User:Egm6322.s09.mafia/HW1

 See my comments below. On-going work, not finished yet.

Please don't remove this comment box; if you make any corrections, modifications, etc. you can add comments to this comment box to explain what you did.

Egm6322.s09 19:13, 27 January 2009 (UTC)

Added signature and comment in box below as requested.

--Egm6322.s09.lapetina 14:35, 28 January 2009 (UTC)

Added a section regarding contributions.

--Egm6322.s09.lapetina 22:28, 30 January 2009 (UTC)

Added signature for member XYZ of Team Mafia

--Egm6322.s09.xyz 22:01, 8 February 2009 (UTC)

 Added the list of contributing members after the deadline of 22:00 UTC. The difference between the present version (before adding this comment box) and the official version before the deadline is given here.

This page is best viewed with the "Math" settings in the "Preferences" Section of Wikipedia placed on "Always Render PNG". Otherwise, math environments may not appear as math environments.

Team coordinator signs here.--Egm6322.s09.lapetina 14:35, 28 January 2009 (UTC)

 Nice tip on the Math settings to view math environments.

Problems: (1) Looking at the history page of this report, one can see that there was too much imbalance in the contribution of different team members to the report: Essentially, one team member did all the work (about 21,000 bytes), another one contributing about 300 bytes (1.4%), and another team member about 200 bytes (1%). You want the history page to show that everyone contributed more or less equally.

(2) Another problem is that user "EGM6322S09TIAN" may not exist at all. First, the username did not correspond to the convention explained in class (see the username of your other teammates); second, this user may not have created his/her account (registered).

Egm6322.s09 15:12, 28 January 2009 (UTC)

=Preface=

--Egm6322.s09.xyz 18:29, 24 April 2009 (UTC)

Partial differential equations (PDEs) are a class of mathematical expressions that include one or more independent variables and their respective derivatives. These equations describe a host of physical occurrences in the real world. Examples of PDEs in everyday life can be as simple as tracking the change in temperature for a can of soda that is placed on a kitchen counter after being removed from the refrigerator (see Heat Conduction Equation, ). More complex examples of PDEs can be found in Schrodinger's Equation which describes the quantum state of a system, , or, the Monge Ampere equation that occurs most often in studies of differential geometry. A brief introduction to the basic tenets in the study of partial differential equations is presented here. Information given on this Wiki page are from lecture notes of EGM 6322 taught by Dr. Vu-Quoc.

=Introduction=

Before discussing PDEs, it is best to review a few basic mathematical topics, and introduce the notations which will be used throughout the article.

Variables and Vector Notation
--Egm6322.s09.lapetina 17:39, 24 April 2009 (UTC)

N independent variables can be expressed as:

$$  \left \{ {x}_{i} \right \}=   \left \{ {x}_{1},...,{x}_{n}   \right \} $$.

An example of this would be single dimensional case with a time variable.

$$ \left ( x,t \right)=  \left ( {x}_{1}, { x}_{2 } \right )  $$

where $$ {x}_{1} = {x}  $$, a spatial variable

and $$ {x}_{2} = {t}  $$, a temporal variable.

A more complex situation would be a three dimensional, time dependent problem:

$$  \left (x,y,z,t  \right )=  \left({x}_{1},...,{x}_{4}  \right )$$.

In this case, $$ {x}_{1} = {x}  $$, $$  {x}_{2} = {y}  $$, $$  {x}_{3} = {z}  $$, and $$  {x}_{4} = {t}  $$.

Functions and Partial Derivatives
--Egm6322.s09.lapetina 01:50, 17 April 2009 (UTC)

$$ n $$ unknown functions $$ {u}_{i} $$ can be expressed as:

$$ {u}_{1}, {u}_{2} ,... , {u}_{n}$$

For our present discussion, this can be restricted to one unknown function $$ u $$, containing $$ n $$ independent variables $$ {{x}_{i}} i=1,...,n $$

The mth partial derivative of this function can be expressed as:

$$ \frac{\partial^m u}{\partial x_i\ ,...,\partial x_m} $$ where

$$ {{i}_{1},...,{i}_{m}} $$ is the subset of m indices among n possible indices

$$ {i}_{i}, ..., {i}_{m}=1,...,n $$.

 Use "Image:" command, instead of "File:" command to insert images so you can include the image caption. Also improve image quality in the future.

Egm6322.s09 13:04, 4 February 2009 (UTC)

The unknown function $$ u $$ maps numbers from the domain $$ \Omega $$ to the real numbers $$ \mathbb{R} $$ In two dimensions, this means:

$$ \left( x,y \right) \in \Omega $$

$$ u \left( x,y \right) \in \mathbb{R} $$

and

$$ \Omega \rightarrow \mathbb{R}$$

$$ \left( x,y \right) \mapsto u \left( x,y \right) $$

which means that $$\Omega $$ is the domain of u, $$ \mathbb{R}$$ is the range of u, and $$u \left ( \Omega \right )$$ is the image of $$\Omega $$ under the mapping of $$ u $$.

An example of this is the Navier-Stokes Equations in 3-D:

$$ {u}_{i} ( \left \{  {x}_{j} \right \} ) $$

where $$ {u}_{i} $$ is the velocity field in x, y, z; $$ {u}_{1}, {u}_{2}, {u}_{3} $$, dependent upon variables $$ {x}_{1}, ... {x}_{4}$$, where $$ {x}_{1} = {x}  $$, $$  {x}_{2} = {y}  $$, $$  {x}_{3} = {z}  $$, and $$  {x}_{4} = {t}  $$.

More on Mapping

AFFINE MAPPING

Definition: "In geometry, an affine transformation or affine map or an affinity (from the Latin, affinis, 'connected with') between two vector spaces (strictly speaking, two affine spaces) consists of a linear transformation followed by a translation:" Source:Affine_transformation

LINEAR MAPPING

Definition: "In mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. The expression "linear transformation" is in particularly common use, especially for linear maps from a vector space to itself (endomorphisms). In advanced mathematics, the definition of linear function coincides with the definition of linear map." Source :Linear_transformation

 Nice contribution with proper references. You want to make the title of this collapsible box clearer by specifying that the mappings discussed in the box are affine mapping and linear mapping. Egm6322.s09 19:16, 27 January 2009 (UTC)

Tensor Notation
--Egm6322.s09.lapetina 01:51, 17 April 2009 (UTC)

--EGM6322.S09.TIAN 17:48, 24 April 2009 (UTC)



Often we will use vector operators and tensor notation when studying PDEs.

For example, the function:

$$u_xx+u_yy+f(x,y)=0$$

can be rewritten as div( $$\kappa$$ $$\cdot$$$$grad u$$$$+f(x,y)=0$$

Here, $$\kappa$$ is a continuity tensor (2nd order tensor).

$$\kappa$$ = $$\kappa$$ $$e_i$$ $$\otimes$$ $$e_j$$ 2nd-order tensor

$$v$$ = $$v_i$$ $$e_i$$ vector,1st-order tensor

A 0th order tensor is a scaler.

Vector Operators
--EGM6322.S09.TIAN 17:48, 24 April 2009 (UTC)

There are many examples of vector operators, which will be expanded upon in discussions of order and linearity

$$grad$$ $$u$$ $$=$$ $$\frac{\partial u}{\partial x_i}$$ $$e_i$$

$$div$$ $$v$$ $$=$$ $$\frac{\partial v_i}{\partial x_i}$$ $$e_i$$

$$\kappa$$ $$\cdot$$ $$grad$$ $$u$$ =( $$\kappa_{ij}$$ $$e_i$$ $$\otimes$$ $$e_j$$ ) $$\cdot$$ ($$\frac{\partial u_k}{\partial x_k}$$ $$e_k$$ )

=$$\kappa_{ij}$$ $$\frac{\partial u}{\partial x_k}$$ ( $$e_i$$ $$\otimes$$ $$e_j$$ ) $$\cdot$$ $$e_k$$

Here, $$e_i$$ $$\otimes$$ $$e_j$$ $$\cdot$$ $$e_k$$ = $$e_i$$ ( $$e_j$$ $$\cdot$$ $$e_k$$ )

Vector operators often act differently depending upon the order of tensor they act upon.

For example:

$$u(x,y)$$ is a scaler function $$\Rightarrow$$ 0th order tensor $$gradu$$ is a vector field $$\Rightarrow$$ 1st order tensor

while

$$grad$$  ($$\cdot$$) increases tensor order by 1

but

$$div$$  ($$\cdot$$) decreases tensor order by 1

We can let $$v$$ : = $$\kappa$$ $$\cdot$$ $$gradu$$

$$div$$ $$v$$ = $$\frac{\partial }{\partial x_i}$$ $$\kappa_{jk}$$ $$\frac{\partial u}{\partial x_j}$$

Kronecker delta
--EGM6322.S09.TIAN 17:48, 24 April 2009 (UTC)

Definition: If $$e_j$$ $$\otimes$$ $$e_k$$ = $$\delta_{jk}$$, $$\delta_{jk}$$ is called Kronecker delta.

$$ \delta_{jk} = \begin{cases} 1 & for \ j=k \\ 0 & for \ j \neq k \end{cases} $$ = $$\kappa_{jk} \frac{\partial u}{\partial x_i} $$ $$e_i$$

Leopold Kronecker (1823 - 1891)

--Egm6322.s09.xyz 18:31, 24 April 2009 (UTC)

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:

 Good idea and contribution to find out well-known historical figures who contributed to this classical field of PDEs and related fields.

Egm6322.s09 13:04, 4 February 2009 (UTC)

=Order=

Definition
The order of a PDE is the order of highest derivative in the PDE. This is the highest number of derivatives in any term of the equation.

 I added the section References at the end of the report, and the command to render the references.

Egm6322.s09 15:44, 28 January 2009 (UTC)

Examples of First Order PDE's
--EGM6322.S09.TIAN 17:50, 24 April 2009 (UTC)

Linear
$$5u_x-7u_y=0$$

Non-Linear
--Egm6322.s09.xyz 18:32, 24 April 2009 (UTC)

$$6(u_x)^3+2(u_y)^2+(u)^{1/2}+x^2+sin xy=0$$

Prove Non-Linearity --Egm6322.s09.xyz 18:33, 24 April 2009 (UTC)

We can assume that $$D(u)=6(u_x)^3+2(u_y)^2+(u)^{1/2}$$

Show D(u) is not linear.

(Solution)

Again, $$D \left( u \right) = 6 \left( u_x \right)^3 + 2 \left( u_y \right)^2 + \left( u \right)^{1/2}$$.

D(u) is linear if the following formula holds:

$$D \left( \alpha\cdot u + \beta\cdot v \right) = \alpha\cdot D \left( u \right) + \beta\cdot D \left( v \right) $$

OR,

$$6 \left( \alpha\cdot u + \beta\cdot v \right)^3 + 2 \left( \alpha\cdot u + \beta\cdot v  \right)^2 +  \left( \alpha\cdot u + \beta\cdot v  \right)^{1/2} = \alpha\cdot \left [ 6 \left( u_x  \right)^3 + 2 \left( u_y  \right)^2 +  \left( u  \right)^{1/2} \right ] + \beta\cdot \left [ 6 \left( v_x  \right)^3 + 2 \left( v_y  \right)^2 +  \left( v  \right)^{1/2} \right ]$$

The expressions on either side of the equality sign in the above equation are NOT identical. Therefore, D(u) is NOT linear.

More Examples of 1st Order PDEs --Egm6322.s09.xyz 18:33, 24 April 2009 (UTC)

Examples of 1st order linear and nonlinear PDEs are given below. For more information, see the Wiki page for Partial Differential Equations.

(LINEAR) $$ \frac{du}{dx}\ + \frac{du}{dy}\ + \frac{du}{dz}\ = 0 $$ (LINEAR) $$ \dot x \ + 42\dot y \ = 0 $$ (LINEAR) $$ t \frac{dy}{dt}\ + 4 \frac{dy}{dx}\ = 0 $$ (NON-LINEAR) $$ \left(\frac{du}{dx}\ \right)^2 + \left(\frac{du}{dy}\ \right)^3  = 0 $$ (NON-LINEAR) $$ u + \sqrt[3]{\left(\frac{du}{dy}\ \right)}  = 0 $$

As an addition proof, it should be noted that the linear PDEs given above are also separable (the non-linear PDEs are not separable expressions). More information about the linearity of PDEs is given in the following references:,.

Examples of Second Order PDEs
--EGM6322.S09.TIAN 17:52, 24 April 2009 (UTC)

$$div(gradu)+f(x,y)=0$$

Note: tensorial notation = coord free notation.

$$\kappa$$ $$= const$$ $$\Rightarrow$$ is a linear second order PDE

$$\kappa$$ $$= \underline{\kappa} (x,y)$$ $$\Rightarrow$$ is also a linear second order PDE

$$\kappa$$ $$= \underline{\kappa} (x,y,u)$$ $$\Rightarrow$$ is a quasilinear second order PDE

Another example of a second order PDE is the Laplace Equation in two dimension:

$$ F \left ( \left (x,y \right) ,u,\left ({u}_{x},{u}_{y} \right), \left ({u}_{xx},{u}_{xy},{u}_{yy} \right ), ...   \right )=0 $$

also expressed as:

$$ {u}_{xx}+{u}_{yy}=0$$.

Another second order PDE is the equation:

$$ {u}_{yy} +u^2=7x $$.

$$div$$ $$v$$ = $$\frac{\partial }{\partial x_i}$$ $$\kappa_{ij}$$ $$\frac{\partial u}{\partial x_j}$$

HW：Matrix Expansion --EGM6322.S09.TIAN 17:54, 24 April 2009 (UTC)

We can expand it as: $$div$$ $$v$$ = $$\frac{\partial \kappa_{ij} }{\partial x_i}$$ $$\frac{\partial u }{\partial x_j}$$ + $$\kappa_{ij}$$ $$\frac{\partial u^2 }{\partial x_i x_j}$$

 Not exactly what was asked; you want to expand the summation indices, which take values in $$\displaystyle \{1,2\}$$. See, e.g., Team Bit.

Egm6322.s09 12:44, 30 January 2009 (UTC)

Other second order PDE's are examined in the Linearity section.

=Linearity=

Definition
--Egm6322.s09.lapetina 01:52, 17 April 2009 (UTC)

If $$ u $$ is an unknown function, the operator $$ L $$ is considered a linear operator with respect to $$ u $$ if:

$$ L \left ( \alpha u + \beta v \right ) = \alpha L \left ( u \right )+ \beta L \left ( v \right ) $$

Steps for Checking for Linearity --Egm6322.s09.lapetina 01:52, 17 April 2009 (UTC)

$$ 2{u}_{xx}+3{u}_{yy}-7{x}^{2}+x=0 $$

The operator is :

$$ \mathcal{L}= 2 \frac{\partial^2 }{\partial x^2} + 3 \frac{\partial^2 }{\partial y^2} -7 x^2-x $$

where $$ -7 x^2-x:= f(x) $$

To check the linearity, we assume $$\alpha$$ and $$\beta$$ are real number constants, and evaluate: $$ \mathcal{L} (\alpha u+ \beta v)= 2 \frac{\partial^2 (\alpha u+ \beta v)}{\partial x^2} + 3 \frac{\partial^2 (\alpha u+ \beta v)}{\partial y^2} + f(x)$$

which is equivalent to:

$$ \mathcal{L}(\alpha u+ \beta v)=2 {\left (\alpha u + \beta v \right )}_{xx}+3 {\left (\alpha u +\beta v\right )}_{yy}+f(x) $$

If we distribute the integers, as well as the second derivative operators, the equation becomes:

$$ \mathcal{L}(\alpha u+ \beta v)=2 \alpha {u}_{xx}+2 \beta {v}_{xx}+3 \alpha {u}_{yy} + 3 \beta {v}_{yy}+f(x) $$.

Grouping the terms by the constants $$\alpha$$ and $$\beta$$ concludes:

$$\mathcal{L}(\alpha u+ \beta v)= \alpha \left [2 {u}_{xx}+3{u}_{yy} \right ] +\beta \left [ 2 {v}_{xx} +3 {v}_{yy} \right ] +f(x) $$.

The operator $$ \mathcal{L} $$ is nonlinear because

$$ \alpha L(u)+ \beta L(v)= \alpha \left [2 {u}_{xx} +3 {u}_{yy} +f(x) \right ] +\beta \left [2 {v}_{xx}+3 {v}_{yy} + f(x) \right ] $$

and

$$ f(x) \ne \alpha f(x) +\beta f(x) $$

Alternative Definition of Linearity

--Egm6322.s09.lapetina 01:52, 17 April 2009 (UTC)

An alternative means of defining linearity is provided by A.N. Kolmogorov & S.V. Fomin in Introductory Real Analysis, 1975. Assuming $$x,y \in \mathbb{R}$$, they define numerical functions on linear spaces as "functionals", and describe these as "additive" if:

$$f(x+y)=f(x)+f(y)$$ for all $$x$$ and $$y$$, and "homogeneous" if

$$f(\alpha x)=\alpha f(x)$$ for all $$x$$ where $$\alpha $$ is a real constant. This can be extended to complex numbers, but is out of our present scope.

This can be extended to define linearity for operators in the following fashion:

Given:

$$f(\alpha x)=\alpha f(x)$$ where $$x,y \in \mathbb{R}$$ and $$\alpha $$ and $$\beta$$ are real constants,

$$\alpha f(x)+\beta f(x)=f(\alpha x)+f(\beta x)$$

In turn, one of the $$x$$ can be replaced with a $$y$$, yielding:

$$\alpha f(x)+\beta f(y)=f(\alpha x)+f(\beta y)$$

Since $$\alpha $$ and $$\beta$$ do not alter the real nature of the domain, then

$$\alpha x = q$$

and

$$\beta y = r$$

where

$$q,r \in \mathbb{R}$$.

The function is still additive, and:

$$f(q)+f(r)= f(q+r)$$.

or

$$f(\alpha x)+f(\beta y)= f(\alpha x+ \beta y)$$.

Using the transitive property:

$$\alpha f(x)+\beta f(y)= f(\alpha x+ \beta y)$$.

Therefore, operators which are homogeneous and additive are also linear. The definitions are equivalent.

 See better proof in class.

I added the section References at the end of the report, and the command to render the references.

Rewrite the above reference in the collapsible box so it looks better in the list of references.

Egm6322.s09 15:44, 28 January 2009 (UTC)

Second Order Linear PDEs
Second order linear PDEs can be expressed in matrix form. If $$\kappa$$ is a two by two matrix, and $$u $$ is a vector of differentials, then the operator $$\mathcal{D} \left ( u\right)= \kappa u $$ is linear if $$ \kappa=\kappa \left ( x_1, x_2 \right ) $$ where $${x}_{1}$$ and $${x}_{2}$$ are independent variables.

 Error: The diffusion operator is $$ \displaystyle \mathcal D (u) :=   {\rm div} \left( \boldsymbol{\kappa} \cdot {\rm grad} \, u \right) =  \frac{\partial}{\partial x_i} \left(     \kappa_{ij}      \frac{\partial u}{\partial x_j}   \right) $$, where $$\displaystyle \mathcal D (\cdot)$$ denotes the differential operator. You want to show that this diffusion operator is 2nd-order and linear.

Egm6322.s09 12:44, 30 January 2009 (UTC)

Proof a Tensor Operator is a Linear Second Order PDE

--Egm6322.s09.lapetina 17:42, 24 April 2009 (UTC)

Let $$ u= \nabla w $$, the gradient of w and $$ v= \nabla z $$, the gradient of z.

$$\mathcal{D} \left ( u\right)= \kappa u $$

$$\mathcal{D} \left ( u\right) =$$ $$ \begin{bmatrix} u_1 c_1 f_1(x_1,x_2)+ u_2 c_2 f_2(x_1,x_2) \\ u_1 c_3 f_3(x_1,x_2)+ u_2 c_4 f_4(x_1,x_2) \end{bmatrix} $$

where $${c}_{i} $$ are constants and $${f}_{i}$$ are functions of independent variables $$x_1$$ and $$x_2$$.

Linearity is checked using its definition

$$\mathcal{D} \left ( \alpha u +\beta v \right) = \alpha \mathcal{D} \left ( u\right )+ \beta \mathcal {D} \left ( v \right ) $$

$$\mathcal{D} \left ( \alpha u +\beta v \right)= \mathcal{D}

\begin{bmatrix} \alpha u_1+\beta v_1\\ \alpha u_2+\beta v_2 \end{bmatrix}

$$

Let $$\alpha u_1+\beta v_1 =\theta$$ and $$\alpha u_2+\beta v_2 =\phi$$

$$\mathcal{D} \left ( \alpha u +\beta v \right) =

\begin{bmatrix} \theta c_1 f_1(x_1,x_2) + \phi c_2 f_2(x_1,x_2) \\ \theta c_3 f_3(x_1,x_2) + \phi c_4 f_4(x_1,x_2) \end{bmatrix} $$

while

$$ \alpha \mathcal{D} \left ( u\right )+ \beta \mathcal {D} \left ( v \right ) =

\begin{bmatrix} \alpha u_1 c_1 f_1(x_1,x_2)+ \alpha u_2 c_2 f_2(x_1,x_2) \\ \alpha u_1 c_3 f_3(x_1,x_2)+ \alpha u_2 c_4 f_4(x_1,x_2) \end{bmatrix} + \begin{bmatrix} \beta v_1 c_1 f_1(x_1,x_2)+ \beta v_2 c_2 f_2(x_1,x_2) \\ \beta v_1 c_3 f_3(x_1,x_2)+ \beta v_2 c_4 f_4(x_1,x_2) \end{bmatrix}

$$

$$ =

\begin{bmatrix} (\alpha u_1 + \beta v_1 ) c_1 f_1(x_1,x_2)+ (\alpha u_2 + \beta v_2 )c_2 f_2(x_1,x_2) \\ (\alpha u_1 + \beta v_1 ) c_3 f_3(x_1,x_2)+ (\alpha u_2 + \beta v_2 )c_4 f_4(x_1,x_2) \end{bmatrix}

$$

$$ =

\begin{bmatrix} \theta c_1 f_1(x_1,x_2)+ \phi c_2 f_2(x_1,x_2) \\ \theta c_3 f_3(x_1,x_2)+ \phi c_4 f_4(x_1,x_2) \end{bmatrix}

$$

Therefore, tensors of the type

$$ \kappa \left ( x_1, x_2 \right ) $$ where $${x}_{1}$$ and $${x}_{2}$$ are independent variables are linear operators. In this case it is a second order operator.

Infinite examples of second order linear PDEs can come from defining the functions and constants of a the two dimensional tensor $$ \kappa \left ( x_1, x_2 \right ) $$.

 Error: Not correct, confusing notation. See my comment on the error above.

Egm6322.s09 12:44, 30 January 2009 (UTC)

Quasi-Linear PDEs
These were covered in Meeting 7, and will be part of HW2.

Non-Linear PDEs
--EGM6322.S09.TIAN 17:53, 24 April 2009 (UTC)

An example of a non-linear PDE is:

$$ F \left ( \left \{ {x}_{i} \right \}, \left \{ \frac{\partial u}{\partial x_i} \right \}, \left \{ \frac{\partial^2 u}{\partial x_i \partial x_j} \right \}, ... \right ) =0$$

where there are $$n $$ independent variables $$  \left \{ {x}_{i} \right \}=   \left \{ {x}_{1},...,{x}_{n}   \right \} $$ ,

and one unknown function $$ u $$.

$$ \left \{ {x}_{i} \right \}$$ contains $$ n $$ arguments,

$$ \left \{ \frac{\partial u}{\partial x_i} \right \}$$ are the components of the $$ \nabla u$$, and

$$ \left \{ \frac{\partial^2 u}{\partial x_i \partial x_j} \right \}$$ are the components of the Hessian of $$ u $$.

Examples of first order non-linear PDE is:

$$u_x u_y u_t =7 $$

and

$$4x + 6y^3 + xy{u}_{x}=10 $$

Other non-linear PDEs are discussed in the Order Section.

The Hessian
--Egm6322.s09.lapetina 01:52, 17 April 2009 (UTC)

The Hessian Matrix is a symmetric $$ n \times n $$ matrix, thus H = HT

and $${H}_{n \times n}$$ := $$ {\left [ {H}_{ij}\right ]}_{n \times n}$$.

The Hessian is defined as:

$$ {H}_{ij} := \left \{ \frac{\partial^2 u}{\partial x_i \partial x_j} \right \}$$.

 For matrices, use square brackets, instead of curly brackets; also use a boldface symbol for a matrix quatity, i.e.,

$$ \mathbf H = \left[ {H}_{ij} \right]_{n \times n} = \left[ \frac{\partial^2 u}{\partial x_i \partial x_j} \right]_{n \times n}$$.

Egm6322.s09 15:44, 28 January 2009 (UTC)

This has many interesting properties, and is often used in optimization problems,.

 I added the section References at the end of the report, and the command to render the references.

Rewrite these references so they look better in the list of references.

Egm6322.s09 15:44, 28 January 2009 (UTC)



The Hessian contains $$ \frac{n^2+n}{2} $$ independent terms. The upper diagonal, minus the main diagonal, contains $$ \frac{n^2-n}{2} $$, while the main diagonal contains $$ n$$ terms, and

$$ \frac{n^2-n}{2} + n= \frac{n^2+n}{2}$$.

=List of Contributing Members=

--Egm6322.s09.lapetina 22:17, 26 January 2009 (UTC) Andrew Lapetina

--EGM6322S09TIAN 22:22, 26 January 2009 (UTC)Miao Tian

--EGM6322S09TIAN 22:22, 26 January 2009 (UTC)Sydni Credle, signed by Miao and Andy in her absence.

--Egm6322.s09.xyz 22:00, 8 February 2009 (UTC)

Contributions
Collaboration on this page was more substantial than indicated on the history page. Contributions were as follows:

Preface-S

Introduction- A--- and M---, with M--- doing the box on mapping and S doing the box on Mr. Kronecker.

Order- S did the proof box and the example box, and M--- did the second order examples. M--- also did the expansion box.

Linearity-A--- did the steps for checking linearity and the alternative definition box, as well as the proof of a tensor operator as a linear PDE. A--- also did the section on non-linear PDEs and the Hessian.

= References =

 Redo the references so they look better. For example, Ref.[2] should read as follows:

Kolmogorov & Fomin, Introductory Real Analysis, Dover, 1975. p.124

Egm6322.s09 17:24, 28 January 2009 (UTC)