Difference between revisions of "MAT2214"

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* Absolute maxima and minima on closed and bounded regions
 
* Absolute maxima and minima on closed and bounded regions
 
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|Week 9  
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|Week 8/9  
  
 
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<div style="text-align: center;">Chapter 5</div>
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<div style="text-align: center;">5.1</div>
  
 
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[[Double and Iterated Integrals over Rectangles]]
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[[Double Integrals over Rectangular Regions]]
  
 
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* Double Integrals
+
* Double Integrals over Rectangular Regions.
* Fubini's Theorem (part 1)
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* Interated Integrals.
 
+
* Fubini's Theorem (part 1).
 
 
 
 
 
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<div style="text-align: center;">Chapter 5</div>
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<div style="text-align: center;">5.2</div>
  
 
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* [[Continuity]] <!-- 1214-3.5 -->
 
* [[Continuity]] <!-- 1214-3.5 -->
 
* [[Determining Volumes by Slicing]] <!-- 1224-2.2 -->
 
* [[Determining Volumes by Slicing]] <!-- 1224-2.2 -->
* [[Double and Iterated Integrals over Rectangles]] <!-- 2214-15.1 -->
+
* [[Double and Iterated Integrals over Rectangular regions]] <!-- 2214-5.1 -->
  
 
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* Double integrals over bounded, nonrectangular regions
+
* Double integrals over bounded, general regions
* Volumes of solid regions
+
* Properties of double Integrals
 
* Fubini's theorem (part 2)  
 
* Fubini's theorem (part 2)  
* Finding the limits of integration for regions in the plane
+
* Changing the order of Integration.
* Properties of double Integrals
+
* Calculating Volumes, Areas and Average Values
 
 
 
 
 
 
 
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<div style="text-align: center;">Chapter 5</div>
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<div style="text-align: center;">5.3</div>
  
 
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* Double Integrals in Polar Coordinates: 
* Integrals in Polar Form
+
Double Integrals over rectangular polar  regions, and
* Finding limits of integration for polar coordinates
+
Double Integrals over general polar regions.
* Changing Cartesian Integrals into Polar Integrals
+
* Changing Cartesian Integrals into Polar Integrals.
 
+
* Finding Volumes, Areas using Double Integrals in Polar Coordinates.
 
 
 
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<div style="text-align: center;">Chapter 5</div>
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<div style="text-align: center;">5.4</div>
  
 
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<div style="text-align: center;">Chapter 5</div>
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<div style="text-align: center;">Chapter 5.4</div>
  
 
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* [[Double and Iterated Integrals over Rectangles]] <!-- 2214-15.1 -->
+
* [[Double Integrals]] <!-- 2214-5.1,  2214-5.2 -->
 
* [[Area by Double Integration]] <!-- 2214-15.3 -->
 
* [[Area by Double Integration]] <!-- 2214-15.3 -->
 
* '''[[Change of Variables]]''' <!-- DNE (recommend 1073) -->
 
* '''[[Change of Variables]]''' <!-- DNE (recommend 1073) -->
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* Triple Integrals
+
* Triple Integrals over general bounded regions.
* Volume of a region in space
+
* Finding Volumes by evaluating Triple Integrals.
* Finding the limits of integration for triple integrals
+
* Average value of a function in space.
* Average value of a function in space
+
* Changing Integration Order and Coordinate systems.
 
 
 
 
 
 
 
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<div style="text-align: center;">Chapter 5</div>
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<div style="text-align: center;">5.5</div>
  
 
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* [[Double Integrals in Polar Form]]  <!-- 2214-15.4 -->
 
* [[Double Integrals in Polar Form]]  <!-- 2214-15.4 -->
* [[Parametric Equations| Polar Form]] <!-- 1093-5.1 -->
 
 
* [[Triple Integrals in Rectangular Coordinates]] <!-- 2214-15.5 -->
 
* [[Triple Integrals in Rectangular Coordinates]] <!-- 2214-15.5 -->
  
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* Spherical coordinates and integrations
 
* Spherical coordinates and integrations
 
* Equations relating spherical coordinates to Cartesian and cylindrical coordinates
 
* Equations relating spherical coordinates to Cartesian and cylindrical coordinates
 
 
 
 
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<div style="text-align: center;">Chapter 5</div>
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<div style="text-align: center;">5.6</div>
  
 
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[[Applications of Triple Integrals]]
+
[[Applications of Multiple Integrals]]
  
 
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* [[Moments and Center of Mass]] <!-- 1224-2.6 -->
+
* [[Double Integral]] <!-- 2214-5.2,2214-5.-->
* [[Triple Integrals in Rectangular Coordinates]] <!-- 2214-15.5 -->
+
* [[Triple Integrals]] <!-- 2214-5.3,2214-5.4 -->
* [[Partial Derivatives]] <!-- 1214-14.3 -->
 
  
 
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* Masses and First moments
+
* Finding Masses, Moments, Centers of Masses, Moments of Inertia
* Moments of Inertia
+
in Two Dimensions.
 
+
* Finding Masses, Moments, Centers of Masses, Moments of Inertia
 
+
in Three Dimensions.
 
 
 
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Revision as of 22:25, 25 October 2020

Topics List

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 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 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 in Polar Coordinates:

Double Integrals over rectangular polar regions, and Double Integrals over general polar regions.

  • Changing Cartesian Integrals into Polar Integrals.
  • Finding Volumes, Areas using Double Integrals in Polar Coordinates.
Week 11
5.4

Applications of Double Integrals

  • Masses and First moments
  • Moments of Inertia


Week 11
Chapter 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

  • 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
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
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