Science

Fermat's Principle. Let C3C3 denote the set of all continuous curve segments in three dimensions. Let points AA and BB in three dimensions be given. Suppose that a light ray begins at point AA and ends at point BB, and suppose that the path of the light ray is constrained to not lie in some subset C3C3, then the path that light takes between AA and BB is a critical point of the travel time functional for any variation of paths contained in the set C3C3.
We can use this principle to prove either of the following two statements, all three of which one might be inclined to call the Law of Reflection.
Law of Reflection 1. If light is emitted in a given direction towards a mirror, then (I) the light will travel in a straight line towards the mirror along the initial direction, (ii) it will hit the mirror, (iii) it will reflect in a straight line, and (iv) the angle of incidence will equal the angle of reflection.
Law of Reflection 2. If light is emitted from a point above a mirror, and if the light makes contact with the mirror, then (I) the light will travel in a straight line from its initial point to the point of contact, (ii) it will reflect in a straight line, and (iii) the angle of incidence will equal the angle of reflection.
Notice that in both of these cases, there is a constraint that one needs to take into consideration when determining the path of least time. In the first statement above, the constraint set is the set of all continuous paths whose initial directions do not coincide with that of the specified initial direction. In the second statement of the law, the constraint set is the set of all continuous paths that do not make contact with the mirror.
Note that if you don't include a constraint, and if you simply pick any two points above the mirror, then, of course Fermat's Principle tells you that the path followed by light is the straight line segment joining those two points. But that's fine, because the Law of Reflection doesn't answer the question "given any two points AA and BB above a mirror, and given that a light ray goes from AA to BB, what is the path that the light ray must take?" In fact, this question doesn't have a unique answer. The answer depends on the constraints.

A matrix version of Kirchhoff's current law is the basis of most circuit simulation software, such as SPICE. Kirchhoff's current law combined with Ohm's Law is used in nodal analysis.
KCL is applicable to any lumped network irrespective of the nature of the network; whether unilateral or bilateral, active or passive, linear or non-linear.
The principle of conservation of energy implies that:
The directed sum of the electrical potential differences (voltage) around any closed network is zero, or:
More simply, the sum of the emfs in any closed loop is equivalent to the sum of the potential drops in that loop, or:
The algebraic sum of the products of the resistances of the conductors and the currents in them in a closed loop is equal to the total emf available in that loop.
Similarly to KCL, it can be stated as:
{\displaystyle \sum _{k=1}^{n}V_{k}=0} \sum _{k=1}^{n}V_{k}=0
Here, n is the total number of voltages measured. The voltages may also be complex:
{\displaystyle \sum _{k=1}^{n}{\tilde {V}_{k}=0} \sum _{k=1}^{n}{\tilde {V}_{k}=0
This law is based on the conservation of energy whereby voltage is defined as the energy per unit charge. The total amount of energy gained per unit charge must be equal to the amount of energy lost per unit charge, as energy and charge are both conserved.

Balancing chemical equations isn't difficult, once you know the way to do it. Start by finding out how many atoms of each type are on each side of the equation. Some teachers recommend making a little table listing the numbers of each atom for the left hand side and for the right hand side.
Next, look for an element which is in only one chemical on the left and in only one on the right of the equation. (But it is usually a good idea to leave hydrogen and oxygen until you've done the others first).
To balance that element, multiply the chemical species on the side which doesn't have enough atoms of that type by the number required to bring it up to the same as the other side. The number is called the coefficient.
BUT
If you have to multiply by, say, 2 1/2, do so, THEN multiply EVERYTHING on each side of the equation by two to get rid of the half.
We don't like having halves in equations, as you can't get half a molecule.
Now look for the next element or species that is not balanced and do the same thing.
Repeat until you are forced to balance the hydrogen and oxygens.
If there is a complex ion, sometimes called a polyatomic ion, on each side of the equation that has remained intact, then that can often be balanced first, as it is acts as a single species. The ions NO3- and CO32- are examples of a complex ion.
A VERY useful rule is to leave balancing oxygen and hydrogen to the last steps as these elements are often in more than one chemical on each side, and it is not always easy to know where to start. Some people also say you should leave any atom or species with a valency of one one until the end, and also generally leave anything present as an element to the end.
In Example 1 above, you would balance the carbons first, by putting a 3 in front of the CO2, then balance the hydrogens by putting a 4 in front of H2O and finally the oxygens (which are in more than one compound on the right, so we leave them until last) by putting a 5 in front of the O2.