Difference between revisions of "MAT2214"

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The textbook for this course is
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[https://openstax.org/details/books/calculus-volume-3 Calculus (Volume 3) by Gilbert Strang, Edwin Herman, et al.]
 +
 
==Topics List==
 
==Topics List==
 
{| class="wikitable sortable"
 
{| class="wikitable sortable"
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|Week 1
+
|Week 1
  
 
||
 
||
 
+
1.1
<div style="text-align: center;">1.1</div>
 
 
 
 
||
 
||
 
          
 
          
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||
 
||
* [[Trigonometric Functions: Unit Circle Approach]] <!-- 1093-2.2 -->
+
* [[Trigonometric Functions: Unit Circle Approach]]  
* [[Inverse Trigonometric Functions]] <!-- 1093-3.1 -->
+
* [[Inverse Trigonometric Functions]]  
 
||
 
||
 
* Plot points using polar coordinates and find several polar coordinates of a single point
 
* Plot points using polar coordinates and find several polar coordinates of a single point
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|Week&nbsp;1
+
|Week 1
  
 
||
 
||
 
+
1.2
<div style="text-align: center;">1.2</div>
 
 
 
 
||
 
||
 
          
 
          
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||
 
||
  
* [[Two-dimensional coordinate systems]] <!-- 2233-3.3 -->
+
* [[Two-dimensional coordinate systems]]  
* [[Solving Equations and Inequalities| Algebraic Expressions]] <!-- 1073-Mod R -->
+
* [[Solving Equations and Inequalities| Algebraic Expressions]]  
  
  
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|Weeks&nbsp;1/2   
+
|Weeks 1/2   
  
 
||
 
||
 
+
2.1
<div style="text-align: center;">2.1</div>
 
 
 
 
||
 
||
 
    
 
    
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||
 
||
  
* [[Linear Equations|Line Segments]] <!-- 1073-Mod R -->
+
* [[Linear Equations|Line Segments]]  
* [[Distance Formula| Distance Formula]] <!-- DNE (recommend pairing with discussion of absolute value function) -->
+
* [[Distance Formula| Distance Formula]]  
  
 
||
 
||
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|Week&nbsp;2
+
|Week 2
  
 
||
 
||
 
+
2.3
<div style="text-align: center;">2.3</div>
 
 
 
 
||
 
||
 
    
 
    
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||
 
||
  
* [[Trigonometric Functions|Basic Trig Functions]]  <!-- 1093-2.2 -->
+
* [[Trigonometric Functions|Basic Trig Functions]]   
* [[Vectors]]  <!-- 2214-12.2 -->  <!-- 2233-1.1 -->
+
* [[Vectors]]   
  
 
||
 
||
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|Week&nbsp;2
+
|Week 2
  
 
||
 
||
 
+
2.4
<div style="text-align: center;">2.4</div>
 
 
 
 
||
 
||
 
    
 
    
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||
 
||
  
* [[Trigonometric Functions|Basic Trig Functions]]  <!-- 1093-2.2 -->
+
* [[Trigonometric Functions|Basic Trig Functions]]   
* [[Determinants]] <!-- 2233-6.1,6.2 -->
+
* [[Determinants]]  
 
* [[Vectors]]
 
* [[Vectors]]
  
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* Area of a parallelogram
 
* Area of a parallelogram
 
* Cross product as a determinant
 
* Cross product as a determinant
 
||
 
 
  
  
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|Week&nbsp;3
+
|Week 3
  
 
||
 
||
 
+
2.5
<div style="text-align: center;">2.5</div>
 
 
 
 
||
 
||
 
    
 
    
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||
 
||
  
* [[The Dot Product]] <!-- 2214-12.3 -->
+
* [[The Dot Product]]  
 
* [[The Cross Product]]  
 
* [[The Cross Product]]  
* [[Quadratic Functions]] <!-- 1073-Mod R -->
+
* [[Quadratic Functions]]  
* [[Parametric Equations]] <!-- 1224-7.1 -->
+
* [[Parametric Equations]]  
  
 
||
 
||
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* Find the distance from a point to a given plane.
 
* Find the distance from a point to a given plane.
  
||
 
  
 
|-
 
|-
  
  
|Week&nbsp;3
+
|Week 3
  
 
||
 
||
 
+
2.6
<div style="text-align: center;">2.6</div>
 
 
 
 
||
 
||
 
    
 
    
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||
 
||
  
* [[Quadratic Functions]] <!-- 1073-Mod R -->
+
* [[Quadratic Functions]]  
* [[Parametric Equations]] <!-- 1224-7.1 -->
+
* [[Parametric Equations]]  
* '''[[Conics]]''' <!-- DNE (recommend 1093) -->
+
* [[Conics]]  
  
 
||
 
||
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* Understand basic quadratic surfaces
 
* Understand basic quadratic surfaces
 
* Understand general quadratic surfaces
 
* Understand general quadratic surfaces
 
||
 
 
 
  
  
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|Weeks&nbsp;3/4
+
|Weeks 3/4
  
 
||
 
||
 
+
3.1, 3.2
<div style="text-align: center;">3.1, 3.2</div>
 
 
 
 
||  
 
||  
  
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||
 
||
  
* [[Parametric Equations]] <!-- 1224-7.1 -->
+
* [[Parametric Equations]]  
* [[Vectors]]  <!-- 2214-12.2 -->  <!-- 2233-1.1 -->
+
* [[Vectors]]   
* [[The Derivative as a Function]] <!-- 1214-3.2 -->
+
* [[The Derivative as a Function]]  
* [[The Limit of a Function]] <!-- 1214-2.2 -->
+
* [[The Limit of a Function]]  
* [[Continuity]] <!-- 1214-2.4 -->
+
* [[Continuity]]  
* [[The Dot Product]] <!-- 2214-12.3 -->
+
* [[The Dot Product]]  
* [[The Cross Product]] <!-- 2214-12.4 -->
+
* [[The Cross Product]]  
  
 
||
 
||
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* Differentiation rules for vector functions
 
* Differentiation rules for vector functions
 
* Curves and paths in space
 
* Curves and paths in space
 
||
 
  
  
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|Week&nbsp;4
+
|Week 4
  
 
||
 
||
 
+
3.3
<div style="text-align: center;">3.3</div>
 
 
 
 
||
 
||
 
    
 
    
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||
 
||
  
* '''[[Distance Formula| The Length of a Line Segment]]''' <!-- DNE (recommend pairing with discussion of absolute value function) -->
+
* [[Distance Formula| The Length of a Line Segment]]
* [[Curves in Space and Vector Functions|Vector Functions]] <!-- 1224-7.1 -->
+
* [[Curves in Space and Vector-Valued Functions|Vector Functions]]  
* [[Integrals of Vector Functions]] <!-- 2214-13.2 -->
+
* [[Line Integrals|Integrals of Vector Functions]], [[Derivatives of Vector Functions]]
  
 
||
 
||
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* The arc Length of a vector function
 
* The arc Length of a vector function
 
* Arc length parameterization
 
* Arc length parameterization
||
 
 
 
  
 
|-
 
|-
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|Weeks&nbsp;4/5
+
|Weeks 4/5
  
 
||
 
||
 
+
3.4
<div style="text-align: center;">3.4</div>
 
 
 
 
||
 
||
 
    
 
    
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||
 
||
* [[Vectors]] <!-- 1214-12.2 -->
+
* [[Vectors]]
* [[Parametric Equations]] <!-- 1224-7.1 -->
+
* [[Parametric Equations]]  
* [[The Cross Product]] <!-- 2214-12.4 -->
+
* [[The Cross Product]]  
* [[Derivatives of Vector Functions]] <!-- 1224-7.1 -->
+
* [[Derivatives of Vector Functions]]  
 
||
 
||
 
* The Unit tangent vector
 
* The Unit tangent vector
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* The tangential and normal components of acceleration
 
* The tangential and normal components of acceleration
 
* The Torsion
 
* The Torsion
||
 
 
 
  
 
|-
 
|-
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|Week&nbsp;5/6   
+
|Week 5/6   
  
 
||
 
||
 
+
4.1
<div style="text-align: center;">4.1</div>
 
 
 
 
||
 
||
 
    
 
    
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||
 
||
  
* [[Domain of a Function]] <!-- 1073-Mod 1.2 -->
+
* [[Domain of a Function]]  
* [[Range of a Function]] <!-- 1073-Mod 1.2 -->
+
* [[Range of a Function]]  
* [[Solving Equations and Inequalities]] <!-- 1073-Mod R -->
+
* [[Solving Equations and Inequalities]]  
* [[Graphs| Graphing a Function]] <!-- 1073-Mod R -->
+
* [[Graphs| Graphing a Function]]  
  
 
||
 
||
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|Week&nbsp;6
+
|Week 6
  
 
||
 
||
 
+
4.2
<div style="text-align: center;">4.2</div>
 
 
 
 
||
 
||
 
    
 
    
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||
 
||
  
* [[Continuity] <!-- 1214-2.2, 2.4 -->
+
* [[Continuity]]
* [[The Limit Laws for Functions]] <!-- 1214-2.3 -->
+
* [[The Limit Laws]]  
* [[Composition of Functions]] <!-- 1073-Mod 7.1 -->
+
* [[Composition of Functions]]  
* [[The Dot Product]] <!-- 2214-12.3 -->
+
* [[The Dot Product]]  
  
 
||
 
||
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|Week&nbsp;6
+
|Week 6
  
 
||
 
||
 
+
4.3
<div style="text-align: center;">4.3</div>
 
 
 
 
||   
 
||   
  
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||
 
||
  
* [[The derivative and second derivative of a function ]] <!-- 1214-3.2 -->
+
* [[The Derivative as a Function|The first]] and [[The Second Derivative|second derivative]] of a function  
* [[The limit and continuity for a function of several variables]] <!-- 2214-14.2 -->
+
* [[Limit and Continuity of Function of Several Variables]]  
  
  
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|Week&nbsp;7   
+
|Week 7   
  
 
||
 
||
 
+
4.4
<div style="text-align: center;">4.4</div>
 
 
 
 
||   
 
||   
  
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||
 
||
* [[Trigonometric Functions]] <!-- 1093-2.2 -->
+
* [[Trigonometric Functions]]  
* [[Vectors, Unit Vectors]]  <!-- 2214-12.2 -->  <!-- 2233-1.1 -->
+
* [[Vectors, Unit Vectors]]   
* [[Partial Derivatives]]  <!-- 1214-14.3 -->
+
* [[Partial Derivatives]]   
* Gradients
+
* [[Gradients]]
* [[The Dot Product]] <!-- 2214-12.3 -->
+
* [[The Dot Product]]  
 
||
 
||
 
* Directional derivatives for functions of two variables
 
* Directional derivatives for functions of two variables
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|Week&nbsp;7  
+
|Week 7  
  
 
||
 
||
 
+
4.5
<div style="text-align: center;">4.5</div>
 
 
 
 
||   
 
||   
 
[[Tangent Plane]],
 
[[Tangent Plane]],
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||
 
||
* [[Partial Derivatives]]  <!-- 1214-14.3 -->
+
* [[Partial Derivatives]]   
* [[Parametric Equations of Lines]]
+
* [[Parametric Equations]] of Lines
* [[Equations of Planes]]  
+
* [[Equations of Lines, Planes and Surfaces in Space]]  
  
 
||
 
||
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|Week&nbsp;7  
+
|Week 7  
  
 
||
 
||
 
+
4.6
<div style="text-align: center;">4.6</div>
 
 
 
 
||   
 
||   
  
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||
 
||
* [[Differentiation Rules of Functions]] <!-- 1214-3.3 -->
+
* [[Differentiation Rules]]  
* [[The Chain Rule of Functions]] <!-- 1214-3.6 -->
+
* [[The Chain Rule]]  
* [[Partial Derivatives]]  <!-- 1214-14.3 -->
+
* [[Partial Derivatives]]   
  
 
||
 
||
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* The general chain rule for functions of several independent variables   
 
* The general chain rule for functions of several independent variables   
 
|-
 
|-
|Week&nbsp;8  
+
|Week 8  
  
 
||
 
||
 
+
4.7
<div style="text-align: center;">4.7</div>
 
 
 
 
||   
 
||   
  
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||
 
||
* [[Limit and Continuity for a Function of several variables|Extreme values on closed and bounded domains]] <!-- 2214-14.2 -->
+
* [[Extreme values on closed and bounded domains]]  
* [[Partial Derivatives]]  <!-- 1214-14.3 -->
+
* [[Partial Derivatives]]   
* [[Maxima, Minima and Critical Points of a Function]] <!-- 1214-4.3 -->
+
* [[Maxima and Minima|Maxima, Minima and Critical Points of a Function]]  
* [[Continuity of functions with two variables]] <!-- 2214-14.2 -->
+
* [[Limit and Continuity of Function of Several Variables]]  
  
 
||
 
||
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* Absolute maxima and minima on closed and bounded regions
 
* Absolute maxima and minima on closed and bounded regions
 
|-
 
|-
|Week&nbsp;8/9  
+
|Week 8/9  
  
 
||
 
||
 
+
4.8
<div style="text-align: center;">4.8</div>
 
 
 
 
||   
 
||   
  
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||
 
||
  
* [[Partial Derivatives]]  <!-- 1214-14.3 -->
+
* [[Partial Derivatives]]   
* [[Critical Points of a Function]] <!-- 1214-4.3 -->
+
* [[Critical Points of a Function]]  
  
 
||
 
||
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|Week&nbsp;9/10
+
|Week 9/10
  
 
||
 
||
 
+
5.1
<div style="text-align: center;">5.1</div>
 
 
 
 
||   
 
||   
  
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||
 
||
  
* [[Approximating Areas]] <!-- 1214-5.1 -->
+
* [[Approximating Areas]]  
* [[The Definite Integral|Limits of Riemann Sums]] <!-- 1214-5.2 -->
+
* [[The Definite Integral|Limits of Riemann Sums]]  
  
 
||
 
||
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|Week&nbsp;10
+
|Week 10
  
 
||
 
||
 
+
5.2
<div style="text-align: center;">5.2</div>
 
 
 
 
||   
 
||   
  
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||
 
||
  
* [[Continuity]] <!-- 1214-3.5 -->
+
* [[Continuity]]  
* [[Determining Volumes by Slicing]] <!-- 1224-2.2 -->
+
* [[Determining Volumes by Slicing]]  
* [[Double and Iterated Integrals over Rectangular regions]] <!-- 2214-5.1 -->
+
* [[Multiple Integrals|Double and Iterated Integrals over Rectangular regions]]  
  
 
||
 
||
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|Week&nbsp;11     
+
|Week 11     
  
 
||
 
||
 
+
5.3
<div style="text-align: center;">5.3</div>
 
 
 
 
||   
 
||   
  
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||
 
||
  
* [[Multiple Integrals|Double Integrals over General Regions]] <!-- 2214-15.2 -->
+
* [[Multiple Integrals|Double Integrals over General Regions]]  
* [[Parametric Equations| Polar Coordinates]] <!-- 1093-5.1 -->
+
* [[Polar Coordinates]]  
  
 
||
 
||
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|Week&nbsp;11  
+
|Week 11  
  
 
||
 
||
 
+
5.4
<div style="text-align: center;"> 5.4</div>
 
 
 
 
||  
 
||  
  
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||
 
||
  
* [Multiple Integrals|[Double Integrals]] <!-- 2214-5.1,  2214-5.2 -->
+
* [[Multiple Integrals|Double Integrals]]  
* [[Area by Double Integration]] <!-- 2214-15.3 -->
+
* [[Multiple Integrals|Area by Double Integration]]  
* '''[[Change of Variables]]''' <!-- DNE (recommend 1073) -->
+
* [[Change of Variables]]  
  
 
||
 
||
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|Week&nbsp;12
+
|Week 12
  
 
||
 
||
 
+
5.5
<div style="text-align: center;">5.5</div>
 
 
 
 
||  
 
||  
  
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||
 
||
  
* [[Multiple Integrals|Double Integrals in Polar Form]]  <!-- 2214-15.4 -->
+
* [[Multiple Integrals|Double Integrals in Polar Form]]   
* [[Multiple Integrals|Triple Integrals in Rectangular Coordinates]] <!-- 2214-15.5 -->
+
* [[Multiple Integrals|Triple Integrals in Rectangular Coordinates]]  
  
 
||
 
||
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|Week&nbsp;13   
+
|Week 13   
  
 
||
 
||
 
+
5.6
<div style="text-align: center;">5.6</div>
 
 
 
 
||
 
||
 
    
 
    
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||
 
||
  
* [[Multiple Integrals|Double Integral]] <!-- 2214-5.2,2214-5.3  -->
+
* [[Multiple Integrals|Double Integral]]  
* [[Multiple Integrals|Triple Integrals]] <!-- 2214-5.3,2214-5.4 -->
+
* [[Multiple Integrals|Triple Integrals]]  
  
 
||
 
||
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* Finding Masses, Moments, Centers of Masses, Moments of Inertia in Three Dimensions.
 
* Finding Masses, Moments, Centers of Masses, Moments of Inertia in Three Dimensions.
 
|-
 
|-
|Week&nbsp;13/14   
+
|Week 13/14   
  
 
||
 
||
 
+
5.7
<div style="text-align: center;">5.7</div>
 
 
 
 
||
 
||
 
    
 
    
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||
 
||
  
* [[Multiple Integrals|Double Integral]] <!-- 2214-5.2,2214-5.3  -->
+
* [[Multiple Integrals|Double Integral]]  
* [[Multiple Integrals|Triple Integrals]] <!-- 2214-5.3,2214-5.4 -->
+
* [[Multiple Integrals|Triple Integrals]]  
  
 
||
 
||
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* Evaluate a triple integral using a change of variables.
 
* Evaluate a triple integral using a change of variables.
 
|-
 
|-
|Week&nbsp;14  
+
|Week 14  
  
 
||
 
||
 
+
6.1
<div style="text-align: center;">6.1</div>
 
 
 
 
||
 
||
  
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* [[The Dot Product]] <!-- 2214-12.3 -->
 
* [[The Dot Product]] <!-- 2214-12.3 -->
* [[Directional Derivatives and Gradient Vectors]] <!-- 2214-14.5 -->
+
* [[Directional Derivatives and Gradient Vectors]]  
 
||
 
||
 
* Vector Fields in a plane.
 
* Vector Fields in a plane.
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|Week&nbsp;14  
+
|Week 14  
  
 
||
 
||
 
+
6.2
<div style="text-align: center;">6.2</div>
 
 
 
 
||   
 
||   
  
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||
 
||
  
* [[Parametric Equations]] <!-- 1224-7.1 -->
+
* [[Parametric Equations]]  
* [[Curves in Space and Vector Functions]] <!-- 2214-13.1 -->
+
* [[Curves in Space and Vector-Valued Functions]]  
* [[Arc Lengths]] <!-- 1224-13.3 -->
+
* [[Arc Length]]  
  
 
||
 
||
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|Week&nbsp;14/15   
+
|Week 14/15   
  
 
||
 
||
 
+
6.3
<div style="text-align: center;">6.3</div>
 
 
 
 
||
 
||
  
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||
 
||
  
* [[Vector Fields and Line Integrals]] <!-- 2214-16.2 -->
+
* [[Vector Fields and Line Integrals]]  
* [[Partial Derivatives]]  <!-- 1214-14.3 -->
+
* [[Partial Derivatives]]   
  
 
||
 
||
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|Weeks&nbsp;14/15   
+
|Weeks 14/15   
  
 
||
 
||
 
+
6.4
<div style="text-align: center;">6.4</div>
 
 
 
 
||
 
||
  
 
[[Green's Theorem]]
 
[[Green's Theorem]]
 +
 +
[[Stokes' Theorem]]
  
 
||
 
||
  
* [[Vector Fields and Line Integrals]] <!-- 2214-16.2 -->
+
* [[Vector Fields]]
* [[Partial Derivatives]]  <!-- 1214-14.3 -->
+
* [[Line Integrals]]  
* [[Dot Product]] <!-- 2214-16.3 -->
+
* [[Partial Derivatives]]   
* [[Path Independence and Conservation Fields]] <!-- 2214-16.3 -->
+
* [[The Dot Product]]  
 +
* [[Line Integrals|Path Independence]]
 +
* [[Conservative Vector Fields]]  
  
 
||
 
||
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* Flux Form of Green’s Theorem.
 
* Flux Form of Green’s Theorem.
 
* Applying Green's Theorem to find Work, Flux.
 
* Applying Green's Theorem to find Work, Flux.
|-
+
 
 +
|}

Latest revision as of 10:10, 21 June 2023

The textbook for this course is Calculus (Volume 3) by Gilbert Strang, Edwin Herman, et al.

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 Space.
  • Standard Equation for a Sphere.
Weeks 1/2

2.1


Vectors in The Plane, 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 Integral is the limit of Double Sums.
  • 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 over rectangular polar regions.
  • Double Integrals over general polar regions.
  • Changing Cartesian Integrals into Polar Integrals.
  • Using Double Integrals in Polar Coordinates to find Volumes, Areas.
Week 11

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

  • Integrations in Cylindrical Coordinates.
  • Equations relating rectangular and cylindrical coordinates.
  • Changing Cartesian integrations into Cylindrical integrations.
  • Integrations in Spherical coordinates.
  • Equations relating spherical coordinates to Cartesian and cylindrical coordinates.
  • Changing Cartesian integrations into Cylindrical integrations.
Week 13

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 13/14

5.7

Change of Variables in Multiple Integrals

  • Determine the image of a region under a given transformation of variables.
  • Compute the Jacobian of a given transformation.
  • Evaluate a double integral using a change of variables.
  • Evaluate a triple integral using a change of variables.
Week 14

6.1

Vector Fields

  • Vector Fields in a plane.
  • Vector Fields in Space.
  • Potential Functions.
  • Gradient Fields, Conservative Vector Fields.
  • The Cross-Partial Test for Conservative Vector Fields.
  • Determining Whether a Vector Field is conservative.
Week 14

6.2

Line Integrals

  • Line Integrals of functions a long a smooth curves in a planer or in space
  • Line Integrals of of vector fields along an oriented curves in a plane or space..
  • Properties of Vector Line Integrals.
  • Evaluating Line Integrals.
  • Applications of line integrals: Calculating Arc Length, the Mass of a wire, Work done by a force, Flux across a curve, Circulation of a force.
Week 14/15

6.3

Conservative Vector Fields

  • Describe simple and closed curves
  • Define connected and simply connected regions.
  • Explain how to test a vector field to determine whether it is conservative.
  • Find a potential function for a conservative vector field.
  • Use the Fundamental Theorem for Line Integrals to evaluate a line integral of a vector field.
Weeks 14/15

6.4

Green's Theorem

Stokes' Theorem

  • Circulation form of Green's Theorem.
  • Flux Form of Green’s Theorem.
  • Applying Green's Theorem to find Work, Flux.