MAT2214

From Department of Mathematics at UTSA
Revision as of 19:10, 12 August 2020 by James.kercheville (talk | contribs) (Altered the table to match the new list of Topics (several large changes to contents))
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Topics List

Date Sections Topics Prerequisite Skills Student Learning Outcomes
Week 1
Chapter 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
Chapter 1

Three-Dimensional Coordinate Systems


  • Three-dimensional coordinate systems
  • Distance Formula in R3
  • Standard Equation for a Sphere


Weeks 1/2
Chapter 2


Vectors in The Plane and in Three Dimensions

  • Vector Algebra Operations
  • Magnitude of a vector
  • Unit Vectors
  • Midpoint of a Line Segment
  • Angle between vectors
  • Orthogonal vectors
  • Vector projection
Week 2
Chapter 2


Vectors in Space

  • Vector Algebra Operations
  • Magnitude of a vector
  • Unit Vectors
  • Midpoint of a Line Segment
  • Angle between vectors
  • Orthogonal vectors
  • Vector projection
Week 2
Chapter 2

The Dot Product


  • Find the area of plane regions bounded by the graphs of functions.
Week 2
Chapter 2

The Cross Product

  • Define the cross product
  • Properties of the cross product
  • Area of a parallelogram
  • Cross product as a determinant


Week 3
Chapter 2


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
Chapters 2 and 3

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
Chapter 3

Vector-Valued Functions: Arc Length

  • Length of a curve in R3
  • General arc length formula
  • Arc length for parameterized curves
  • The Unit tangent vector


Weeks 4/5
Chapter 3

Curvature and Normal Vectors

  • Curvature in R2
  • Formula for curvature
  • Definition of Principal unit normal
  • Curvature and normal vectors for higher dimensions.


Weeks 4/5
Chapter 3


Tangential and Normal Components of Acceleration

  • Binormal Vectors
  • Definition of tangential and normal components of acceleration
  • Torsion


Week 6
Chapter 4


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 6
Chapter 4


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 6
Chapter 4

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 7
Chapter 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 7
Chapter 4

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 7
Chapter 4

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 8
Chapter 4

Maxima and Minima Problems

  • 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
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