Functions and its conditions

Mathematics is the language of physics and Functions is the heart of mathematics. 
       Function is a mathematical representation of an artificial machine which gets some input and gives some output.

       But how functions were first formulated and how it was first defined by the people of physics and mathematics?  

The need of functions:

       In Physics, we always have the situation of relating ‘dependent and independent variables’.

Most of the physical terms are denoted using the relation between one or two or three or more variables.

Eg.

“Speed (s) = Distance (d) / Time (t)” or simply using variables as “s = d/t”

“Acceleration (a) = Velocity (v) / Time (t)” or simply using variables as “a= v/t”

“Gravitational force between two masses (F) = G*m1*m2 / r^2”, where G= gravitational constant, m1=mass of object1, m2=mass of object2, r^2=square of the distance between them. 

      

 We can give so many examples and all of the terms in Physics have the same formulation like above.

       Here Speed changes with the change in Distance and Time. Similarly Acceleration changes with the change in Velocity and Time. That is why Speed and Acceleration is called the dependent variables and distance, time are called the independent variables.  

       Physics itself is measured using these independent and dependent variables.

       The knowledge of the relations between these dependent and independent variables will give us a generalized study of Physics using Mathematics. Using the generalized study we can calculate any physical quantity using Mathematics.

       That is how people started to measure Physics using Mathematics in the name of functions.

       From all of the above physical terms, we can get some common thing.

       Speed changes with distance.

       Speed changes with time.

       Acceleration changes with velocity.

       Acceleration changes with time.

       The change in dependent variable makes change in independent variable.

       Let us look the first and the easiest physics term 'speed'. If we change the distance or time then speed changes.

If you take Speed as the output and time, distance as the input then the above statement changes to,

For the values of input [time, distance] there is an output value of speed.

       Change in input will give a change in output.

       In a mathematical way, it was stated as

Speed is a function of distance and time.

Sentences are not always worth to use and symbols were introduced for the sake of simple and compact representation,

s= f(d , t)

Here f( ) means “the function of” and “s= speed , d=distance, t=time”.  

       But before considering the functions of two independent variables, it should be taken care of the functions that varies with one single variable. Studying the single variable functions, gives practice to look into multivariable functions.

Single Variable functions:

For the generalized version, the variables were taken as “x” and “y”.

Let’s say “x” is the independent variable and y is the dependent variable.

Then, the functional statement is given as “y is the function of x” or

“y=f(x)”.

But all those things that vary with some variables cannot be considered as a function. What about its properties?

There should be some conditions imposed on them to proceed further. And so, there are two major conditions for any function and it is given by,

Condition 1:

To study anything in mathematics, the data should be clear.

In functions, it is necessary to have the complete data of the dependent and the independent variables. Only if the complete information of a function is known, we can use it for any physical or mathematical operations related with measurements.

       Without the complete knowledge, we cannot create any working machine or anything for any of our purpose.

       Let us imagine a machine that gives some output for every input. We should get output for every input.

       If there is no output for some input, it means the machine has some problems and it will give error to become useless.

       Similarly for the functions, there should be an output for every input. Otherwise it will not be said as a “function”.

In the E.g. of y=f(x)

“x” is the input and “y” is the output.

Thus the condition1 for functions was stated as,

For each value of input “x” there should be a value of output “y”.

 Condition2  

What if an input value of “x” has more than one output “y” values?

Does it have any problem? Is it allowed?

The same example of speed was taken again. As we are caring about the functions of one independent variable function, here we can take account of only speed and time variables.

Speed changes with time where time is the input value and speed is the output.

If we defined the function that “one input can take two outputs” then,

       It also allows the possibility of “one time input can take two speed outputs”.

How an object can have two speeds at one time? It is as same as two objects at one time.

How an object can be in two places at one time?

As far as we concerned it is not possible in our real world. So the possibility of “two or more outputs for a single input” is removed from the definition of function. 

That is how we arrived to the second condition for a function which can be stated as,

For each value of input “x” there should be one and only value of output “y”.

 Thus two conditions were formulated for a function. 

     

       Therefore only the mathematical relationships which follow the above conditions are called functions otherwise it is called Relations.  

       Still functions need to be defined in a complete mathematical way using Number system so that it could be quantified. Those kind of extra conditions and its structure is defined depending on its role in various situations. As far as concerned about the definition, these two conditions are the necessary and sufficient one.  

Dipole Moment

     It is very important to understand the mathematical beauty of the terms occurring in the Multipole expansion. Since higher order terms vanish faster than the monopole, dipole terms, they are important only when the need of the accuracy is high. 
     As a consequence, in most of the common problems, dipole terms plays the most significant role after the monopole term . 
    
    The dipole term in the expansion is , 
$$ V(\vec{r})_{dipole} = \frac{1}{4\pi\epsilon_0} \frac{1}{r^2} \int_{V'} r'cos\theta' \rho(\vec{r'}) \,dV'  ...\ldots eq.(1)$$
To make the integrand a vector quantity, we know that
 $\hat{r} \cdot \vec{r'} = r' cos\theta' $
and so the dipole term becomes, $$ V(\vec{r})_{dipole} = \frac{\hat{r}}{4\pi\epsilon_0 r^2}\cdot \int_{V'} \vec{r'} \rho(\vec{r'}) \,dV'$$ Terms they depend only on r' is separated and called as dipole moment, $$ \vec{p} = \int_{V'} \vec{r'} \rho(\vec{r'}) \,dV'         ...\ldots eq.(2)$$
and the dipole potential becomes, $$ V_{dipole}(\vec{r}) = \frac {\vec{p}\cdot \hat{r}}{4\pi\epsilon_0 r^2}           ...\ldots eq.(3)$$ From eq.(2) it is known that the dipole moment defined only in terms of r' and it implies that, 'dipole moment of a volume charge depends only on it distribution'.
For point charges, $$ \vec{p} = \sum_{i=1}^N q_i \vec{r'_i}$$ and for a physical dipole it becomes, $ \vec{p} = q(\vec{r'_+} - \vec{r'_-}) = q\vec{d}$ where d is the vector from -q to +q.
    But we should remember that dipole moment doesn't mean there should be only two charges. Our definition is general for any charge distribution. It so happens the physical dipole has similar kind of representation. It is always possible to ask for the dipole moment of any number of charges e.g. three charges in a triangle.
    Some properties of the dipole moment are, Change in coordinate system usually changes the dipole moment except when the total charge is zero. And if we place a physical dipole in a uniform Electric field E, it will experiences a torque and if it is non-uniform it will experience an additional force other than the torque given by, $  \vec{F} = (\vec{p}\cdot\nabla)\vec{E} $
   Each problem will give more insight. 
Similar kind of multipole expansion for a vector potential reveals that the dipole term for a vector potential as,
$$ \vec{A_{dipole}(\vec{r})} = \frac{\mu_0}{4\pi} \frac {\vec{m}\times\hat{r}}{r^2} $$ where $\vec{m}$ is the magnetic dipole moment $$ \vec{m} = I \int \,d{\vec{a}} = I\vec{a} $$
'a' is the area enclosed by the loop and 'I' is the current. Magnetic dipole moment is always independent of the coordinate system since they don't play any role. 
   As we did in Electric dipole, certain properties for a magnetic dipole are obtained and are,
In a uniform field, the net force on any loop is zero. In a non-uniform magnetic field, an infinitesimal loop of dipole moment 'm' will experience a force, $$ \vec{F} = \nabla (\vec{m}\cdot\vec{B})$$ Thus, the concept of polarization explains the newer ideas namely, bound charges and bound currents.