Friday, 31 January 2014

Some Important Differential Terms

 


Logarithmic Differentiation :- 

In order to differentiate a function of the form     v power on u    , where    u, v   are both variables , it is necessary to take its logarithm and then differentiate . This process which is known as Logarithmic Differentiation is also useful when the function to be differentiated is the product of a number of factors .


Preliminary Transformation :- 

In some cases , a preliminary transformation of the function to be differentiated facilitates the process differentiation a good deal .


Differentiation "ab into" :- 

To differentiate "ab into" or from first principles means that the process of differentiation is to be performed without making any use of the theorems on the differentiations of sums , products functions and functions etc. nor is any use to be made of the differential co-officiants of standard  forms  .


The   n th  derivative of the product of the powers of  sines and cosines :-

In order to find out   n th  derivative of such a product , we have to express it as the some of the sines and cosines of multiples of the independent variables .



Saturday, 25 January 2014

Polar Equation of Curves

 


Any explicit or implicit relation between    r    and    theta      will give a curve determined by the points whose co-ordinates satisfies that relation .

Thus the equations ;

r=f(theta)   or   F(r , theta)=0

determine curves .

The co-ordinates of two points symmetrically situated about the initial line are of the form   (r,Theta)    and    (r,-theta)   so that their vertical angles differ in sign only .

Hence a curve will be symmetrical about the initial line if on changing    theta    to     -theta    its equation does not change . For instance the curve

                    r=a[1+cos(theta)]   

is symmetrical about the initial line , for

        r=a[1+cos(theta)]=a[1+cos(-theta)]

It may be noted that

r=a    represents a circle with its center at pole and radius   a ;  and

theta=b    represents a line through the pole obtained by revolving the initial line through the angle   b   .

A few important curves will not be traced . To trace polar curves , we generally consider the variations in     r    as    theta    varies .


Thursday, 23 January 2014

Approximate Value

 


Approximate Value :-

Let    y=f(x)

Now ,   Lt delta  x--->0 (delta  y)/(delta  x)=dy/dx ,

therefore from the definition of the limit

as    delta  x   approaches   0  ,

(delta  y)/(delta  x)  approaches  dy/dx

therefore ,  (delta  y)/(delta  x)=dy/dx

[approx . when   delta  x   is small]

or,  (delta  y)=dy/dx  (delta  x)

Thus ,  (delta  y)=dy/dx  (delta  x)

              [approximately]

Important Result  Essential For Problems :-

If   x   and   y   are functions of time ,  t  then

(dx/dt)=change of  x  in unit time 

         =rate of change of    x    .

and   (dy/dt)=change in   y  in unit time 

                  =rate of change of    y  .

Also , (dy/dx)=(dy/dt)/(dx/dt)


Tuesday, 21 January 2014

Existence of Tangent

 


Let   y=f(x)    be any function and

dy/dx   exists at     x=a    

Then at the corresponding point    [a , f(a)]    of the function tangent to the curve exists and if the tangent

makes an angle    theta     with the positive direction of    x-axis    we have

tan(theta)=dy/dx    at     x=a .

tan(theta) = gradient of the tangent to the curve

      y=f(x)     at    x=a 

therefore ; if    f(x)   is differentiable at    x=a    then tangent to the curve at

x=a    must exist and it must be unique .

In any graph of    /x/   tangent is not unique at    x=0   .

i.e. ; tangent at point    x=0   when     x-->0    from left is not same as the tangent

x=0    when   x-->0   from right .

when   x-->0   from left , gradient of the tangent is

tan135degree=-1    and when   x-->0  from right , gradient of the tangent is

tan45degree=1.

Hence tangent is not unique at    x=0    and consequently    /x/  is not differentiable at     x=0    .



Monday, 20 January 2014

Formulas For Differential Co-officiant





Before find differential co-officiant of any mathematical term it is important to remember these formulas . These formulas are very helpful to find differential co-officiant .

1)          (d/dx)(c)=0

                   where    c    is a constant 

2)          (d/dx)(u+v)=(du/dx)+(dv/dx)

3)          (d/dx)(u-v)=(du/dx)-(dv/dx)

4)          (d/dx)(u.v)=(vdu/dx)+(udv/dx)

5)          (d/dx)(u/v)=[(vdu/dx)+(udv/dx)]/v.v

6)          (d/dx)(cu)=cdu/dx

                    where    c      is a constant .

7)          dy/dx=(dy/du)(du/dx)

8)          dy/dx=1/(dx/dy)

9)          dy/dx= (dy/du)(du/dv)(dv/dx)  



Saturday, 18 January 2014

Angle Between Curves

 


Here is some Working Rule on Angle between Curves ;

1)  If Equations of two curves are given . Find the value of    dy/dx     fom the equation of two curves .

2)  If the value of     dy/dx    obtained from the two equations are equal then , angle between the curve is    0 degree    i.e. they touch each other at each point  i.e. the two curves are same .

     If product of the values of     dy/dx     for two curves is    -1,   then the two curves cut each other orthogonally (perpendicularly) .

3)  If given condition (2) is not true , find the co-ordinates of the points of intersection of the two curves by solving the equation of the two curves .

  Then find the values of     dy/dx    from the equation of two curves at one point of intersection .

 This will give gradient of the tangent to the two curves
i.e.    m1     and    m2    .

Then find the angle     A       between the curves by the formula

                tanA= +and- [m1 -m2]/1 + m1m2    .

Do this fo every point of intersection .



Thursday, 16 January 2014

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Tuesday, 14 January 2014

Differentiation as rate measurer




dy/dx   as rate measurer  ;

dy/dx=(dy/dt)/(dx/dt)   , if any two of

dy/dx  ,  dy/dt  and  dx/dt   are

given the value of the third can be determined

    For working out problems there will be two variables    and   y   .   From given condition  a relation between    x    and    y    can be found and differentiating this expression we can find

                   dy/dx   ,

Either     dy/dt    or    dx/dt    is given

Thus if   dy/dt   is given we can find    dx/dt     and vice versa  .

If there is only one variable and its rate of change is given , then we can find the value of the variable in term of time    t    by integrating this expression .

We already know that ;

dy/dx=(dy/dt)/(dx/dt)=rate of change of (y)/rate of change of (x)

thus differential coefficient of   y    with respect to    x     is equal to the ratio of the rate of change of    y     and rate of change of      x      . 



Monday, 13 January 2014

Exact Equation

 

The differential equation     Mdx+Ndy=0    ,

where both    and   N   are functions of     and   y     is said to be exact when there is a function     u     of     x, y    such that 

Mdx+Ndy=du ,

i.e. ,  when   Mdx+Ndy   becomes a partial differential .

Now , we know from Differential Calculus that ;

Mdx+Ndy   should be a perfect differential if ;

DM/Dy=DN/Dx  ,   Hence the condition that

Mdx+Ndy=0  should be an exact differential equation is

DM/Dy=DN/Dx

The method of solving and exact equation of the type

Mdx+Ndy=0   .

First integrate the term in   Mdx   as if    were constant then integrate the terms in   Ndy   considering    x   as constant and rejecting the terms already obtained equate the sum of these integrals to a constant .



Saturday, 11 January 2014

Solutions of Differential Equation

A solution of a differential equation with arbitrary constant equal in number to the order if differential equation  is called General Solution .

Other solutions obtained by  giving particular values to the arbitrary constant in the general solution are called Particular Solutions .

Also we know that the general solution of a function contains an arbitrary constant . Therefore the solution of differential equations , resulting as they do from the operations of integration , must contain arbitrary constants, equal in number to the number of time the integration is involved in obtaining the solution which is equal to the order of differential equation .

Thus we see that the most general solution of a differential equation of the   nth   order must contain       and only    n    independent arbitrary constants .

An ordinary differential equation  is a differential equation in which the unknown function is a function of a single independent variable. In the simplest form, the unknown function is a real or complex valued function, but more generally, it may be vector-valued or matrix-valued: this corresponds to considering a system of ordinary differential equations for a single function.


Ordinary differential equations are further classified according to the order of the highest derivative of the dependent variable with respect to the independent variable appearing in the equation. The most important cases for applications are first-order and second-order differential equations. For example, Bessel's differential equation
x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + (x^2 - \alpha^2)y = 0 
 is a second-order differential equation. In the classical literature a distinction is also made between differential equations explicitly solved with respect to the highest derivative and differential equations in an implicit form. Also important is the degree, or (highest) power, of the highest derivative in the equation . A differential equation is called a nonlinear differential equation if its degree is not one.
A partial differential equation  is a differential equation in which the unknown function is a function of multiple independent variables and the equation involves its partial derivatives. The order is defined similarly to the case of ordinary differential equations, but further classification into elliptic, hyperbolic, and parabolic equations, especially for second-order linear equations, is of utmost importance. Some partial differential equations do not fall into any of these categories over the whole domain of the independent variables and they are said to be of mixed type.



Wednesday, 8 January 2014

Functions in R

 

A rule of correspondence   f   is called a real value function if it associates each member    of the set    to an uniquely determined member    y    (called image of   x ) of   B  , a subset of    . is denoted by


          f: A--> B   or    y=f(x) ,


indicating that the real number     corresponds to 

                                                   is element of    A

                                             under the rule      f      .


The set  A  is called domain of the function and

set   f(A)={ y/ y=f(x) ;   x   is element of  A}

i.e. the set of images of  A  is called the range of   f  .

Also   x   is called the independent variable and    y   is called a dependent variable .

Features of a function    y=f(x)   in    R   are as follows ;


1)   y    is uniquely determined for every value of   x   from the domain . The value of     corresponding to the value    a    of    x    is denoted by     f(a)  .


2)   For two or more values of    x ,   y    can have the same value .


3)   The image i.e. the value of   y    for a prescribed value of    x    is obtained by substituting the value of    x    in the equation connecting    x , y    to describe the rule    f    .



Monday, 6 January 2014

Laws Of Motion

 

Laws of Motion discovered by Newton , They are as follows ;

Law 1  :-     Every body  perseveres in its state of rest or of uniform motion in a straight line unless it will compelled to change that state by impressed force .

Law 2  :-     The rate of change of momentum is proportional to the impressed force and take place in the direction of the straight line in which the force acts .

Law 3  :-     To every section there is always an equal and opposite reaction or the mutual action of any two bodies are always equal and oppositely directed .

The first law implies :

1)-----  That the internal force in a body do not play a part in the state of rest or of uniform motion .

2)-----  The definition of an external force as that which tends to change the inertia of a body on which it acts .

The second law enables us to measure force . If a mass    m     moves with a velocity    v     then its momentum is   mv     .

According to second law ;

F  (an external force)   varies as     (d/dt) (mv) 

therefore   F=K.(d/dt)(mv)  where   K   is a suitable constant

                  =K.m.dv/dt  ,  if    m   is a constant .

                  =K.m.f  ,  where    f    is the acceleration .

We now define a unite of force as that force which acting on unit mass produces unit acceleration .

therefore    1=k.1.1  so that      K=1.

therefore     F=mf

                     = mass .  acceleration    .


Sunday, 5 January 2014

Properties of Continuous Function



Theorem :- 

A continuous function which has opposite signs at two points vanishes at least once between these points , that is if   f(x)   be continuous in the closed interval    [a,b]   and    f(a)    and    f(b)   have opposite signs , then there is at least one value of   x   between    and   b   for which    f(x)=0  .  

Proof :-

For the sake of definiteness , let us suppose that ;

F(a)<0   and    f(b)>0   .

Science   f(x)  is continuous and   f(a)<0  , therefore   f(x)   will be negative in the neighborhood of   a   .
Again Since    f(x)   is  continuous and     f(b)>0   therefore   f(x)   will be positive in the neighborhood of   b   .
The set of values of   x   between    and   b   which make   f(x)    positive is bounded below by  a  and hence possesses an exact lower bound    k  .

Hence ;   a<k<b 

In this way we find that   f(x)   is positive in the interval
k<x<b   and is negative or zero in the interval
a   is less then and equal to   x<k  .

Since   f(x)  is continuous at  x=k  , therefore by the definition of continuity,
f(k-0)=f(k)=f(k+0)  .

Since   f(x)   is negative or zero in
a   is less then and equal to    x<k ,   therefore    f(k-0)   must be negative and zero , therefore    f(k)    which is equal to    f(k-0)    must be negative or zero .

We shall now show that    f(k)   can not be negative  .

Since    f(x)    is positive in the interval   k<x<b   , that is ;
b>x>k   , therefore    f(k+0)   can not be negative and since ;
f(k)=f(k+0)   , therefore   f(k)   can not be negative

Hence it follows that    f(k)=0     and the Theorem is therefore proved .


Friday, 3 January 2014

Intervals in R

 

At the time of analyze  the Real Number Set   R   it is essential to associates subsets of     with real numbers . This purpose is served by taking subsets of    in a spacial way as follows .

Open and Closed Intervals :-

An Open Interval of real number is  a subset of   defined as

 {x/  a<x<b}  i.e.  it is the set of all real numbers that are greater then   a   but less then    b    . This open interval is denoted by
                               (a ,b)   or  ]a, b[    .

Thus ;  (a, b) = {x/  a<x<b , x   is element of  R }

and ;   y   is element of (a, b) => a<y<b .

A closed interval of real number is subset of    defined as

{ x/  a  is less then an equal to   x   is less then and equal to  b }

i.e.  it is the set of all real numbers that are equal to or greater then   but equal to or less then    b   . This closed interval is denoted by    [a , b] . 
Thus;

[a, b]= {x/  a  is less then and equal to   x   is less then and equal to  b  ,
                                                               where   x   is element of   R}

and is element of  [a, b]=>  a  is less then and equal to   y  
                                     and   y  is less then and equal to   b .

Clearly open and closed intervals are bounded sets in   R   . While in case of     (a, b)   there is no maximum or  minimum  element , in case of      [a, b]     the maximum element is   b   and  the minimum element is    .

Here is examples of open , closed ,semi open and semi closed intervals .


 \begin{align}
(a,b) = \mathopen{]}a,b\mathclose{[} &= \{x\in\R\,|\,a<x<b\}, \\{}
[a,b) = \mathopen{[}a,b\mathclose{[} &= \{x\in\R\,|\,a\le x<b\}, \\{}
(a,b] = \mathopen{]}a,b\mathclose{]} &= \{x\in\R\,|\,a<x\le b\}, \\{}
[a,b] = \mathopen{[}a,b\mathclose{]} &= \{x\in\R\,|\,a\le x\le b\}.
\end{align}


Thursday, 2 January 2014

Differentiability Theorem




Theorem :-  A function   f    is differentiable  at     x=a     if and only if there exists a number   l    .
                   such that ;

                    f(a+h)-f(a)=lh+hn

                   Where     n    denotes a quantity which tends to    0     as   h-->0     .

Proof :-     Let    f     be differentiable at    x=a    . Then there exists a number   l    
                Such that ;

                lim x-->a  [f(x)-f(a)]/[x-a]=l

               putting ;     x=a+h ,

               lim h-->0  [f(a-h)-f(a)]/h=l

           or;    lim h-->0 [{f(a+h)-f(a)/h}-l]=0

              therefore   [{f(a+h)-f(a)}/h]-l     is equal to    n  

            where    n-->0    as     h-->0    

          Therefore f(a+h)-f(a)=lh+hn 

          where    n-->0    as    h-->0     .

          Thus it is the necessary condition .

As the argument is reversible , the condition is also sufficient .


Wednesday, 1 January 2014

Polar Co-ordinates

 

Beside the Cartesian , there are other systems also for  representing points and curve analytically . Polar system which is one of them .

---------In this system we started with a fixed line    OX   called the Initial Line and a fixed point on it , called the Pole .

---------If   P   be any given point, the distance   OP=r   is called the Radius Vector and    Angle XOP=theta    the Vectorial Angle . The two together are referred to as the Polar Co-ordinates of     .

Unrestricted Variation of Polar Co-ordinates :-

If we concerned with  assigning polar co-ordinates to only individual points in the plane , then it would clearly be enough to consider the radius vector to have positive values only and the vectorial angle    theta     to lie between    0 and 2pi   .

Transformation of Co-ordinates :-

Take the initial line   OX    of the polar system as the positive direction of   X-axis and the Pole   O    as origin for the Cartesian system .The positive direction of  Y-axis is to be such that the line   OX    after revolving through     Half Pi   is counter - clockwise direction comes to consider with it .

Polar Equation of Curves :-

Any explicit and implicit relation between   r    and   theta      will give a curve determined by the points whose co-ordinates satisfy that relation .

       ---------  Thus the Equations .

                    r=f(theta)       or       F(r,theta)    

                   determines  Curves .

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