back to top

Bibliography: (Apostol, 1997; Marsden and Tromba, 2011)

Topics

1. Double integrals over rectangles and general regions; iterated integrals (Fubini's theorem)
2. Double integrals in polar coordinates and change-of-variables formula
3. Triple integrals in Cartesian coordinates and applications to volume and mass
4. Triple integrals in cylindrical and spherical coordinates
5. Applications: area, volume, centers of mass, and moments of inertia

Learning Outcomes

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]

5.16.4.5. Line Integrals and Vector Fields (10 hours) [Skills AG-C07] ↑ Back to top

Bibliography: (Apostol, 1997; Marsden and Tromba, 2011)

Topics

1. Vector fields: definition, visualization, divergence, and curl
2. Line integrals of scalar functions: arc length and mass
3. Line integrals of vector fields: work and circulation
4. Conservative fields, potential functions, and path independence
5. Green's Theorem and its applications in the plane

Learning Outcomes

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]

5.16.4.6. Surface Integrals and Fundamental Theorems (12 hours) [Skills AG-C07] ↑ Back to top

Bibliography: (Apostol, 1997; Marsden and Tromba, 2011)

Topics

1. Parametric surfaces, tangent plane, and surface area
2. Surface integrals of scalar functions
3. Flux integrals: surface integrals of vector fields
4. Stokes' Theorem: relation between line integrals and surface integrals
5. Divergence Theorem (Gauss): relation between flux and volume integral

Learning Outcomes

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]

5.16.5. Bibliography ↑ Back to top

Apostol, T. M. (1997). Calculus, Vol. II: Multi-Variable Calculus and Linear Algebra with Applications. John Wiley & Sons, 2nd edition.

Marsden, J. E. and Tromba, A. (2011). Vector Calculus. W. H. Freeman and Company, 6th edition.