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:

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.