The changes at this blog:
In case you haven’t noticed it already, notice [what else?] that the layout of this blog has undergone a change. Hopefully for the better!
In particular, I’ve made the following changes:
- This blog is now concerned not only with the more transient writings of mine, but also with the less transient ones! … Accordingly, I have made a new page which holds links to my less transient writings, too, whether the write-ups were published here or elsewhere. See that page here [^].
- The tagline too now reflects the change in the purpose of this blog.
- I have added a header image, too. As of now, it holds some of the equations that have come to grab my attention for a long while. This may change in future. (See the separate section below.)
- A more minor change is the one made to the font.
A note for reading on the mobile:
In case you read this blog on a mobile phone, then to see the “less transient” page, you will have to press the menu button appearing at the top to get to the new page. On a desktop, however, the menu is by default seen as expanded.
The image at the top:
Just for the record, the equations in the top image, as of today (13 August 2018, 11:31 hrs), are the following:
- The inner product and the outer product of two vectors, expressed using the more familiar notation of matrices.
- Definitions of the grad of scalars and vectors, and the div of vectors and tensors.
- The Taylor series expansion
- The Fourier series expansion
- The generic conservation equation for a scalar quantity, in the Eulerian form
- The conservation equation for momentum, in the Eulerian form. (NB: The source term is in terms of i.e. the conserved quantity itself, whereas the rest of the terms have the mass-specific term in them. This is correct.)
- Definition of stress. (See the note for this equation below.)
- Definitions of the displacement gradient tensor, the strain tensor, and the rotation tensor.
- Cauchy’s formula (the relation between stress and the net force)
- The Planck-Einstein relations
- The most general form of the Schrodinger equation
- The time-dependent Schrodinger equation in
- The inner product defined over a Hilbert space, and expansion of a function in terms of its basis set defined in a Hilbert space
An important note on the definition of stress as given in the header image:
I haven’t yet seen this definition in any solid/fluid/continuum mechanics text. So, please treat it with caution.
Also, please do drop me a line if you find it erroneous, problematic, or simply not general enough.
On the other hand, if you run into this definition anywhere, then please do bring the reference to my attention; thanks in advance. [This definition is a part of my planned paper on stress and strain.]
Some of the equations that got left out:
The equations which I would have liked to have in the header, but which got left out for a lack of space, are the following (in no particular order):
- Newton’s second law defining force
- Definitions of action (as momentum-dot-displacement and energy-times-time); action as an integral; action as a functional
- The general equation for the methods of the weighted residuals, and the particular equations for the commonly used test functions (i.e., the Galerkin, the pseudospectral, the least-squares, the method of moments, and the collocation)
- The Euler identity
Perhaps also, things like:
- The wavefunction normalization principle, and the Born equation for finding probabilities
- Structure of probability: simultaneous vs. subsequent events
- The wave, diffusion and potential equations (juxtaposed with the Schrodinger equation)
On the other hand, some of the equations that are generally of great importance, but which have not come to preoccupy me a lot, are the following:
- The Euler-Lagrange equations for classical mechanics
- The Maxwell equations of electrodynamics, supplemented with the “fifth” (i.e. the Lorentz) force equation
- Boltzmann’s equation, and other equations from statistical mechanics
I must have left out quite a few more in both the lists.
However, I am sure that the three laws of thermodynamics probably would not make it to the header image, despite all their grandeur, their all-encompassing scope.
The reason is this: a computational modeler like me seldom works in a very direct manner with the laws of thermodynamics themselves. These laws do inform his theory; the derivation of the equations he uses indeed are based on them, even if only indirectly. However, the equations he works with happen to be much more detailed (and of far more delimited scope). For instance: the Navier-Stokes system (CFD)—an expression of the first law; the stress-strain fields (FEM)—which makes for merely a part of the internal energy; or the Maxwell system (FDTD)—ditto. Etc.
Further change may be coming:
All in all, I am not quite happy with the top image as it exists right now. … It is too crowded, and speaking from a visual aesthetics point of view, its layout is not well-balanced.
So, on both these counts (too much crowding already, and too many good equations being left out), I am thinking of a further idea: may be I should create a sequence of images, each containing only a few equations, and let the server show you one of them at random. Whaddaya think?
Do check out the “less transient” page:
But yes, if you are interested, check out the “less transient” page too, and let me know if something I wrote in the past should be there or not.
So… does that mean that I’ve gone “mathy”?
Though I exclusively include only equations in the header image—no pictures or visualizations at all, no code, and not much text either—it doesn’t mean that I have gone “mathy”. … Hell, no! Not at all! … Just check out my less transient page [^].
A song I like:
(Hindi) “aankhon aankhon mein hum tum, ho gaye…”
Singers: Kishore Kumar, Asha Bhosale
Lyrics: Anand Bakshi