User:Egm6322.s09.xyz

=HMK #1: Create a Wiki version of Lecture #3 (1/12/09)=

page: 3-1

note to students: Create individual Wikiversity username

Wiki Editing (Shortcuts)
note: (M) is for MediaWiki commands and (F) is for Firefox commands


 * Edit (M): alt + shift + E
 * Go back to previous page (F): alt + ←
 * Create link (M):  (Example:  Lecture Plan)
 * Preview (M): alt + shift + P
 * Save (M): alt + shift + S
 * Title (M): = = (with subheading 1 denoted by == ==, and subheading 2 denoted by === ===, etc)

General Non-linear PDEs
(x,t) = (x1, x2) where x1 = x (spatial var.) and x2 = t (temporal var.)
 * n independent variables: {xi} = {x1, x2, ...xn }
 * Ex: 1D case with time variable (1D time-dependent problem)

page: 3-2 (x, y, z, t) = (x1, x2, x3, x4) ui( {xj} ) where i = 1,2,3 and j = 1,2,3,4 and (u1, u2, u3 ) → velocity field along x,y,z coords. and (x1, x2, x3 ) → (x, y, z)
 * 3D time-dependent problems
 * Unknown functions: u1, u2, ...
 * Ex. Navier Stokes in 3D


 * Restrict to one (1) unknown function u
 * n partial derivative: $$\frac{\partial^m u}{\partial x_i1,...,\partial x_im}$$
 * {i1, ..., im } subset of m indices among n possible indices; note: i1, ..., im = 1, ..., n
 * Hmk #1 - re-create today's lecture using Wikiversity

Leopold Kronecker (1823 - 1891)

Kronecker Delta was named after Leopold Kronecker, a 19th century mathematician from Prussia. Born in 1823, Kronecker studied at the University of Berlin and later went on to teach at what is now known as Humboldt University in 1883. He specialized in the areas of number theory, finitism, and continuity before dying in Berlin in 1891. source:.

Clarification of topics from R1 (Repeat R1)
It has been observed by Dr. Vu-Quoc that some of the basic concepts concerning the topic of linearity and linear operators covered in R1 were not fully understood by students. These concepts were presented a second time by Dr. Vu-Quoc in hopes that he will be more effective in communicating these ideas to students. Linearity (and the linear operator) is important in the study of PDEs. It forms the basis of study for Linear Transformation of Coordinates and future topics. A detailed review of these concepts is given here:

$$\blacktriangleright$$ Students were asked to expand the following expression, $$ D(u):= div \left [ \mathbf{k} \cdot grad(u) \right ] $$    (Coordinate form)
 * $$= \frac{\partial }{\partial x_i} \left [ \mathbf{k_{ij}} \cdot \frac{\partial u}{\partial x_j} \right ]$$ where $$i,j \in {1,2}$$    (Component form)

Most students expanded this equation using Leibniz Rule (aka "Product rule"). To do this would have been incomplete. Dr. Vu-Quoc wanted an expansion of  the two indices (i,j) in the folowing manner:
 * $$= \sum_{i}\sum_{j} \left [ \mathbf{k_{ij}} \cdot \frac{\partial u}{\partial x_j} \right ]$$
 * $$= \frac{\partial }{\partial x_1} \left ( k_{11} \frac{\partial u}{\partial x_1} \right ) + \frac{\partial }{\partial x_1} \left ( k_{12} \frac{\partial u}{\partial x_2} \right ) + \frac{\partial }{\partial x_2} \left ( k_{21} \frac{\partial u}{\partial x_1} \right ) + \frac{\partial }{\partial x_2} \left ( k_{22} \frac{\partial u}{\partial x_2} \right )$$

note: the above expression is the correct solution to the homework given in R1.

R6: Photos of Student Interaction

A main objective of the co-operative learning scheme is to enhance student interaction.

The following photos were taken during a study session amongst students as they completed the respective assignments given in Report 6.

As shown in the photos, students were able to use the co-operative learning framework as a tool to discuss relevant concepts, share homework solutions, and cultivate a sense of camaraderie with each other.