Drawing in Latex

Here are some pictures I have created using LaTex and TikZ.

Figure 1

Figure 2

The code TEX used for generation of the images is shown below:

	\usepackage{tikz}
	\usepackage{verbatim}
	\usetikzlibrary{arrows,shapes}
	\usepackage{pgfplots}
	\pgfplotsset{compat=1.3}
	\usepackage{amsmath}
	\usetikzlibrary{chains,positioning}
	...
	%%fig 1
	\begin{tikzpicture}[scale = 0.7]
	% Draw medias
	\fill[blue!10] (0,6) rectangle (10,8);
	\fill[blue!40] (0,4) rectangle (10,6);
	\draw[dashed,line width=1pt] (0,4)--(10,4);
	\fill[blue!40] (0,1) rectangle (10,4);
	\fill[brown!100] (0,0) rectangle (10,1);
	%draw media labels
	\draw(5,0)node[above]{Soil};
	\draw(5,2)node[above]{River};
	\draw(5,6)node[above]{Air};
	%exchange fluxes
	\draw(4, 1) node [above left]{$F_{2,b}$};
	\draw[<->,line width=1pt] (4,0.5)--(4,1.5);
	\draw(4, 4) node [above left]{$F_{1,2}$};
	\draw[<->,line width=1pt] (4,3.5)--(4,4.5);
	\draw(4, 6) node [above left]{$F_{a,1}$};
	\draw[<->,line width=1pt] (4,5.5)--(4,6.5);
	%draw state variables
	\draw(7, 5) node {$h_1, T_1$};
	\draw(7, 2.5) node {$h_2, T_2$};
	%draw radiation flux
	\draw(8, 7) node [above left]{$LW + SW$};
	\draw[->,line width=1pt] (8,7.5)--(8,6.5);
	%draw precipitation flux
	\draw(9, 7) node [above left]{$P$};
	\draw[->,line width=1pt] (9,7.5)--(9,6.5);
	%runoff
	\draw(0, 6) node [above right]{$I$};
	\draw[->,line width=1pt] (0,6)--(1,6);
	%inflow
	\draw(0, 5) node [above right]{$Q^{in}_{1}$};
	\draw[->,line width=1pt] (0,5)--(1,5);
	\draw(0, 2.5) node [above right]{$Q^{in}_{2}$};
	\draw[->,line width=1pt] (0,2.5)--(1,2.5);
	%outflow
	\draw(10, 5) node [above left]{$Q^{out}_{1}$};
	\draw[->,line width=1pt] (9,5)--(10,5);
	\draw(10, 2.5) node [above left]{$Q^{out}_{2}$};
	\draw[->,line width=1pt] (9,2.5)--(10,2.5);
	\end{tikzpicture}
	%%fig 2
	\begin{tikzpicture}[scale = 0.6]
	\begin{axis}[ xlabel = $h$, ylabel=$T$, xticklabels={,,}, yticklabels={,,}]
	\addplot[color=black,line width=2pt] coordinates{ (-1,-1) (-0.5, 0) (0,0) (1,1)};
	\addplot[dashed] coordinates{(-0.5,-1) (-0.5, 0)};
	\addplot[dashed] coordinates{(0,-1) (0, 0)};
	\addplot[dashed] coordinates{(-0.9,0) (1, 0)};
	\node at (axis cs:-1,0){$0 ^\circ C $};
	\node[coordinate,pin=above:{$h=-L_{i,w} m_{total}$}] at (axis cs:-0.5,0) {};
	\node[coordinate,pin=below right:{$h=0$}] at (axis cs:0,0) {};
	%\node at (axis cs:-0.5,-1){$(-L_{i\rightarrow w} m_{total})$};
	\end{axis}
	\end{tikzpicture}

view raw draw.tex hosted with ❤ by GitHub

Gradient of scalar product or divergence of a tensor.

When you read a scientific work with equations there could be surprises and confusion even in the notation. Here I want to speak about the one I encountered recently, when looking at the Navier-Stokes equation.

First I thought, the result of the operation should be a vector since it appears as a term in a vectorial equation so it was natural for me to consider it as a gradient of a scalar product of two vectors. Then I started remembering the formula for the gradient of a scalar product.

Here the star represents the vector on which the nabla operator acts. Those could be expressed as follows( (*,*) - scalar product, [*,*] - vector product ):

and swapping a and b we also get

Then combining the last 3 equations we get:

Which differs from the first equation. Then It came to me that the term

is a flux of a momentum, that is a flux of a vector quantity. This means that each coordinate of the quantity should be multiplied by each coordinate of the velocity, which gives us the following tensor:

Then each component of the divergence D of the flux will have the following form

which is equivalent to

Which, I suppose, is exactly what the author had in mind.