Date |
Sections |
Topics |
Prerequisite Skills |
Student Learning Outcomes
|
Week 1
|
1.1
|
Polar Coordinates
|
|
- 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
|
|
- Three-dimensional coordinate systems.
- Distance Formula in Space.
- Standard Equation for a Sphere.
|
Weeks 1/2
|
2.1
|
Vectors in The Plane, Space
|
|
- 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
|
|
- Definition of Dot Product
- Properties of Dot Product
- Angle between vectors
- Orthogonal vectors
|
Week 2
|
2.4
|
The Cross Product
|
|
- 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
|
|
- 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
|
|
- 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
|
|
- 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 arc Length of a vector function
- Arc length parameterization
|
|
Weeks 4/5
|
3.4
|
Motion in Space
|
|
- 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
|
|
- 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
|
|
- 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
|
|
- 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
|
|
- 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
|
|
- 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
|
|
- 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
|
|
- 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
|
|
- Lagrange Multipliers with One Constraint
- Lagrange Multipliers with Two Constraints
|
Week 9/10
|
5.1
|
Double Integrals over Rectangular Regions
|
|
- 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
|
|
- 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 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
|
|
- 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
|
|
- 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
|
|
- 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 14
|
6.1
|
Vector Fields
|
|
- 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
|
|
- 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
|
Conservation Fields
|
|
- Closed Curve, Simple Curve.
- Simply connected regions.
- Line integrals in Conservation fields.
- Conservative Field is independent of path.
- Finding potentials for conservative fields.
|
Weeks 14/15
|
6.4
|
Green's Theorem
|
|
- Circulation form of Green's Theorem.
- Flux Form of Green’s Theorem.
- Applying Green's Theorem to find Work, Flux.
|