Please help me in reading a mathematical handwriting

[Evin Baxter]

Mod note: Thread moved from HW section
1. Homework Statement
My native language is not English,and I'm not good in reading hand writtings.I got a short passage on mathematics.Please tell me if you could help me with that.I've attached it as
"Lif.pdf".

Homework Equations

The Attempt at a Solution

 

[Mark44]

Mod note: Thread moved from HW section
1. Homework Statement
My native language is not English,and I'm not good in reading hand writtings.I got a short passage on mathematics.Please tell me if you could help me with that.I've attached it as
"Lif.pdf".

The handwriting is close to illegible, but whoever wrote it is apparently doing a Taylor expansion of a function of two variables.
The expressions such as $\rho u_y$ and $(\rho u)_{yy}$ are first and second partial derivatives with respect to y.
Hope that helps.

 

[DrClaude]

Here is what I deciphered. I cannot vouch that it is exact. There is one word, repeated, that I couldn't figure out.

Your Chap 5.2 cont. eqn example
Assume 2D and smooth $\rho$, $u$, $v$
Consider first term in (5.1), use Taylor exp.
$$
\rho u (x,y) dy + (\rho u)_y (x,y) \frac{dy^2}{2} + (\rho u)_{yy}(x, y
+ \theta_3) \frac{dy^3}{6}
$$
$$
-( \rho u ( x + dx,y) dy) + (\rho u)_y (x+dx,y) \frac{dy^2}{2} + (\rho
u)_{yy} ( x, y + \theta_4) \frac{dy^3}{6}
$$
$$
(*) = (\rho u)_x(x+\theta_1,y)dxdy+(\rho u)_{xy}(x+\theta_2,y)
\frac{dxdy^2}{2} + \mbox{--- " ---}
$$
with $0 < \theta_j < dy=dx$. Divide with $dxdy = dy^2$
Do same with remaining terms of (5.1)
Standard ?: Take limit $dx=dy \rightarrow 0 \Rightarrow (5.3)$
Nonstandard ?: By the transfer property
$\frac{(*)}{dxdy}$ can be considered for $dx=dy$ infinitesimal
$$
\mbox{const.} \Rightarrow (\rho u)(\theta_1,y) \approx (\rho u)_x
(x,y)
$$
$$
\begin{align*}
(\rho u)_{xy} (x + \theta_2, y) dy \approx 0, (\rho u)_{yy} (x, y+
\theta_3) dy &\approx 0 \\
(\rho u)_{yy}(x+dx, y +dy) &\approx 0
\end{align*}
$$
Analogously for other terms of (5.1) $\Rightarrow$
$$
\left.
\begin{align*}
\mbox{(5.3) holds modulo infinitesimals} \\
\mbox{(5.3) is standard}
\end{align*}
\right\} \Rightarrow \mbox{(5.3) holds}
$$

 

[Mark44]

@DrClaude, the word you indicated is "math" I believe.

 

[DrClaude]

@DrClaude, the word you indicated is "math" I believe.

I was trying to find a word with an o in it, but "math" would make sense.

 

[Evin Baxter]

I was trying to find a word with an o in it, but "math" would make sense.

Thanks a lot,

 

[Evin Baxter]

Here is what I deciphered. I cannot vouch that it is exact. There is one word, repeated, that I couldn't figure out.

Thanks a lot,

 

[Mark44]

I was trying to find a word with an o in it, but "math" would make sense.

And "moth" probably doesn't make sense.

 

[Erland]

Unreadable, but it seems as it deals with Nonstandard analysis: https://en.wikipedia.org/wiki/Non-standard_analysis