- 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:
- implicit representation (examples)
- graph of functions (examples)
- parametric representation (examples)
- Representation of Surfaces
Resources: Lecture Notes, pp 15--17,19 | in-class screen | bb2 | Graph Demo | Parametrisation Demo
What to Know:
- implicit representation (examples)
- graph of functions (examples)
- parametric representation (examples)
- Tangent Vector to Parametric Curve
Resources: Lecture Notes, pp 13--14 | in-class screen | bb2 | geogebra
What to Know:
- how to find tangent vector to curve given parametrically
- Position, Velocity and Acceleration
Resources: in-class screen | bb2 | geogebra
What to Know:
- how to find velocity of particle given by a position vector
- how to find acceleration of particle given by a position vector
- difference between speed and velocity
- Orthonormal Bases
Resources: wiki
What to Know:
- definition of orthonormal basis and orthogonal basis
- how to find representation of a vector with respect to orthonormal basis
- Vector Cross Product
Resources: wiki
What to Know:
- algebraic formula via components for vector cross product
- geometrical properties of vector cross product
- formula for magnitude of vector cross product
- Circle
Resources: in-class screen | bb2 | wiki
What to Know:
- implicit equation (interpretation of coefficients)
- parametric representation
- completion of squares
- Ellipse
Resources: in-class screen | bb2 | geogebra | wiki
What to Know:
- implicit equation (interpretation of coefficients, intercepts)
- parametric representation
- completion of squares
- Hyperbola
Resources: in-class screen | bb2 | geogebra | wiki
What to Know:
- implicit equation (interpretation of coefficients, intercepts, asymptotes)
- parametric representation
- completion of squares
- Lines
Resources: in-class screen | bb2 | Normal Demo | Parametrisation Demo | wiki
What to Know:
- lines through parametrisation (any dimension)
- lines through Cartesian equation (2d)
- interpretation of coefficients in Cartesian equation (2d)
- 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:
- how to construct and identify
- Interior and Boundary Points
Resources: Lecture Notes, pp 17 | wiki
What to Know:
- definition of boundary point
- definition of interior point
- ability to proof, using definition, that a point interior
- 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:
- open interval is open (proof, Q26)
- closed interval is closed (proof, Q26)
- 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:
- union of open subsets is open (countable allowed)
- intersection of closed subsets is closed (countable allowed)
- finite union of closed subsets is closed
- finite intersection of open subsets is open
- Union, Intersection of Open, Closed Subsets, II
Resources: in-class screen | bb2
What to Know:
- example showing that countable union of closed subsets is not necessarily closed
- 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:
- 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:
- see Limits in Rn I, II, III, IV
- Limits in Rn
Resources: in-class screen | bb2
What to Know:
- definition of limit of vector function, one variable
- definition of limit of vector sequence
- definition of limit of function of several variables
- Limits in Rn, II
Resources: in-class screen | bb2
What to Know:
- Limit of vector function and vector sequence via components
- Show that a limit does not exist, for function of several variables
- Examples of limit which does not exist
- Limits in Rn, III
Resources: in-class screen | bb2
What to Know:
- direct argument showing that limit exists, numerical function of two variables
- Limits in Rn, IV
Resources: in-class screen | bb2
What to Know:
- general statement of definition of limit, the case of vector map of several variables
- 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:
- statement of the principle
- example of usage
- Continuous Functions
Resources: Lecture Notes, pp 31--36 | in-class screen | bb2 | wiki
What to Know:
- definition of continuous function at point
- definition of continuous function over domain
- how to find a limit of a function which is continuous at the point where the limit is needed
- example of discontinuous function
- Elementary Functions
Resources: in-class screen | bb2 | wiki
What to Know:
- what is an elementary function
- continuity, differentiability of elementary function
- Limits and Taylor Expansions
Resources: wiki
What to Know:
- 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:
- why not to use it
- Preimage of Subset
Resources: in-class screen | bb2 | wiki
What to Know:
- the statement of the preimage theorem
- ability to use the preimage theorem (e.g., Q28)
- Preimage of Subset, II
Resources: in-class screen | bb2
What to Know:
- 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:
- definition of bounded subset
- proof that closed interval is bounded
- proof that an open ball (higher dimensions) is bounded
- proof that rectangle is bounded
- example of unbounded subset (with proof)
- Compact subset
Resources: Lecture Notes, pp 43
What to Know:
- definition of compact subset
- 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:
- definition of path-connected subset
- proof that a closed interval and closed rectangle are path-connected
- 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:
- compact, path-connected subsets are mapped to compact, path-connected subsets, respectively
- Parallelepiped
Resources: in-class screen | bb2
What to Know:
- parallelepiped is path-connected (proof in one and two dimensions)
- Polar Map
Resources: in-class screen | bb2 | geogebra
What to Know:
- definition of polar map
- images of rectangles under polar map
- formula for inverse of polar map
- 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:
- formula for tangent line to graph
- 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:
- formula for finding tangent plane to graph
- the role of Gradient
- Gradient
Resources: Lecture Notes, pp 24
What to Know:
- see 'Tangent Plane' for the role of gradient in tangent plane equation (graph of function)
- see 'Tangent plane via Gradient' the role of gradient in tangent plane equation (surface given implicitly)
- the role of gradient in tangent plane equation (curve given implicitly)
- see 'Directional derivative' for the role of gradient in Directional derivative
- 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:
- formula for affine approximation
- the role of Jacobian matrix
- difference from tangent plane
- existence of affine approximation is synonym to differentiability (see differentiability topic)
- Tangent Plane, II
Resources: in-class screen | bb2
What to Know:
- existence of tangent plane is synonym for differentiability (see differentiability topic)
- direct argument showing that tangent plane exists
- Differentiability
Resources: Lecture Notes, pp 2--7, 17--23 | in-class screen | bb2
What to Know:
- definition of differentiability
- result guaranteeing differentiability of C^1 functions
- example of non-differentiable function (e.g., Q74, Q75)
- direct argument of differentiability (e.g., Q73)
- Partial derivatives
Resources: Lecture Notes, pp 8--14
What to Know:
- compute partial derivatives via formulae of differentiation of elementary functions
- compute partial derivatives via definition
- definition of partial derivatives
- connection with directional derivative
- Jacobian matrix
Resources: Lecture Notes, pp 15--16
What to Know:
- how to find Jacobian Matrix, numerical and general
- the role of Jacobian matrix in Affine Approximation
- the role in definition of differentiability
- Clairaut's Theorem
Resources: Lecture Notes, pp 13--14
What to Know:
- when mixed second order partial derivatives are the same
- example where second order mixed partial are different (Q67)
- Chain rule
Resources: Lecture Notes, pp 27--34
What to Know:
- chain rule in matrix form
- chain rule in scalar form
- Directional derivative
Resources: Lecture Notes, pp 35--39
What to Know:
- formula for the directional derivative via Gradient
- definition of directional derivative
- Directional Derivative and Gradient
Resources: in-class screen | bb2 | video comments
What to Know:
- formula for the directional derivative via Gradient
- when gradient formula works and when it does not
- direction of fastest change for a function
- direction of no-change for a function
- Taylor Series
Resources: Lecture Notes, pp 50--60
What to Know:
- method of finding Taylor polynomial upto square term using partial derivatives
- method of finding Taylor polynomial upto square term using know Taylor expansions of elementary functions
- connection between Gradient and Taylor polynomial
- connection between Hessian and Taylor polynomial
- Absolute Max/Min
Resources: Lecture Notes, pp 63--67
What to Know:
- two stage method of finding absolute max/min over domain
- 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:
- Q118
- Efficiency trick
- Constrained Max/Min, Lagrange Multipliers,
Resources: Lecture Notes, pp 91--105
What to Know:
- method of finding extremum under one constraint
- Tangent Plane via Gradient
Resources: Lecture Notes, pp 40--49 | in-class screen | bb2
What to Know:
- method of finding the tangent to a curve/surface given implicitly
- gradient perpendicular to curve/surface given implicitly
- Implicit Function Theorem
Resources: Lecture Notes, pp 115--127
What to Know:
- test which ensures that implicit function exists
- how to find derivative/Jacobian of implicit function
- Inverse Function Theorem,
Resources: Lecture Notes, pp 106--114
What to Know:
- test which ensures that inverse function exists
- how to find derivative/Jacobian of inverse
- Polar Map, Global Inverse
Resources: in-class screen | bb2
What to Know:
- one example of polar inverse
- the seam of discontinuity of polar inverse
- proof that global inverse does not exist
- Riemann Integration
Resources: Lecture Notes, pp 2--8
What to Know:
- definition of Riemann integration
- how to find integral by definition (Q139)
- Geometrical Interpretation of Integral
Resources: Lecture Notes, pp 18
What to Know:
- geometrical meaning of double integral when the function is constant
- geometrical meaning of double integral with general function
- geometrical meaning of triple integral when the function is constant
- Fubini's Theorem on Rectangles
Resources: Lecture Notes, pp 9--12
What to Know:
- difference between double/triple integral and repeated integration
- 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:
- see 'Fubini's Theorem on Rectangles
- definition of x-simple and y-simple domains
- Leibniz's rule
Resources: Lecture Notes, pp 28--38 | wiki
What to Know:
- how to use Leibniz's rule