Course Overview

ENGM2101 Applied Vector Calculus is a university-level course focusing on multivariable and vector calculus, with applications in engineering. The course covers advanced calculus topics beyond single-variable calculus, including vectors, vector-valued functions, functions of several variables, multiple integration, and vector calculus theorems.

Course Structure and Main Topics

• Credit Hours: 3

• Prerequisites: MATH1290.03 (Multivariable Calculus)

• Textbook: Calculus: Early Transcendental, 3rd edition, by William L. Briggs et al. (Chapters 13–17 covered)

Major Topics and Subtopics

Vectors and the Geometry of Space

• Vectors in the Plane and Space: Definitions, properties, and operations (addition, scalar multiplication).

• Dot Product: Calculation, geometric interpretation, and applications (e.g., projections, work).

• Cross Product: Calculation, geometric meaning (area, orthogonality), and applications.

• Lines and Planes in Space: Parametric and vector equations, intersections, and distances.

• Cylinders and Quadric Surfaces: Equations and classification of surfaces in three dimensions.

Vector-Valued Functions

• Vector-Valued Functions: Definition, domain, and range.

• Calculus of Vector-Valued Functions: Differentiation and integration of vector functions.

• Motion in Space: Position, velocity, and acceleration vectors; tangent and normal components.

• Length of Curves: Arc length formula for space curves.

• Curvature and Normal Vectors: Definitions and calculations for curvature, unit tangent, and normal vectors.

Functions of Several Variables

• Graphs and Level Curves: Visualization of functions of two variables.

• Limits and Continuity: Definitions and criteria for continuity in several variables.

• Partial Derivatives: Computation and interpretation.

• The Chain Rule: Differentiation of composite functions.

• Directional Derivatives and the Gradient: Calculating rates of change in arbitrary directions.

• Tangent Planes and Linear Approximation: Approximating surfaces near a point.

• Maximum/Minimum Problems: Finding extrema for functions of several variables.

• Lagrange Multipliers: Method for constrained optimization.

Multiple Integration

• Double Integrals over Rectangular and General Regions: Setting up and evaluating double integrals.

• Double Integrals in Polar Coordinates: Changing variables for integration over circular regions.

• Triple Integrals: Setting up and evaluating triple integrals in Cartesian, cylindrical, and spherical coordinates.

• Integrals for Mass Calculations: Applications to center of mass and moments.

• Change of Variables in Multiple Integrals: Jacobian determinant and coordinate transformations.

Vector Calculus

• Vector Fields: Definition and examples (e.g., velocity fields, force fields).

• Line Integrals: Integration of scalar and vector fields along curves.

• Conservative Vector Fields: Criteria and potential functions.

• Green’s Theorem: Relationship between a line integral around a simple closed curve and a double integral over the region it encloses.

• Divergence and Curl: Definitions and physical interpretations.

• Surface Integrals: Integration over surfaces in space.

• Stokes’ Theorem: Generalization of Green’s Theorem to surfaces in space.

• Divergence Theorem: Relates the flux of a vector field through a closed surface to the divergence over the volume inside.

Key Definitions and Formulas

• Dot Product:

• Cross Product:

• Arc Length:

• Gradient:

• Double Integral (Rectangular):

• Triple Integral (Cartesian):

• Change of Variables (Jacobian):

• Line Integral (Vector Field):

• Green’s Theorem:

• Divergence:

• Curl:

• Stokes’ Theorem:

• Divergence Theorem:

Course Assessment Overview

Weekly Topic Schedule (Summary)

Additional Information

• Applications: The course applies vector calculus to engineering problems such as heat flow, electrostatics, and fluid flow.

• Software Tools: Programming skills are developed for solving practical problems.

• Academic Integrity: Strict policies on plagiarism, cheating, and unauthorized collaboration.

• Accessibility: Accommodations available for students with documented needs.

Additional info: The course content aligns with standard multivariable calculus curricula, extending single-variable calculus to higher dimensions and vector fields, which are foundational for advanced studies in engineering and physical sciences.