User:EGM6322S09TIAN

One case
$$\kappa$$ $$= \underline{\kappa} (x,y,u,u_x,u_y)$$ $$\Rightarrow$$

Definition of quasilinear: For a PDE of order n (i.e. highest derivative terms are of order n),coeffients of nth order derivative are functions $$= (x,y,u,u_x,u_y,\frac{\partial^m u}{\partial {x^p y^q}})$$

Here, we assume there are two independent variables, mth order derivative, and $$p+q=m,m<n$$

Another Case
PDEs linear with respect to 2nd derivative, but still non-linear in general:

$$div($$ $$\kappa$$   $$\cdot grad u)+$$ $$f(x,y,u,u_x,u_y)=0$$

Here, $$\kappa =$$ $$\kappa (x,y)$$

$$f(x,y,u,u_x,u_y)=0$$ is nonlinear with respect to argues in general, e.g.

$$div($$ $$\kappa$$   $$\cdot grad u)+$$ $$ax^2+by+\sqrt{u}+(u_x)^4+2(u_y)^2=0$$

Homework
Show it linear to 2nd derivative, but non-linear in general.

To make it clear,

$$D_1(\cdot):=$$ $$div[$$  $$\kappa(x,y)$$   $$\cdot grad (\cdot)]$$

We can do this:

$$D_2(\cdot):=$$ $$D_1(\cdot)+$$ $$(\cdot)^{1/2}+$$ $$[(\cdot)_x]^4+$$ $$[2(\cdot)_y]^2$$

$$D_3(\cdot):=$$ $$D_2(\cdot)+$$ $$ax^2+by$$

P11－2
In $$div(\cdot)$$ an example of   for $$R^m \to R^n$$ $$(M:R^m \to R^n)$$ and $$M$$ ??????????

$$R^m, R^n$$ are space vectors (tensors, matrix). Divide maps vector field (vector-valued function) into a scalar function. In other words, domain and range of $$div(\cdot)$$ are function of spaces.

Coordinate Transformation (continued)
Eq(1)P.9.1

Eq(2)P.8.1

$$x= \phi (\bar{x},\bar{y})$$

$$y= \psi (\bar{x},\bar{y})$$

Linear coordinate transformation Eq(5) P.9-1

$$u(x,y)=$$ $$u($$ $$x= \phi (\bar{x},\bar{y})$$, $$y= \psi (\bar{x},\bar{y}))$$

$$=u(\bar{x},\bar{y})$$,  which is an abuse of notation by using "u".

$$=\bar{u}(\bar{x},\bar{y})$$,  which is a more rigorous notation.

Example
Let $$u(x)=ax+b$$

Condition, $$x=\phi (\bar x)$$ $$=sin \bar x$$

$$u(x)=$$ $$u(\phi (\bar x))$$ $$=asin \bar {x} + b$$

=$$u(\bar x)$$ $$\gets$$  abuse of notation

=$$\bar {u} (\bar x)$$  $$\gets$$  more rigorous

Another One
$$u_x(x,y)=$$ $$\frac{\partial u}{\partial x}(x,y)$$ $$=u_x(\phi(\bar{x},\bar{y}),\psi(\bar{x},\bar{y}))$$ $$= \frac{\partial }{\partial x}$$ $$\bar{u}(\bar{x},\bar{y})$$ $$=\frac{\partial \bar u}{\partial \bar x}$$ $$\frac{\partial \bar x}{\partial x}$$ $$+$$ $$\frac{\partial \bar u}{\partial \bar y}$$ $$\frac{\partial \bar y}{\partial y}$$ $$= \bar{u}_\bar {x}$$ $$\frac{\partial \bar x}{\partial x}$$ + $$\bar{u}_\bar {y}$$ $$\frac{\partial \bar y}{\partial x}$$

Define:

$$\bar x$$ = $$\bar {x} (x,y)$$ = $$\bar {\phi} (x,y)$$

$$\bar y$$ = $$\bar {y} (x,y)$$ = $$\bar {\psi} (x,y)$$

$$u_y(x,y)=$$ $$\frac{\partial }{\partial y}\bar u(\bar {x},\bar {y})$$ = $$= \bar{u}_\bar {x}$$ $$\frac{\partial \bar x}{\partial y}$$ + $$\bar{u}_\bar {y}$$ $$\frac{\partial \bar y}{\partial y}$$

Matrix Form
$$\partial_x u=$$ $$\big\lfloor$$ $$\frac{\partial \bar x}{\partial x}$$ $$\frac{\partial \bar y}{\partial x}$$ $$\big\rceil$$ $$\begin{Bmatrix} \partial_ {\bar x} \\ \partial_ {\bar y} \end{Bmatrix} $$ $$(\bar u)$$

HW
Likewise for $$\partial_y $$

$$\partial_y u=$$ $$\big\lfloor$$ $$\frac{\partial \bar x}{\partial y}$$ $$\frac{\partial \bar y}{\partial y}$$ $$\big\rceil$$ $$\begin{Bmatrix} \partial_ {\bar x} \\ \partial_ {\bar y} \end{Bmatrix} $$ $$(\bar u)$$

Then,

$$\begin{Bmatrix} \partial_ x \\ \partial_ y \end{Bmatrix} $$ = $$\begin{bmatrix} \frac{\partial \bar x}{\partial x} & \frac{\partial \bar y}{\partial x} \\ \frac{\partial \bar x}{\partial y} & \frac{\partial \bar y}{\partial y} \end{bmatrix} $$ $$\begin{Bmatrix} \partial_ {\bar x} \\ \partial_ {\bar y} \end{Bmatrix} $$

$$\begin{bmatrix} \frac{\partial \bar x}{\partial x} & \frac{\partial \bar y}{\partial x} \\ \frac{\partial \bar x}{\partial y} & \frac{\partial \bar y}{\partial y} \end{bmatrix} $$ is known as the Jacobian matrix(sometimes it is defined as transposition of Jacobian matrix).

Easier and more general,

$$(x_1,\cdots,x_n)$$ $$\to$$ $$(\bar{x_1},\cdots,\bar{x_n})$$

Initial notation:$$\bar{x_i}$$ = $$\bar{x_i}$$ $$(x_1,\cdots,x_n)$$

$$J_{n \times n}$$ = $$\begin{bmatrix} \frac{\partial \bar x_i}{\partial x_j} \end{bmatrix}_{n \times n} $$

Here, "$$i$$"is the row index while "$$j$$"is the column index.

Carl Gustav Jacob Jacobi(1804-1851)

Jacobian Matrix was named after Carl Gustav Jacob Jacobi, a 19th century mathematician from Prussia. Born in 1804, Jacobi studied at Berlin University and later went on to teach at Königsberg University. He is known for Jacobi's elliptic functions,Jacobian,Jacobi symbol, and Jacobi identity. He died in Berlin in 1851. source:

=Signatures=

--EGM6322S09TIAN 20:35, 5 February 2009 (UTC) Andy L