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 R3
- Standard Equation for a Sphere
|
Weeks 1/2
|
2.1
|
Vectors in The Plane
|
|
- 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
|
|
- 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 and
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 9
|
4.8
|
Lagrange Multipliers
|
|
- Lagrange Multipliers with One Constraint
- Lagrange Multipliers with Two Constraints
|
Week 9/10
|
Chapter 5
|
Double and Iterated Integrals over Rectangles
|
|
- Double Integrals
- Fubini's Theorem (part 1)
|
Week 10
|
Chapter 5
|
Double Integrals over General Regions
|
|
- 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
|
|
- 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
|
|
- Masses and First moments
- Moments of Inertia
|
Week 11
|
Chapter 5
|
Triple Integrals in Rectangular Coordinates
|
|
- 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
|
|
- 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
|
|
- Masses and First moments
- Moments of Inertia
|
Week 14
|
Chapter 6
|
Line Integrals of Scalar Functions
|
|
- Evaluating a Line Integral
- Additivity of Line Integrals
- Mass and Moments
- Line Integrals in the plane
|
Week 14
|
Chapter 6
|
Vector Fields
|
|
- 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
|
|
- 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
|
|
- 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
|