MAT2213

Topics List

Date	Sections	Topics	Prerequisite Skills	Student Learning Outcomes
Week 1	1.1	Polar Coordinates	Trigonometric Functions: Unit Circle Approach Inverse Trigonometric 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 Space. Standard Equation for a Sphere.
Weeks 1/2	2.1	Vectors in The Plane, Space	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	2.3	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-Valued Functions	Parametric Equations Vectors The Derivative as a Function The Limit of a Function Continuity 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	Arc Length	The Length of a Line Segment Vector Functions Integrals of Vector Functions, Derivatives of Vector Functions	The arc Length of a vector function Arc length parameterization
Weeks 4/5	3.4	Motion in Space	Vectors Parametric Equations The Cross Product Derivatives of Vector Functions	The Unit tangent vector The curvature The Principal Unit Normal Vector 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 Equations and 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	Continuity The Limit Laws 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 first and second derivative of a function Limit and Continuity of 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 Lines, Planes and Surfaces in Space	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 The Chain Rule 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 Limit and Continuity of Function of Several 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 8/9	4.8	Lagrange Multipliers	Partial Derivatives Critical Points of a Function	Lagrange Multipliers with One Constraint Lagrange Multipliers with Two Constraints
Week 9/10	5.1	Double Integrals over Rectangular Regions	Approximating Areas Limits of Riemann Sums	Double Integral is the limit of Double Sums. Double Integrals over Rectangular Regions. Interated Integrals. Fubini's Theorem (part 1).
Week 10	5.2	Double Integrals over General Regions	Continuity Determining Volumes by Slicing Double and Iterated Integrals over Rectangular regions	Double integrals over bounded, general regions. Properties of double Integrals. Fubini's theorem (part 2) Changing the order of Integration. Calculating Volumes, Areas and Average Values
Week 11	5.3	Double Integrals in Polar Coordinates	Double Integrals over General Regions Polar Coordinates	Double Integrals over rectangular polar regions. Double Integrals over general polar regions. Changing Cartesian Integrals into Polar Integrals. Using Double Integrals in Polar Coordinates to find Volumes, Areas.
Week 11	5.4	Triple Integrals in Rectangular Coordinates	Double Integrals Area by Double Integration Change of Variables	Triple Integrals over general bounded regions. Finding Volumes by evaluating Triple Integrals. Average value of a function in space. Changing Integration Order and Coordinate systems.
Week 12	5.5	Triple Integrals in Cylindrical and Spherical Coordinates	Double Integrals in Polar Form Triple Integrals in Rectangular Coordinates	Integrations in Cylindrical Coordinates. Equations relating rectangular and cylindrical coordinates. Changing Cartesian integrations into Cylindrical integrations. Integrations in Spherical coordinates. Equations relating spherical coordinates to Cartesian and cylindrical coordinates. Changing Cartesian integrations into Cylindrical integrations.
Week 13	5.6	Applications of Multiple Integrals	Double Integral Triple Integrals	Finding Masses, Moments, Centers of Masses, Moments of Inertia in Two Dimensions. Finding Masses, Moments, Centers of Masses, Moments of Inertia in Three Dimensions.
Week 13/14	5.7	Change of Variables in Multiple Integrals	Double Integral Triple Integrals	Determine the image of a region under a given transformation of variables. Compute the Jacobian of a given transformation. Evaluate a double integral using a change of variables. Evaluate a triple integral using a change of variables.
Week 14	6.1	Vector Fields	The Dot Product Directional Derivatives and Gradient Vectors	Vector Fields in a plane. Vector Fields in Space. Potential Functions. Gradient Fields, Conservative Vector Fields. The Cross-Partial Test for Conservative Vector Fields. Determining Whether a Vector Field is conservative.
Week 14	6.2	Line Integrals	Parametric Equations Curves in Space and Vector-Valued Functions Arc Length	Line Integrals of functions a long a smooth curves in a planer or in space Line Integrals of of vector fields along an oriented curves in a plane or space.. Properties of Vector Line Integrals. Evaluating Line Integrals. Applications of line integrals: Calculating Arc Length, the Mass of a wire, Work done by a force, Flux across a curve, Circulation of a force.
Week 14/15	6.3	Conservative Vector Fields	Vector Fields and Line Integrals Partial Derivatives	Describe simple and closed curves Define connected and simply connected regions. Explain how to test a vector field to determine whether it is conservative. Find a potential function for a conservative vector field. Use the Fundamental Theorem for Line Integrals to evaluate a line integral of a vector field.
Weeks 14/15	6.4	Green's Theorem Stokes' Theorem	Vector Fields Line Integrals Partial Derivatives The Dot Product Path Independence Conservative Vector Fields	Circulation form of Green's Theorem. Flux Form of Green’s Theorem. Applying Green's Theorem to find Work, Flux.