Course Modules

MODULE 1 - Elementary PDEs; change of variables; examples of BVPs; classification

MODULE 1 - Elementary PDEs; change of variables; examples of BVPs; classification MODULE 1 - Elementary PDEs; change of variables; examples of BVPs; classification

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Lecture 1 (01/25) - ODEs, ICs, IVP=ODE+IC; really elementary PDEs Lecture 1 (01/25) - ODEs, ICs, IVP=ODE+IC; really elementary PDEs 1

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Page

Lecture 1 (01/25) - Video recording Lecture 1 (01/25) - Video recording 2

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Attachment

Lecture 2 (01/27) - Really elementary PDEs, additional conditions, change of variables Lecture 2 (01/27) - Really elementary PDEs, additional conditions, change of variables 3

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Page

Lecture 2 (01/27) - Video recording Lecture 2 (01/27) - Video recording 4

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Attachment

Exercise: change-of-variables, chain rule Exercise: change-of-variables, chain rule 5

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Lecture 3 (01/29) - Chain Rule, FTC, unidirectional wave equation, wave propagation Lecture 3 (01/29) - Chain Rule, FTC, unidirectional wave equation, wave propagation 6

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Lecture 3 (01/29) - Video recording Lecture 3 (01/29) - Video recording 7

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Assignment

Homework 1, due by 11:59 pm on Feb 5 (Fri) Homework 1, due by 11:59 pm on Feb 5 (Fri) 15 8

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Attachment

Solution to Homework 1 Solution to Homework 1 9

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Assignment

Practice homework (not graded) Practice homework (not graded) 15 10

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Attachment

Lecture 4 (02/01) - Notations and terminology; linearity; setting up BVPs for Laplace and heat eqns Lecture 4 (02/01) - Notations and terminology; linearity; setting up BVPs for Laplace and heat eqns 11

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Lecture 4 (02/01) - Video recording Lecture 4 (02/01) - Video recording 12

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Attachment

Recommended reading: Integral theorems of calculus Recommended reading: Integral theorems of calculus 13

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Lecture 5 (02/03) - Heat eqn in 2 dim, wave eqn; more on linearity; classification of linear 2nd order PDEs Lecture 5 (02/03) - Heat eqn in 2 dim, wave eqn; more on linearity; classification of linear 2nd order PDEs 14

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Lecture 5 (02/03) - Video recording Lecture 5 (02/03) - Video recording 15

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MODULE 2 - Heat equation, separation of variables

MODULE 2 - Heat equation, separation of variables MODULE 2 - Heat equation, separation of variables

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Lecture 6 (02/05) - Derivation of the heat eqn in 3 dim Lecture 6 (02/05) - Derivation of the heat eqn in 3 dim 1

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Lecture 6 (02/05) - Video recording Lecture 6 (02/05) - Video recording 2

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Attachment

Recommended reading: Derivation of the heat equation (read Sec. 1, 2.1) Recommended reading: Derivation of the heat equation (read Sec. 1, 2.1) 3

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Assignment

Homework 2, due by 11:59 pm on Feb 12 (Fri) Homework 2, due by 11:59 pm on Feb 12 (Fri) 15 4

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Attachment

Solution to Homework 2 Solution to Homework 2 5

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Attachment

Lecture 7 (02/08) - BCs for the heat equation Lecture 7 (02/08) - BCs for the heat equation 6

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Discussion Topic

Lecture 7 (02/08) - Video recording Lecture 7 (02/08) - Video recording 7

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Attachment

Recommended reading: Derivation of the heat equation Recommended reading: Derivation of the heat equation 8

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Attachment

Lecture 8 (02/10) - Asymptotic temperature; Poisson and Laplace eqns; separation of variables Lecture 8 (02/10) - Asymptotic temperature; Poisson and Laplace eqns; separation of variables 9

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Lecture 8 (02/10) - Video recording Lecture 8 (02/10) - Video recording 10

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Attachment

Lecture 9 (02/12) - Separation of variables for the heat equation with Dirichlet BCs Lecture 9 (02/12) - Separation of variables for the heat equation with Dirichlet BCs 11

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Lecture 9 (02/12) - Video recording Lecture 9 (02/12) - Video recording 12

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Assignment

Homework 3, due by 11:59 pm on Feb 19 (Fri) Homework 3, due by 11:59 pm on Feb 19 (Fri) 15 13

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Attachment

Solution to Homework 3 Solution to Homework 3 14

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Page

Reading assignment (mandatory): Logan, Sec. 4.1 (pages 156-166) Reading assignment (mandatory): Logan, Sec. 4.1 (pages 156-166) 15

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Attachment

Reading assignment: Separation of variables for the heat equation with Neumann BCs Reading assignment: Separation of variables for the heat equation with Neumann BCs 16

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Lecture 10 (02/15) - Lecture cancelled due to weather Lecture 10 (02/15) - Lecture cancelled due to weather 17

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Lecture 11 (02/17) - Lecture cancelled due to weather Lecture 11 (02/17) - Lecture cancelled due to weather 18

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Attachment

Lecture 12 (02/19) - Separation of variables for the heat equation with Neumann BCs, and with Dirichlet/Robin BCs Lecture 12 (02/19) - Separation of variables for the heat equation with Neumann BCs, and with Dirichlet/Robin BCs 19

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Lecture 12 (02/19) - Video recording Lecture 12 (02/19) - Video recording 20

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Assignment

Homework 4, due by 11:59 pm on Feb 26 (Fri) Homework 4, due by 11:59 pm on Feb 26 (Fri) 15 21

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Attachment

Solution to Homework 4 Solution to Homework 4 22

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MODULE 3 - Orthogonal expansions, Fourier series

MODULE 3 - Orthogonal expansions, Fourier series MODULE 3 - Orthogonal expansions, Fourier series

Attachment

Lecture 13 (02/22) - Linear spaces, norms, inner products, spaces of polynomials, bases Lecture 13 (02/22) - Linear spaces, norms, inner products, spaces of polynomials, bases 1

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Page

Lecture 13 (02/22) - Video recording Lecture 13 (02/22) - Video recording 2

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Attachment

Reading assignment: Vector spaces and structures in them Reading assignment: Vector spaces and structures in them 3

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Attachment

Lecture 14 (02/24) - orthogonal expansions (Logan, pages 131-137 of Sec. 3.2) Lecture 14 (02/24) - orthogonal expansions (Logan, pages 131-137 of Sec. 3.2) 4

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Lecture 14 (02/24) - Video recording Lecture 14 (02/24) - Video recording 5

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Attachment

Lecture 15 (02/26) - Expansions of square integrable functions; classical Fourier series (Logan, pages 138-142, 145-149 of Sec. 3.2, 3.3) Lecture 15 (02/26) - Expansions of square integrable functions; classical Fourier series (Logan, pages 138-142, 145-149 of Sec. 3.2, 3.3) 6

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Lecture 15 (02/26) - Video recording Lecture 15 (02/26) - Video recording 7

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Assignment

Homework 5, due by 11:59 pm on Mar 5 (Fri) Homework 5, due by 11:59 pm on Mar 5 (Fri) 15 8

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Solution to Homework 5 Solution to Homework 5 9

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Examples of different types of convergence of sequences of functions Examples of different types of convergence of sequences of functions 10

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Lecture 16 (03/01) - Convergence of Fourier series (FS); computing sums of series by using FS (Logan, pages 151-153 of Sec. 3.3) Lecture 16 (03/01) - Convergence of Fourier series (FS); computing sums of series by using FS (Logan, pages 151-153 of Sec. 3.3) 11

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Lecture 16 (03/01) - Video recording Lecture 16 (03/01) - Video recording 12

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MODULE 4 - Sturm-Liouville theory

MODULE 4 - Sturm-Liouville theory MODULE 4 - Sturm-Liouville theory

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Lecture 17 (03/03) - Linear operators, matrix elements, symmetric operators, eigenectors and eigenvalues, projection operators Lecture 17 (03/03) - Linear operators, matrix elements, symmetric operators, eigenectors and eigenvalues, projection operators 1

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Lecture 17 (03/03) - Video recording Lecture 17 (03/03) - Video recording 2

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Homework 5a, not to be turned in; solutions will be posted on Mar 10 (Wed) Homework 5a, not to be turned in; solutions will be posted on Mar 10 (Wed) 3

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Solution to Homework 5a Solution to Homework 5a 4

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Lecture 18 (03/08) - Sturm-Liouville theory: statement of theorem, Lagrange's and Green's identities, proof or orthogonality of eigenfunctions (Logan, pages 167-171 of Sec. 4.2) Lecture 18 (03/08) - Sturm-Liouville theory: statement of theorem, Lagrange's and Green's identities, proof or orthogonality of eigenfunctions (Logan, pages 167-171 of Sec. 4.2) 5

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Lecture 18 (03/08) - Video recording Lecture 18 (03/08) - Video recording 6

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Lecture 19 (03/10) - Sturm-Liouville theory: proofs that the e-fcns corresponding to different e-values are orthogonal (cont.), the e-values are real, for each e-value there is only one independent e-fcn, (the e-fcns are real) (Logan, pages 171-172 of Sec. 4.2) Lecture 19 (03/10) - Sturm-Liouville theory: proofs that the e-fcns corresponding to different e-values are orthogonal (cont.), the e-values are real, for each e-value there is only one independent e-fcn, (the e-fcns are real) (Logan, pages 171-172 of Sec. 4.2) 7

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Lecture 19 (03/10) - Video recording Lecture 19 (03/10) - Video recording 8

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Lecture 20 (03/12) - Exam 1 Lecture 20 (03/12) - Exam 1 9

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Solution to Exam 1 Solution to Exam 1 10

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Lecture 21 (03/15) - Sturm-Liouville theory: periodic BCs (heat equation on a ring), integration by parts, energy argument for determining the sign of eigenvalues, on the importance of the sign of eigenvalues (Logan, Examples 4.10 and 4.11 of Section 4.2 on pages 172-173) Lecture 21 (03/15) - Sturm-Liouville theory: periodic BCs (heat equation on a ring), integration by parts, energy argument for determining the sign of eigenvalues, on the importance of the sign of eigenvalues (Logan, Examples 4.10 and 4.11 of Section 4.2 on pages 172-173) 11

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Lecture 21 (03/15) - Video recording Lecture 21 (03/15) - Video recording 12

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Lecture 22 (03/17) - Sturm-Liouville theory: generalized SLP, weight function, restatement of the general theorem for generalized SLP, weighted inner product; integrating factor, converting an eigenvalue problem into a generalized SLP; example - heat equation, related to Example 4.15 (Logan, pages 180-182 of Section 4.3) Lecture 22 (03/17) - Sturm-Liouville theory: generalized SLP, weight function, restatement of the general theorem for generalized SLP, weighted inner product; integrating factor, converting an eigenvalue problem into a generalized SLP; example - heat equation, related to Example 4.15 (Logan, pages 180-182 of Section 4.3) 13

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Lecture 22 (03/17) - Video recording Lecture 22 (03/17) - Video recording 14

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Assignment

Homework 6, due by 11:59 pm on Mar 24 (Wed) Homework 6, due by 11:59 pm on Mar 24 (Wed) 15 15

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Attachment

Solution to Homework 6 Solution to Homework 6 16

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Lecture 23 (03/19) - Sturm-Liouville theory: symmetric operator, symmetric BCs; Rayleigh quotient (RQ) for a generalized SLP: definition, each e-value is the RQ of its e-fcn; Theorem: the leading e-value equals the minimum of the RQ taken over all continuous fcns satisfying the BCs; an example of using RQ to give upper and lower bounds on the leading e-value (Logan, pages 174-177 of Sec. 4.2) Lecture 23 (03/19) - Sturm-Liouville theory: symmetric operator, symmetric BCs; Rayleigh quotient (RQ) for a generalized SLP: definition, each e-value is the RQ of its e-fcn; Theorem: the leading e-value equals the minimum of the RQ taken over all continuous fcns satisfying the BCs; an example of using RQ to give upper and lower bounds on the leading e-value (Logan, pages 174-177 of Sec. 4.2) 17

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Lecture 23 (03/19) - Video recording Lecture 23 (03/19) - Video recording 18

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Reading assignment: Logan, Examples 4.12 and 4.13 on pages 173-176 Reading assignment: Logan, Examples 4.12 and 4.13 on pages 173-176 19

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Lecture 24 (03/22) - Sturm-Liouville theory: an example of using RQ to give an upper bound on the leading e-value; an example of using SL theory to solve Laplace's equation in a semi-infinite strip Lecture 24 (03/22) - Sturm-Liouville theory: an example of using RQ to give an upper bound on the leading e-value; an example of using SL theory to solve Laplace's equation in a semi-infinite strip 20

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Lecture 24 (03/22) - Video recording Lecture 24 (03/22) - Video recording 21

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MODULE 5 - Laplace, Poisson, and wave equations

MODULE 5 - Laplace, Poisson, and wave equations MODULE 5 - Laplace, Poisson, and wave equations

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Lecture 25 (03/24) - finishing the example from Lecture 24; Laplace's equation: statement, physical interpretation, uniqueness of the solution of the Dirichlet problem for Laplace's equation (Theorem 4.22), energy functional, the solution of the Laplace's BVP with Dirichlet BCs minimizes the energy functional among all functions satisfying the BCs (Theorem 4.23) (Logan, pages 191-194 of Sec. 4.4) Lecture 25 (03/24) - finishing the example from Lecture 24; Laplace's equation: statement, physical interpretation, uniqueness of the solution of the Dirichlet problem for Laplace's equation (Theorem 4.22), energy functional, the solution of the Laplace's BVP with Dirichlet BCs minimizes the energy functional among all functions satisfying the BCs (Theorem 4.23) (Logan, pages 191-194 of Sec. 4.4) 1

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Lecture 25 (03/24) - Video recording Lecture 25 (03/24) - Video recording 2

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Reading assignment (mandatory): Read (actively) the proof of the Dirichlet Principle (Theorem 4.23) on page 194 of Logan Reading assignment (mandatory): Read (actively) the proof of the Dirichlet Principle (Theorem 4.23) on page 194 of Logan 3

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Lecture 26 (03/26) - Maximum principle for the Laplace equation - statement and physical meaning (Logan, Theorem 1.23 on page 68); uniqueness of the solutions of the heat equation with Dirichlet BCs in any dimension (with proof) Lecture 26 (03/26) - Maximum principle for the Laplace equation - statement and physical meaning (Logan, Theorem 1.23 on page 68); uniqueness of the solutions of the heat equation with Dirichlet BCs in any dimension (with proof) 4

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Lecture 26 (03/26) - Video recording Lecture 26 (03/26) - Video recording 5

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Lecture 27 (03/29) - Wave equation: derivation of the wave equation for planar motion of a string (Logan, pages 49-52 of Sec. 1.5) Lecture 27 (03/29) - Wave equation: derivation of the wave equation for planar motion of a string (Logan, pages 49-52 of Sec. 1.5) 6

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Lecture 27 (03/39) - Video recording Lecture 27 (03/39) - Video recording 7

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Assignment

Homework 7, due by 11:59 pm on Apr 7 (Wed) Homework 7, due by 11:59 pm on Apr 7 (Wed) 15 8

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Solution to Homework 7 Solution to Homework 7 9

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Lecture 28 (03/31) - BCs for the 1D wave equation; general solution of the 1D wave equation on the whole real line by a change of variables; solution of an IVP for the wave equation on R - D'Alembert formula, region of influence (Logan, Sec. 2.2) Lecture 28 (03/31) - BCs for the 1D wave equation; general solution of the 1D wave equation on the whole real line by a change of variables; solution of an IVP for the wave equation on R - D'Alembert formula, region of influence (Logan, Sec. 2.2) 10

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Lecture 28 (03/31) - Video recording Lecture 28 (03/31) - Video recording 11

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Lecture 29 (04/02) - Energy of the string; solving the IBVP for vibration of a string with attached ends by separation of variables Lecture 29 (04/02) - Energy of the string; solving the IBVP for vibration of a string with attached ends by separation of variables 12

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Lecture 29 (04/02) - Video recording Lecture 29 (04/02) - Video recording 13

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Lecture 30 (04/05) - Waves in a guitar string: speed, wavelength, period, angular frequency, (linear) frequency, fundamental frequency, harmonics, dependence of the fundamental frequency on the physical parameters (length, tension, linear density), flageolets (suppressing harmonics by lightly touching the string at certain positions) Lecture 30 (04/05) - Waves in a guitar string: speed, wavelength, period, angular frequency, (linear) frequency, fundamental frequency, harmonics, dependence of the fundamental frequency on the physical parameters (length, tension, linear density), flageolets (suppressing harmonics by lightly touching the string at certain positions) 14

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Page

Lecture 30 (04/05) - Video recording Lecture 30 (04/05) - Video recording 15

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Attachment

Reading assignment (mandatory): Waves in a guitar string Reading assignment (mandatory): Waves in a guitar string 16

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Lecture 31 (04/07) - Vibration of a rectanglar membrane; vibrations of a circular membrane Lecture 31 (04/07) - Vibration of a rectanglar membrane; vibrations of a circular membrane 17

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Lecture 31 (04/07) - Video recording Lecture 31 (04/07) - Video recording 18

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Lecture 32 (04/09) - Vibrations of a circular membrane (cont.) Lecture 32 (04/09) - Vibrations of a circular membrane (cont.) 19

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Lecture 32 (04/09) - Video recording Lecture 32 (04/09) - Video recording 20

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Assignment

Homework 8, due by 11:59 pm on Apr 14 (Wed) Homework 8, due by 11:59 pm on Apr 14 (Wed) 15 21

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Solution to Homework 8 Solution to Homework 8 22

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Lecture 33 (04/12) - Separation of variables in the wave equation in a disk Lecture 33 (04/12) - Separation of variables in the wave equation in a disk 23

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Lecture 33 (04/12) - Video recording Lecture 33 (04/12) - Video recording 24

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Lecture 34 (04/14) - Dealing with source terms in the PDEs within the method of separation of variables - expand the source term in the eigenfunctions of the Sturm-Liouville problem for the spatial function X(x) Lecture 34 (04/14) - Dealing with source terms in the PDEs within the method of separation of variables - expand the source term in the eigenfunctions of the Sturm-Liouville problem for the spatial function X(x) 25

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Page

Lecture 34 (04/14) - Video recording Lecture 34 (04/14) - Video recording 26

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Assignment

Homework 9, due by 11:59 pm on Apr 21 (Wed) Homework 9, due by 11:59 pm on Apr 21 (Wed) 15 27

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Solution to Homework 9 Solution to Homework 9 28

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MODULE 6 - Fundamental solutions, delta function, Fourier transform

MODULE 6 - Fundamental solutions, delta function, Fourier transform MODULE 6 - Fundamental solutions, delta function, Fourier transform

Attachment

Lecture 35 (04/16) - Sound waves in a pipe with open and closed ends; Cauchy problem for the heat equation - Buckingham pi-theorem (Logan, pages 79-81 of Sec. 2.1) Lecture 35 (04/16) - Sound waves in a pipe with open and closed ends; Cauchy problem for the heat equation - Buckingham pi-theorem (Logan, pages 79-81 of Sec. 2.1) 1

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Page

Lecture 35 (04/16) - Video recording Lecture 35 (04/16) - Video recording 2

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Lecture 36 (04/19) - Cauchy problem for the heat equation (cont.) - the period of a pendulum derived from Buckingham pi-theorem, using the pi-theorem to derive the fundamental solution of the heat equation on R, error function (Logan, pages 81-83 of Sec. 2.1) Lecture 36 (04/19) - Cauchy problem for the heat equation (cont.) - the period of a pendulum derived from Buckingham pi-theorem, using the pi-theorem to derive the fundamental solution of the heat equation on R, error function (Logan, pages 81-83 of Sec. 2.1) 3

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Page

Lecture 36 (04/19) - Video recording Lecture 36 (04/19) - Video recording 4

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Attachment

Lecture 37 (04/21) - Fundamental solution of the heat equation on R (cont.); Integral tranforms; example: Laplace transform; integral kernel; Fourier transform (FT) and its inverse; FT of derivatives of a function (Logan pages 83-85 of Sec. 2.1, pages 117, 118, 120, 121 of Sec. 2.7) Lecture 37 (04/21) - Fundamental solution of the heat equation on R (cont.); Integral tranforms; example: Laplace transform; integral kernel; Fourier transform (FT) and its inverse; FT of derivatives of a function (Logan pages 83-85 of Sec. 2.1, pages 117, 118, 120, 121 of Sec. 2.7) 5

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Page

Lecture 37 (04/21) - Video recording Lecture 37 (04/21) - Video recording 6

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Lecture 38 (04/23) - Exam 2 Lecture 38 (04/23) - Exam 2 7

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Solution to Exam 2 Solution to Exam 2 8

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Assignment

Homework 10, due by 11:59 pm on Apr 30 (Fri) Homework 10, due by 11:59 pm on Apr 30 (Fri) 15 9

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Attachment

Solution to Homework 10 Solution to Homework 10 10

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Attachment

Lecture 39 (04/26) - Properties of FT, convolution, the Convolution Theorem, examples; fundamental solution of the heat equation on R by FT (Logan, pages 119-122 of Sec. 2.7) Lecture 39 (04/26) - Properties of FT, convolution, the Convolution Theorem, examples; fundamental solution of the heat equation on R by FT (Logan, pages 119-122 of Sec. 2.7) 11

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Page

Lecture 39 (04/26) - Video recording Lecture 39 (04/26) - Video recording 12

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Attachment

Lecture 40 (04/28) - Functionals, linear functionals, examples; Dirac delta function, shift of the argument in the delta function, the delta function as a limit of a family of functions; derivative of a distribution, derivative of a delta function Lecture 40 (04/28) - Functionals, linear functionals, examples; Dirac delta function, shift of the argument in the delta function, the delta function as a limit of a family of functions; derivative of a distribution, derivative of a delta function 13

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Lecture 40 (04/28) - Video recording Lecture 40 (04/28) - Video recording 14

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Lecture 41 (04/30) - Derivative of the "shifted" Heaviside function, Fourier transform of the delta function, integral representation of the delta function, convolution with the delta function, solving the unidirectional wave equation by using Fourier transform Lecture 41 (04/30) - Derivative of the "shifted" Heaviside function, Fourier transform of the delta function, integral representation of the delta function, convolution with the delta function, solving the unidirectional wave equation by using Fourier transform 15

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Lecture 41 (04/30) - Video recording Lecture 41 (04/30) - Video recording 16

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Assignment

Homework 11, due by 11:59 pm on May 7 (Fri) Homework 11, due by 11:59 pm on May 7 (Fri) 15 17

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Attachment

Solutions to Homework 11 Solutions to Homework 11 18

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Lecture 42 (05/03) - Distributions applied to ODEs; example - the steady-state shape of a string subjected to a load concentrated at a point (Olver, pages 234, 235 of Sec. 6.2) Lecture 42 (05/03) - Distributions applied to ODEs; example - the steady-state shape of a string subjected to a load concentrated at a point (Olver, pages 234, 235 of Sec. 6.2) 19

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Page

Lecture 42 (05/03) - Video recording Lecture 42 (05/03) - Video recording 20

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Lecture 43 (05/05) - Fundamental solutions and Green functions; example - solving the problem of the steady state of a string with distributed load by using the Green function derived in Lecture 42 (Olver, pages 235, 236 of Sec. 6.2) Lecture 43 (05/05) - Fundamental solutions and Green functions; example - solving the problem of the steady state of a string with distributed load by using the Green function derived in Lecture 42 (Olver, pages 235, 236 of Sec. 6.2) 21

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Lecture 43 (05/05) - Video recording Lecture 43 (05/05) - Video recording 22

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Attachment

Lecture 44 (05/07) - Recap of Green functions (GFs), physical meaning of GFs, GF for the heat equation on R, GF for the wave equation on R; visualizing the eigenmodes of vibrations - Chladni's figures (https://www.youtube.com/watch?v=OLNFrxgMJ6E&t=467s); free-boundary problems, Korveweg-De Vries equation; more nonlinear equations - Navier-Stokes equation of fluid dynamics Lecture 44 (05/07) - Recap of Green functions (GFs), physical meaning of GFs, GF for the heat equation on R, GF for the wave equation on R; visualizing the eigenmodes of vibrations - Chladni's figures (https://www.youtube.com/watch?v=OLNFrxgMJ6E&t=467s); free-boundary problems, Korveweg-De Vries equation; more nonlinear equations - Navier-Stokes equation of fluid dynamics 23

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Lecture 44 (05/07) - Video recording Lecture 44 (05/07) - Video recording 24

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MODULE 7 - Collections of lecture notes

MODULE 7 - Collections of lecture notes MODULE 7 - Collections of lecture notes

Attachment

NOTES FROM LECTURES 1-17 NOTES FROM LECTURES 1-17 1

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NOTES FROM LECTURES 18-32 NOTES FROM LECTURES 18-32 2

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NOTES FROM LECTURE 33 NOTES FROM LECTURE 33 3

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NOTES FROM LECTURES 34-44 NOTES FROM LECTURES 34-44 4

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Score at least Must score at least to complete this module item Scored at least Module item has been completed by scoring at least Score at least % Must score at least % to complete this module item Scored at least % Module item has been completed by scoring at least % View Must view in order to complete this module item Viewed Module item has been viewed and is complete Mark done Must mark this module item done in order to complete Marked done Module item marked as done and is complete Contribute Must contribute to this module item to complete it Contributed Contributed to this module item and is complete Submit Must submit this module item to complete it Submitted Module item submitted and is complete