Date |
Sections |
Topics |
Prerequisite Skills |
Student Learning Outcomes
|
Week 1
|
12.1
|
Three-Dimensional Coordinate Systems
|
|
- Three-dimensional coordinate systems
- Distance Formula in R3
- Standard Equation for a Sphere
|
Week 1/2
|
12.2
|
Vectors
|
|
- Vector Algebra Operations
- Magnitude of a vector
- Unit Vectors
- Midpoint of a Line Segment
- Angle between vectors
- Definition of Dot product
- Orthogonal vectors
- Vector projection
|
Week 2
|
12.3
|
The Dot Product
|
|
- Find the area of plane regions bounded by the graphs of functions.
|
Week 2/3
|
12.4
|
The Cross Product
|
|
- Define the cross product
- Properties of the cross product
- Area of a parallelogram
- Cross product as a determinant
|
|
Week 3
|
12.5
|
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
|
|
Week 3/4
|
13.1
|
Curves in Space and Vector Functions
|
|
- Vector functions
- Limits of vector functions
- Continuity of vector functions
- Differentiation of vector functions
- Differentiation rules for vector functions
- Curves and paths in space
|
|
Week 4
|
13.2
|
Integrals of Vector Functions
|
|
- Indefinite integrals of vector functions
- Definite integrals of vector functions
- Vector and parametric equations for ideal projectile motion
|
|
Week 4/5
|
13.3
|
Arc Length
|
|
- Length of a curve in R3
- General arc length formula
- Arc length for parameterized curves
- The Unit tangent vector
|
|
Week 5
|
13.4
|
Curvature and Normal Vectors
|
|
- Curvature in R2
- Formula for curvature
- Definition of Principal unit normal
- Curvature and normal vectors for higher dimensions.
|
|
Week 5/6
|
13.5
|
Tangential and Normal Components of Acceleration
|
|
- Binormal Vectors
- Definition of tangential and normal components of acceleration
- Torsion
|
|
Week 7
|
14.1
|
Functions of Several Variables
|
|
- Domain and range of multivariable functions
- Functions with two variables
- Bounded regions
- Graphs and level curves of two variable functions
- Functions of three variables
- Level surfaces
|
Week 7/8
|
14.2
|
Limits and Continuity in Higher Dimensions
|
|
- Limits of functions of two variables
- Properties of limits of functions of two variables
- Continuity for functions of two variables
- Continuity of composition
- Functions of more than two variables
- Extreme values on closed and bounded sets
|
Week 8
|
14.3
|
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
- Define the derivative for functions of two variables
|
Week 8/9
|
14.4
|
The Chain Rule for Functions of more than One Variable
|
|
- Chain rule for functions of one independent variable and two intermediate variables.
- Chain rule for functions of one independent variable and three intermediate variables.
- Chain rule for functions of two independent variable and two intermediate variables.
- Additional method for implicit differentiation.
- The general chain rule
|
Week 9
|
14.5
|
Directional Derivatives and Gradient Vectors
|
|
- Direction Derivatives in the plane
- Gradients
- Properties of directional derivatives
- Tangents to level curves
- Directional derivatives for functions of three variables
- The chain rule for paths
|
Week 9/10
|
14.6
|
Tangent Planes and Differentials
|
|
- Tangent Planes and Normal lines
- The plane tangent to a surface
- How to linearize a function of two variables
- Differentials for functions of two variables
- Linearization and differentials for functions of more than two variables
|
Week 10
|
14.7
|
Extreme Values and Saddle Points
|
|
- The derivative test for local extreme values
- 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 10/11
|
14.8
|
Lagrange Multipliers
|
|
- Define the convergence or divergence of an infinite series.
- Find the sum of a geometric or telescoping series.
|
Week 12
|
14.9
|
Taylors formula for Two Variables
|
|
- The derivation of the second derivative test
- Taylor's formula for functions of two variables
|
Week 12/13
|
15.1
|
Double and Iterated Integrals over Rectangles
|
|
- Double Integrals
- Fubini's Theorem (part 1)
|
Week 14
|
15.2
|
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 14/15
|
15.3
|
Area by Double Intgration
|
|
- Areas of bounded regions in the plane
- Average value for functions of two or more variables
|
Week 15
|
15.4
|
Double Integrals in Polar Form
|
|
- Integrals in Polar Form
- Finding limits of integration for polar coordinates
- Changing Cartesian Integrals into Polar Integrals
|
Week 15/16
|
15.5
|
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 16
|
15.6
|
Applications of Double and Triple Integrals
|
|
- Masses and First moments
- Moments of Inertia
|
Week 16/17
|
15.7
|
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 17
|
16.1
|
Line Integrals of Scalar Functions
|
|
- Evaluating a Line Integral
- Additivity of Line Integrals
- Mass and Moments
- Line Integrals in the plane
|
Week 18
|
16.2
|
Vector Fields and Line Integrals
|
|
- 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 18/19
|
16.3
|
Path Independence and 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
|
Week 20
|
16.4
|
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
|
Week 21
|
16.5
|
Surfaces and Area
|
|
- Parameterizations of Surfaces
- Surface Area
- Surface Area Differential for a Parameterized Surface
- Implicit Surfaces
|