MAT2214

Topics List

Date	Sections	Topics	Prerequisite Skills	Student Learning Outcomes
Week 1	1.1	Polar Coordinates	Trigonometric Functions: Unit Circle Approach The inverse Sine, Cosine and Tangent functions	Plot points using polar coordinates and find several polar coordinates of a single point Convert polar coordinates to rectangular coordinates and vice versa Transform equations from polar form to rectangular form and vice versa
Week 1	1.2	Three-Dimensional Coordinate Systems	Two-dimensional coordinate systems Algebraic Expressions	Three-dimensional coordinate systems Distance Formula in R3 Standard Equation for a Sphere
Weeks 1/2	2.1	Vectors in The Plane	Line Segments Distance Formula	Vector Algebra Operations The Magnitude of a vector Unit Vectors The Midpoint of a Line Segment The Vector projection
Week 2	2.2	Vectors in Space	Vectors in the Plane Distance Formula	Vector Algebra Operations The Magnitude of a vector Unit Vectors The Midpoint of a Line Segment The Vector projection
Week 2	2.3	The Dot Product	Basic Trig Functions Vectors	Definition of Dot Product Properties of Dot Product Angle between vectors Orthogonal vectors
Week 2	2.4	The Cross Product	Basic Trig Functions Determinants Vectors	Definition of Cross Product Properties of the cross product Area of a parallelogram Cross product as a determinant
Week 3	2.5	Equations of Lines, Planes and Surfaces in Space	The Dot Product The Cross Product Quadratic Functions Parametric Equations	Write the vector, parametric equation of a line through a given point in a given direction, and a line through two given points. Find the distance from a point to a given line. Write the equation of a plane through a given point with a given normal, and a plane through three given points. Find the distance from a point to a given plane.
Week 3	2.6	Cylinders and Quadratic Surfaces	Quadratic Functions Parametric Equations Conics	Find equations for cylinders that are generated by rotating lines that are parallel to a plane Understand basic quadratic surfaces Understand general quadratic surfaces
Weeks 3/4	3.1, 3.2	Curves in Space and Vector Functions	Parametric Equations Vectors The Derivative as a Function The Limit of a Function Continuity of a function The Dot Product The Cross Product	Vector functions Limits of vector functions Continuity of vector functions Differentiation rules for vector functions Curves and paths in space
Week 4	3.3		The Length of a Line Segment Vector Functions Integrals of Vector Functions	The arc Length of a vector function Arc length parameterization The Unit tangent vector
Weeks 4/5	3.4	Curvature and Normal Vectors	Antiderivatives Vectors Parametric Equations The Unit Tangent Vector	The curvature The Principal Unit Normal Vector
Weeks 4/5	3.4	Tangential and Normal Components of Acceleration	The Cross Product Derivatives of Vector Functions Curvature and Normal Vectors	The Binormal Vector The tangential and normal components of acceleration The Torsion
Week 5/6	4.1	Functions of Several Variables	Domain of a Function Range of a Function Solving Inequalities Graphing a Function	Functions of two variables Functions of three variables Domain and range of multivariable functions Bounded regions Graphs and level curves of two variable functions Level surfaces of three variable functions
Week 6	4.2	Limit and Continuity of Function of Several Variables	The Limit and Continuity of a Function The Limit Laws for Functions Composition of Functions The Dot Product	Limits of functions of two variables Limits of functions of more than two variables Properties of limits of functions of several variables Two path test of non-existing of a limit Continuity for functions of several variables Continuity of composition Extreme values on closed and bounded domains
Week 6	4.3	Partial Derivatives	The derivative and second derivative of a function The limit and continuity for a function of several variables	Partial derivatives for functions of two variables Partial derivatives for functions of more than two variables Partial derivatives and continuity Second order partial derivatives Mixed derivative theorem
Week 7	4.4	Directional Derivatives and Gradient Vectors	Trigonometric Functions Vectors, Unit Vectors Partial Derivatives Gradients The Dot Product	Directional derivatives for functions of two variables Gradients Properties of directional derivatives Tangents to level curves Directional derivatives for functions of three variables
Week 7	4.5	Tangent Plane Differentiability	Partial Derivatives Parametric Equations of Lines Equations of Planes	Determine the equation of a plane tangent to a given surface at a point Determine the parametric equation of a normal line to a given surface at a point The linear approximation of a function of two variables at a point The definition of differentiability for a function of two variables Differentiability implies Continuity Continuity of First Partial Derivatives implies Differentiability The definition of total differentiability for a function of two variables Use the total differential to approximate the change in a function of two variables
Week 7	4.6	The Chain Rule for Functions of more than One Variable	Differentiation Rules of Functions The Chain Rule of Functions Partial Derivatives	Chain rule for functions of one independent variable and several intermediate variables. Chain rule for functions of two independent variable and several intermediate variables. Method for implicit differentiation. The general chain rule for functions of several independent variables
Week 8	4.7	Maxima and Minima Problems	Extreme values on closed and bounded domains Partial Derivatives Maxima, Minima and Critical Points of a Function Continuity of functions with two variables	The derivative test for local extreme values Extreme values on closed and bounded domains Critical points and saddle points for functions of two variables Second derivative test for local extreme values Absolute maxima and minima on closed and bounded regions
Week 9	4.8	Lagrange Multiplires	Partial Derivatives Critical Points of a Function	Lagrange Multipliers with One Constraint Lagrange Multipliers with Two Constraints
Week 9/10	Chapter 5	Double and Iterated Integrals over Rectangles	Approximating Areas Limits of Riemann Sums	Double Integrals Fubini's Theorem (part 1)
Week 10	Chapter 5	Double Integrals over General Regions	Continuity Determining Volumes by Slicing Double and Iterated Integrals over Rectangles	Double integrals over bounded, nonrectangular regions Volumes of solid regions Fubini's theorem (part 2) Finding the limits of integration for regions in the plane Properties of double Integrals
Week 11	Chapter 5	Double Integrals in Polar Coordinates	Double Integrals over General Regions Polar Coordinates	Integrals in Polar Form Finding limits of integration for polar coordinates Changing Cartesian Integrals into Polar Integrals
Week 11	Chapter 5	Applications of Double Integrals	Moments and Center of Mass Triple Integrals in Rectangular Coordinates Partial Derivatives	Masses and First moments Moments of Inertia
Week 11	Chapter 5	Triple Integrals in Rectangular Coordinates	Double and Iterated Integrals over Rectangles Area by Double Integration Change of Variables	Triple Integrals Volume of a region in space Finding the limits of integration for triple integrals Average value of a function in space
Week 12	Chapter 5	Triple Integrals in Cylindrical and Spherical Coordinates	Double Integrals in Polar Form Polar Form Triple Integrals in Rectangular Coordinates	Integration in Cylindrical Coordinates Equations relating rectangular and cylindrical coordinates Spherical coordinates and integrations Equations relating spherical coordinates to Cartesian and cylindrical coordinates
Week 12	Chapter 5	Applications of Triple Integrals	Moments and Center of Mass Triple Integrals in Rectangular Coordinates Partial Derivatives	Masses and First moments Moments of Inertia
Week 14	Chapter 6	Line Integrals of Scalar Functions	Parametric Equations Curves in Space and Vector Functions Arc Length	Evaluating a Line Integral Additivity of Line Integrals Mass and Moments Line Integrals in the plane
Week 14	Chapter 6	Vector Fields	The Dot Product Directional Derivatives and Gradient Vectors Parametric Equations Line Integrals of Scalar Functions	Vector Fields Gradient Fields Line Integrals of vector fields Line integrals with respect to each components direction Work done by a force over a curve in space Flow integrals and circulation for velocity fields Flux across a simple closed plane curve
Week 14	Chapter 6	Conservation Fields	Vector Fields and Line Integrals Partial Derivatives Gradient Fields Tangent Planes and Differentials	Path Independence Piecewise smooth curves and connected domains in open regions Line integrals in Conservation fields Finding potentials for conservative fields Exact Differential forms
Weeks 14/15	Chapter 6	Green's Theorem	Vector Fields and Line Integrals Partial Derivatives Dot Product Path Independence and Conservation Fields	Circulation Density Divergence (flux density) of a vector field The two forms of Green's theorem (Tangential and Normal forms) Green's theorem for evaluating line integrals