Difference between revisions of "MAT2214"

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* Finding Masses, Moments, Centers of Masses, Moments of Inertia in Two Dimensions.
 
* Finding Masses, Moments, Centers of Masses, Moments of Inertia in Two Dimensions.
 
* Finding Masses, Moments, Centers of Masses, Moments of Inertia in Three Dimensions.
 
* Finding Masses, Moments, Centers of Masses, Moments of Inertia in Three Dimensions.
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|Week 13 
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<div style="text-align: center;">5.6</div>
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[[Change of Variables in Multiple Integrals]]
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* [[Double Integral]] <!-- 2214-5.2,2214-5.3  -->
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* [[Triple Integrals]] <!-- 2214-5.3,2214-5.4 -->
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* Determine the image of a region under a given transformation of variables.
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* Compute the Jacobian of a given transformation.
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* Evaluate a double integral using a change of variables.
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* Evaluate a triple integral using a change of variables.
 
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|Week&nbsp;14  
 
|Week&nbsp;14  

Revision as of 15:46, 11 December 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 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
5.6

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

Conservation Fields

  • Closed Curve, Simple Curve.
  • Simply connected regions.
  • Line integrals in Conservation fields.
  • Conservative Field is independent of path.
  • Finding potentials for conservative fields.
Weeks 14/15
6.4

Green's Theorem

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