Any mathematical expert can see my attachment I have highlighted few mathematical symbols , what that symbol signifies how to understand that can anyone tell
Dear Naveen, if you have such problems to understand this notation probably the reference you are reading is going to be really hard for you. Anyway I will try to explain you the expressions you have highlighted.
The first and fourth expressions mean that the \psi function (the free surface potentia) is defined for x that live in \Omega (i.e x \in \Omega) (NB: it is written in reverse), and this function takes values in R (it is a real valued function). Where I understand that \Omega \subset R^2. This first one it is clear?
Now your second highlighted expression is the definition of the functional space of functions defined for x \in \Gamma_0 \subset \Omega (I guess that \Gamma_0 is a piece of a 2D boundary, therefore a curve in R^2). This functional space is composed by the real valued functions \eta defined from \Gamma_0 whose line integral in \Gamma_0 is 0. F^1/2(\Gamma_0) it is just the symbol to denote this functional space.
Naveen, you probably did not have good teachers who explain basic ideas of hydrodynamics "on fingers". Imagine water over rigid bottom. If it is incompressible, Laplace equation is valid in every internal point. \psi is velocity potential, and \eta is deviation of free surface from equilibrium. Due to non-compressivility, the total volume of water is preserved, and thus an integral of deviation function over unperturbed surface is zero. The condition on bottom shows that the velocity is locally parallel to the bottom (given by differentiable function); its normal component is zero.
To understand notations. Recall that a symbol resembling Euro symbol means: element belongs to a set. R is a set of real numbers, while other sets in your case are some subsets of R1(real line) or R2 (plane).
You can look at equations (1) in my attached article to see what is what and what one can do with that: https://www.researchgate.net/publication/275581998_Evolution_of_long_nonlinear_waves_on_shelves
Conference Paper Evolution of long nonlinear waves on shelves
thank you peter, I just forgot my maths hardly i remember the notations definitions,
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