A contravariant vector transforms this way :
A covariant vector does it the other way round :
Here’s another way to see the differences between them, using spherical coordinates as an example :
Contravariant components match with covariant vectors, and vice versa.
This set of four fundamental equations may well be the most beautiful in Physics. They encompass every phenomenon in electricity and magnetism, crystallise the intellectual struggle in finding a language for waving fields in spacetime, and from which declares how light is born.
Applying vector calculus techniques, the solutions of these equations are as follows :
(Take note that the charge density (ρ) and current density (j) are taken at the retarded time t-r/c)
Why is the sky blue?
Why do we see clouds?
Why is the sun so reddish at sunrise or sunset?
A quick answer : Rayleigh scattering is behind them all.
Light from the sun strikes the air molecules, causing them to oscillate and re-radiate scattered light. The intensity of the scattered light varies with wavelength : the shorter the wavelength, the higher the intensity. Blue light has shorter wavelength than red, so it is scattered more intensely and that gives us the glorious blue sky.
Thousands of condensed water vapour molecules act in unison as sunlight strikes them. The scattered radiation is proportional to the square of the number of molecules vibrating together in phase, hence producing a much more intense light than what could be produced from individual, randomly distributed water molecules. That’s how clouds stand out in the sky.
At sunrise or sunset, light from the sun passes through a thicker layer of air and dust particles, than at high noon. The longer wavelength red light outlasts the shorter wavelength blue light when the scattering particle size increases. So the blue intensity fades while the red intensity endures, leaving us the beautiful sunset.
Next time we look at the physics equations behind them.
When a fluid particle moves in a non-viscous medium, it follows the following equation of flow of dry water (a term coined by Von Neumann) :
Remembering our earlier nabla trick,
We can replace v·∇(v) by ½(∇(v·v))-v×(∇×v), also g can be replaced by a potential, then the equation of motion becomes :
We can further take a nabla to cross every term. Now all terms involving a gradient will disappear when crossed by a nabla, so what remains is just :
Let’s explore the dot product between a nabla and E cross B, and apply Maxwell’s equations :
Rearranging the terms,
The above equation implies that the rate of field energy decrease is equal to the outflow of electromagnetic energy plus work done on matter.
So Poynting vector is the energy flux of the electromagnetic field. It is always at right angle to both E and B, pointing in the direction of the electromagnetic wave.
We need to find 
First we must accept that 
then using the Nabla trick once more
We find what we look for
This is useful in the equation of motion in Hydrodynamics.
This trick I learnt from Feynman and it’s really useful.
Here’s another one:
They can be used to derive relations involving the Poynting vector.
figkid 