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MATH2111 Chronological List of Topics

Representation of Curves

Resources: Lecture Notes, pp 3-7,18 | in-class screen | bb2 | Graph Demo | Parametrisation Demo (2d) | Parametrisation Demo (3d)

What to Know:

  1. implicit representation (examples)
  2. graph of functions (examples)
  3. parametric representation (examples)

Representation of Surfaces

Resources: Lecture Notes, pp 15--17,19 | in-class screen | bb2 | Graph Demo | Parametrisation Demo

What to Know:

  1. implicit representation (examples)
  2. graph of functions (examples)
  3. parametric representation (examples)

Tangent Vector to Parametric Curve

Resources: Lecture Notes, pp 13--14 | in-class screen | bb2 | geogebra

What to Know:

  1. how to find tangent vector to curve given parametrically

Position, Velocity and Acceleration

Resources: in-class screen | bb2 | geogebra

What to Know:

  1. how to find velocity of particle given by a position vector
  2. how to find acceleration of particle given by a position vector
  3. difference between speed and velocity

Orthonormal Bases

Resources: wiki

What to Know:

  1. definition of orthonormal basis and orthogonal basis
  2. how to find representation of a vector with respect to orthonormal basis

Vector Cross Product

Resources: wiki

What to Know:

  1. algebraic formula via components for vector cross product
  2. geometrical properties of vector cross product
  3. formula for magnitude of vector cross product

Circle

Resources: in-class screen | bb2 | wiki

What to Know:

  1. implicit equation (interpretation of coefficients)
  2. parametric representation
  3. completion of squares

Ellipse

Resources: in-class screen | bb2 | geogebra | wiki

What to Know:

  1. implicit equation (interpretation of coefficients, intercepts)
  2. parametric representation
  3. completion of squares

Hyperbola

Resources: in-class screen | bb2 | geogebra | wiki

What to Know:

  1. implicit equation (interpretation of coefficients, intercepts, asymptotes)
  2. parametric representation
  3. completion of squares

Lines

Resources: in-class screen | bb2 | Normal Demo | Parametrisation Demo | wiki

What to Know:

  1. lines through parametrisation (any dimension)
  2. lines through Cartesian equation (2d)
  3. interpretation of coefficients in Cartesian equation (2d)
  4. parametrisation of straight line segment

Section of Surfaces by Coordinate Planes

Resources: Lecture Notes, pp 20--29 | in-class screen | bb2 | Paraboloid Demo | Cylinder Demo

What to Know:

  1. how to construct and identify

Interior and Boundary Points

Resources: Lecture Notes, pp 17 | wiki

What to Know:

  1. definition of boundary point
  2. definition of interior point
  3. ability to proof, using definition, that a point interior
  4. ability to proof, using definition, that a point is boundary

Open and Closed Subsets

Resources: Lecture Notes, pp 17--19, 21 | in-class screen | bb2 | wiki/open sets | wiki/closed sets

What to Know:

  1. open interval is open (proof, Q26)
  2. closed interval is closed (proof, Q26)
  3. open ball in higher dimensions is open (proof)

One-point Set Boundary

Resources: in-class screen | bb2

What to Know:

    Union, Intersection of Open, Closed Subsets

    Resources: in-class screen | bb2

    What to Know:

    1. union of open subsets is open (countable allowed)
    2. intersection of closed subsets is closed (countable allowed)
    3. finite union of closed subsets is closed
    4. finite intersection of open subsets is open

    Union, Intersection of Open, Closed Subsets, II

    Resources: in-class screen | bb2

    What to Know:

    1. example showing that countable union of closed subsets is not necessarily closed
    2. example showing that countable intersection of open subsets is not necessarily open

    Limit of Function of Several Variables

    Resources: Lecture Notes, pp 23--30 | wiki

    What to Know:

    1. see Limits in Rn I, II, III, IV

    Limit of Vector Function

    Resources: Lecture Notes, pp 7--8, 10--12 | wiki | Lecture Notes, p.9

    What to Know:

    1. see Limits in Rn I, II, III, IV

    Limits in Rn

    Resources: in-class screen | bb2

    What to Know:

    1. definition of limit of vector function, one variable
    2. definition of limit of vector sequence
    3. definition of limit of function of several variables

    Limits in Rn, II

    Resources: in-class screen | bb2

    What to Know:

    1. Limit of vector function and vector sequence via components
    2. Show that a limit does not exist, for function of several variables
    3. Examples of limit which does not exist

    Limits in Rn, III

    Resources: in-class screen | bb2

    What to Know:

    1. direct argument showing that limit exists, numerical function of two variables

    Limits in Rn, IV

    Resources: in-class screen | bb2

    What to Know:

    1. general statement of definition of limit, the case of vector map of several variables
    2. ability to see the definitions on slide 'Limits in Rn' as special cases of this topic

    Pinching Principle

    Resources: in-class screen | bb2 | wiki

    What to Know:

    1. statement of the principle
    2. example of usage

    Continuous Functions

    Resources: Lecture Notes, pp 31--36 | in-class screen | bb2 | wiki

    What to Know:

    1. definition of continuous function at point
    2. definition of continuous function over domain
    3. how to find a limit of a function which is continuous at the point where the limit is needed
    4. example of discontinuous function

    Elementary Functions

    Resources: in-class screen | bb2 | wiki

    What to Know:

    1. what is an elementary function
    2. continuity, differentiability of elementary function

    Limits and Taylor Expansions

    Resources: wiki

    What to Know:

    1. understand the method of Taylor expansions of elementary functions for finding limits (e.g., Q39, Q42)

    l'Hopital's Rule

    Resources: in-class screen | bb2 | wiki

    What to Know:

    1. why not to use it

    Preimage of Subset

    Resources: in-class screen | bb2 | wiki

    What to Know:

    1. the statement of the preimage theorem
    2. ability to use the preimage theorem (e.g., Q28)

    Preimage of Subset, II

    Resources: in-class screen | bb2

    What to Know:

    1. proof of preimage theorem

    Preimage of Boundary

    Resources: in-class screen | bb2 | video comments

    What to Know:

      Bounded subset

      Resources: Lecture Notes, pp 37--38 | in-class screen | bb2

      What to Know:

      1. definition of bounded subset
      2. proof that closed interval is bounded
      3. proof that an open ball (higher dimensions) is bounded
      4. proof that rectangle is bounded
      5. example of unbounded subset (with proof)

      Compact subset

      Resources: Lecture Notes, pp 43

      What to Know:

      1. definition of compact subset
      2. that compact subsets are mapped to compact subsets by continuous maps

      Path-connected subset

      Resources: Lecture Notes, pp 44--45 | in-class screen | bb2

      What to Know:

      1. definition of path-connected subset
      2. proof that a closed interval and closed rectangle are path-connected
      3. example of not path-connected subset (with proof)

      Continuous map and compact and path-connected subsets

      Resources: Lecture Notes, pp 46--47 | in-class screen | bb2

      What to Know:

      1. compact, path-connected subsets are mapped to compact, path-connected subsets, respectively

      Parallelepiped

      Resources: in-class screen | bb2

      What to Know:

      1. parallelepiped is path-connected (proof in one and two dimensions)

      Polar Map

      Resources: in-class screen | bb2 | geogebra

      What to Know:

      1. definition of polar map
      2. images of rectangles under polar map
      3. formula for inverse of polar map
      4. seam of discontinuity of inverse polar map

      Algebra of Continuous Functions

      Resources: Lecture Notes, pp 36

      What to Know:

        Tangent Line

        Resources: in-class screen | bb2

        What to Know:

        1. formula for tangent line to graph
        2. existence of tangent line is synonym to differentiability and existence of derivative (one dimensional case)

        Tangent Plane

        Resources: in-class screen | bb2

        What to Know:

        1. formula for finding tangent plane to graph
        2. the role of Gradient

        Gradient

        Resources: Lecture Notes, pp 24

        What to Know:

        1. see 'Tangent Plane' for the role of gradient in tangent plane equation (graph of function)
        2. see 'Tangent plane via Gradient' the role of gradient in tangent plane equation (surface given implicitly)
        3. the role of gradient in tangent plane equation (curve given implicitly)
        4. see 'Directional derivative' for the role of gradient in Directional derivative
        5. see 'Directional Derivative via Gradient' for the direction of fastest and no change

        Affine approximation

        Resources: Lecture Notes, pp 25--26 | in-class screen | bb2

        What to Know:

        1. formula for affine approximation
        2. the role of Jacobian matrix
        3. difference from tangent plane
        4. existence of affine approximation is synonym to differentiability (see differentiability topic)

        Tangent Plane, II

        Resources: in-class screen | bb2

        What to Know:

        1. existence of tangent plane is synonym for differentiability (see differentiability topic)
        2. direct argument showing that tangent plane exists

        Differentiability

        Resources: Lecture Notes, pp 2--7, 17--23 | in-class screen | bb2

        What to Know:

        1. definition of differentiability
        2. result guaranteeing differentiability of C^1 functions
        3. example of non-differentiable function (e.g., Q74, Q75)
        4. direct argument of differentiability (e.g., Q73)

        Partial derivatives

        Resources: Lecture Notes, pp 8--14

        What to Know:

        1. compute partial derivatives via formulae of differentiation of elementary functions
        2. compute partial derivatives via definition
        3. definition of partial derivatives
        4. connection with directional derivative

        Jacobian matrix

        Resources: Lecture Notes, pp 15--16

        What to Know:

        1. how to find Jacobian Matrix, numerical and general
        2. the role of Jacobian matrix in Affine Approximation
        3. the role in definition of differentiability

        Clairaut's Theorem

        Resources: Lecture Notes, pp 13--14

        What to Know:

        1. when mixed second order partial derivatives are the same
        2. example where second order mixed partial are different (Q67)

        Chain rule

        Resources: Lecture Notes, pp 27--34

        What to Know:

        1. chain rule in matrix form
        2. chain rule in scalar form

        Directional derivative

        Resources: Lecture Notes, pp 35--39

        What to Know:

        1. formula for the directional derivative via Gradient
        2. definition of directional derivative

        Directional Derivative and Gradient

        Resources: in-class screen | bb2 | video comments

        What to Know:

        1. formula for the directional derivative via Gradient
        2. when gradient formula works and when it does not
        3. direction of fastest change for a function
        4. direction of no-change for a function

        Taylor Series

        Resources: Lecture Notes, pp 50--60

        What to Know:

        1. method of finding Taylor polynomial upto square term using partial derivatives
        2. method of finding Taylor polynomial upto square term using know Taylor expansions of elementary functions
        3. connection between Gradient and Taylor polynomial
        4. connection between Hessian and Taylor polynomial

        Absolute Max/Min

        Resources: Lecture Notes, pp 63--67

        What to Know:

        1. two stage method of finding absolute max/min over domain
        2. method of finding max/min on the boundary of domain

        Local Max/Min/Saddle Point

        Resources: Lecture Notes, pp 61, 68--90

        What to Know:

        1. Q118
        2. Efficiency trick

        Constrained Max/Min, Lagrange Multipliers,

        Resources: Lecture Notes, pp 91--105

        What to Know:

        1. method of finding extremum under one constraint

        Tangent Plane via Gradient

        Resources: Lecture Notes, pp 40--49 | in-class screen | bb2

        What to Know:

        1. method of finding the tangent to a curve/surface given implicitly
        2. gradient perpendicular to curve/surface given implicitly

        Implicit Function Theorem

        Resources: Lecture Notes, pp 115--127

        What to Know:

        1. test which ensures that implicit function exists
        2. how to find derivative/Jacobian of implicit function

        Inverse Function Theorem,

        Resources: Lecture Notes, pp 106--114

        What to Know:

        1. test which ensures that inverse function exists
        2. how to find derivative/Jacobian of inverse

        Polar Map, Global Inverse

        Resources: in-class screen | bb2

        What to Know:

        1. one example of polar inverse
        2. the seam of discontinuity of polar inverse
        3. proof that global inverse does not exist

        Riemann Integration

        Resources: Lecture Notes, pp 2--8

        What to Know:

        1. definition of Riemann integration
        2. how to find integral by definition (Q139)

        Geometrical Interpretation of Integral

        Resources: Lecture Notes, pp 18

        What to Know:

        1. geometrical meaning of double integral when the function is constant
        2. geometrical meaning of double integral with general function
        3. geometrical meaning of triple integral when the function is constant

        Fubini's Theorem on Rectangles

        Resources: Lecture Notes, pp 9--12

        What to Know:

        1. difference between double/triple integral and repeated integration
        2. ability to convert double integral to repeated one and vise versa

        Fubini's Theorem in general

        Resources: Lecture Notes, pp 13--17,19--26

        What to Know:

        1. see 'Fubini's Theorem on Rectangles
        2. definition of x-simple and y-simple domains

        Leibniz's rule

        Resources: Lecture Notes, pp 28--38 | wiki

        What to Know:

        1. how to use Leibniz's rule

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