Difference between revisions of "MAT2214"

From Department of Mathematics at UTSA
Jump to navigation Jump to search
Line 371: Line 371:
 
* [[Domain of a Function]] <!-- 1073-Mod 1.2 -->
 
* [[Domain of a Function]] <!-- 1073-Mod 1.2 -->
 
* [[Range of a Function]] <!-- 1073-Mod 1.2 -->
 
* [[Range of a Function]] <!-- 1073-Mod 1.2 -->
* [[Solving Inequalities| Interval Notation]] <!-- 1073-Mod R -->
+
* [[Solving Inequalities]] <!-- 1073-Mod R -->
 
* [[Graphs| Graphing a Function]] <!-- 1073-Mod R -->
 
* [[Graphs| Graphing a Function]] <!-- 1073-Mod R -->
* '''[[Conics|Equation of a Circle]]''' <!-- DNE (recommend 1093) -->
 
  
  
 
||
 
||
 
+
* Functions of two variables
 +
* Functions of three variables
 
* Domain and range of multivariable functions
 
* Domain and range of multivariable functions
* Functions with two variables
 
 
* Bounded regions
 
* Bounded regions
 
* Graphs and level curves of two variable functions
 
* Graphs and level curves of two variable functions
* Functions of three variables
+
* Level surfaces of three variable functions
* Level surfaces
 
  
  
Line 403: Line 401:
 
||
 
||
  
* [[The Limit of a Function]] <!-- 1214-2.2 -->
+
* [[The Limit and Continuity of a Function]] <!-- 1214-2.2, 2.4 -->
* [[Continuity]] <!-- 1214-2.4 -->
+
* [[The Limit Laws for Functions]] <!-- 1214-2.3 -->
* [[The Limit Laws]] <!-- 1214-2.3 -->
 
 
* [[Composition of Functions]] <!-- 1073-Mod 7.1 -->  
 
* [[Composition of Functions]] <!-- 1073-Mod 7.1 -->  
 
* [[The Dot Product]] <!-- 2214-12.3 -->
 
* [[The Dot Product]] <!-- 2214-12.3 -->
* [[The Cross Product]] <!-- 2214-12.4 -->
 
  
 
||
 
||
  
 
* Limits of functions of two variables
 
* Limits of functions of two variables
* Properties of limits of functions of two variables
+
* Limits of functions of more than two variables
* Continuity for functions of 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
 
* Continuity of composition
* Functions of more than two variables
+
* Extreme values on closed and bounded domains
* Extreme values on closed and bounded sets
 
  
  
Line 435: Line 432:
 
||
 
||
  
* [[The Derivative as a Function| Second derivatives]] <!-- 1214-3.2 -->
+
* [[The Derivative and Second derivative of a function ]] <!-- 1214-3.2 -->
* [[Limits and Continuity in Higher Dimensions]] <!-- 2214-14.2 -->
+
* [[Limits and Continuity for functions of several variables]] <!-- 2214-14.2 -->
* [[Functions of Several Variables]] <!-- 1214-14.1 -->
+
 
  
 
||
 
||
Line 446: Line 443:
 
* Second order partial derivatives
 
* Second order partial derivatives
 
* Mixed derivative theorem
 
* Mixed derivative theorem
* Define the derivative for functions of two variables
+
 
  
  
Line 464: Line 461:
 
||
 
||
  
* [[Differentiation Rules]] <!-- 1214-3.3 -->
+
* [[Differentiation Rules of Functions]] <!-- 1214-3.3 -->
* [[The Chain Rule]] <!-- 1214-3.6 -->
+
* [[The Chain Rule of Functions]] <!-- 1214-3.6 -->
* [[Implicit Differentiation]] <!-- 1214-3.8 -->
 
* [[Parametric Equations]] <!-- 1224-7.1 -->
 
 
* [[Partial Derivatives]]  <!-- 1214-14.3 -->
 
* [[Partial Derivatives]]  <!-- 1214-14.3 -->
  
 
||
 
||
  
* Chain rule for functions of one independent variable and two intermediate variables.
+
* Chain rule for functions of one independent variable and several intermediate variables.
* Chain rule for functions of one independent variable and three intermediate variables.
+
* Chain rule for functions of two independent variable and several intermediate variables.
* Chain rule for functions of two independent variable and two intermediate variables.
+
* Method for implicit differentiation.
* Additional method for implicit differentiation.
+
* The general chain rule for functions of several independent variables 
* The general chain rule
 
  
 
|-
 
|-
Line 494: Line 488:
  
 
* [[Trigonometric Functions]]  <!-- 1093-2.2 -->  
 
* [[Trigonometric Functions]]  <!-- 1093-2.2 -->  
* [[The Chain Rule for Functions of more than One Variable]]  <!-- 2214-14.4 -->
+
* [[Vectors, Unit Vectors]]  <!-- 2214-12.2 -->  <!-- 2233-1.1 -->
* [[Vectors|Unit Vectors]]  <!-- 2214-12.2 -->  <!-- 2233-1.1 -->
 
 
* [[Partial Derivatives]]  <!-- 1214-14.3 -->
 
* [[Partial Derivatives]]  <!-- 1214-14.3 -->
 
* [[The Dot Product]] <!-- 2214-12.3 -->
 
* [[The Dot Product]] <!-- 2214-12.3 -->
Line 501: Line 494:
 
||
 
||
  
* Direction Derivatives in the plane
+
* Directional derivatives for functions of two variables
 
* Gradients
 
* Gradients
 
* Properties of directional derivatives
 
* Properties of directional derivatives
 
* Tangents to level curves
 
* Tangents to level curves
 
* Directional derivatives for functions of three variables
 
* Directional derivatives for functions of three variables
* The chain rule for paths
 
  
  
Line 521: Line 513:
 
||   
 
||   
  
[[Tangent Planes and Differentials]]
+
[xx]
  
 
||
 
||
  
* [[Linear Approximations and Differentials]] <!-- 1214-4.2 -->
+
 
* [[Parametric Equations]] <!-- 1224-7.1 -->
 
* [[Partial Derivatives]]  <!-- 1214-14.3 -->
 
* [[Linear Equations|Slope of a Line]] <!-- 1073-Mod R -->
 
* [[Cylinders and Quadratic Surfaces]] <!-- 2214-12.5 -->
 
  
 
||
 
||
  
* 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
 
  
  

Revision as of 15:53, 23 September 2020

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

  • Vector Algebra Operations
  • The Magnitude of a vector
  • Unit Vectors
  • The Midpoint of a Line Segment
  • The Vector projection
Week 2
Chapter 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
Chapter 2

The Dot Product


  • Definition of Cross Product
  • Properties of the cross product
  • Angle between vectors
  • Orthogonal vectors
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


Week 3
Chapter 2


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.


Weeks 3/4
Chapters 2 and 3

Curves in Space and Vector Functions

  • Vector functions
  • Limits of vector functions
  • Continuity of vector functions
  • Differentiation rules for vector functions
  • Curves and paths in space


Week 4
Chapter 3

Vector-Valued Functions: Arc Length

  • The arc Length of a vector function
  • Arc length parameterization
  • The Unit tangent vector


Weeks 4/5
Chapter 3

Curvature and Normal Vectors


  • The curvature
  • The Principal Unit Normal Vector


Weeks 4/5
Chapter 3


Tangential and Normal Components of Acceleration


  • The Binormal Vector
  • The tangential and normal components of acceleration
  • The Torsion


Week 6
Chapter 4


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
Chapter 4


Limits and Continuity in Higher Dimensions

  • 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
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


Week 7
Chapter 4

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 7
Chapter 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
Chapter 4

[xx]




Week 7
Chapter 4

Linear Approximations

  • Determine the equation of a plane tangent to a given surface at a point.
  • Use the tangent plane to approximate a function of two variables at a point.
  • Explain when a function of two variables is differentiable.
  • Use the total differential to approximate the change in a function of 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