Thursday, 26 December 2013

Motion Of A Particle

 

Principles in  the formation of the Equation of Motion of a Particle :-  

For Motion in Straight Line ;

Let the mass of the body (particle) be    m    and let the distance of the particle measured from a suitable origin be   x   at the time  t   .

Then acceleration is      (d/dt)(dx/dt)

therefore by Second Law of Motion

m[(d/dt)(dx/dt)] = Forces in the direction of    x   increasing

The forces usually are ;-

1)---   g    vertically downwards in a gravitational field like earth .
2)---  tension in a string
3)---  reactions or stress at point where the point may be in contact with other particles .
4)---  forces of attraction 
5)---  forces of resistance to motion by any means may be by atmosphere , friction, winds or by any other means .

In  taking the forces we must fix the sign properly if the forces acts along the line .

If a force  F   acts at an angle    A     to the straight line , then the resolved part of the force along the line is    FcosA  . If   acts on on a particle and if the particle is moving along the horizontal line , then   g    , being vertical , has no resolved part horizontally  and    does not effects the Motion .

Also note that velocity,    v=dx/dt     , and acceleration

     (d/dt)(dx/dt) = dv/dt = (dx/dt)(dv/dx) = v(dv/dx)  

For Motions in two dimension;

Let two perpendicular axes are   X      and    Y     at the plane of motion . The forces are resolved in these two directions and respectively

   m(d/dt)(dx/dt)       and       m(d/dt)(dy/dt)


Monday, 23 December 2013

Equations of First Order

 

In this post we will discuss about the equation of first order but not first degree .It is usually denoted
      
                dy/dx    by   p .

Thee are three types of such equations

1) Equations solvable for .
2)Equation solvable for    y  .
3)Equation solvable for    .

Equation Solvable for   p   :- 

examples like this    p.p +2py  cot x = y.y   and its solution is

[y-(c/1+cos x)][y-(c/1-cos x)] = 0

is the equation solvable for   p   .

Equation Solvable for    y   :-

Let in the given differential equation , on solving for   y   , given that ;

                    y=f(x,p)  ---------------(1)

Differentiating with respect to    , we obtain ;

              p=dy/dx=A(x,p,dp/dx)

so that we obtain a new differential equation with variables     and     .

Suppose that it is possible to solve the equation

Let the solution be
                            F(x,p,c)=0  ----------(2)
                          where     is the arbitrary constant .

The equation of  (1)  may be exhibited in either of the two forms . We may either eliminate   p   between  (1)  and  (2)  and obtain  A(x,y,c)   as the required solution or we may solve   (1)  and  (2)  for  x , y   and obtain .

                   x=f'(p,c)   and   y=f"(p,c)

as required solution where   p  is the parameter .

Equations Solvable for    x   :-

Let the given differential equation , on solving for  x   , gives

                                 x=f(p,y)  -----------------------(1)

differentiating with respect to   y  we obtain

                           1/p=dy/dx=A(y,p,dp/dx)  ; say

So that we obtain a new differential equation in variables    y      and   p   , Suppose that it is possible to solve the equation .

Let the solution be
                              F(p,y,c)=0   -------------------(2)

After the elimination    p   between    (1)   and   (2)     will give the solution .  Express   x  and    in terms    of    and   c   where    p   is to be regarded as parameter .


Equations of Tangent and Normal

 

Explicit Cartesian Equations :-

If      be the angle which the tangent at any point   (x, y)   on the curve     y = f (x)     makes with    x   axis then ;
                tan A = dy/dx = f' (x) 

Therefore , the equation of the tangent at any point     (x , y)     on the curve    y = f (x)      is

              Y - y = f' (x) (X - x)  -------------(1)

where   X , Y   are the current co-ordinates of any point on the tangent .

The normal to the curve    y = f (x)    at any point    (x , y)     is the straight line which  passes through that point ans is perpendicular to the tangent to the curve at the point so that its slope is ;

             -1/f (x)

Hence the equation of the normal at   (x , y)    to the curve    y= f (x)    is ;

            (X - x) + f' (x) (Y - y) = 0 

Implicit Cartesian Equations :-

If any point    (x , y) , then the curve   f (x, y) = 0

          Where   Dy/Dx    is not equivalent to  0   .

           dy/dx =  - (Df/Dx) / (Df/Dy)

Hence the equations of the tangent and the normal at any point
(x , y)   on the curve    f (x , y) = 0  are ;

         (X - x)(Df / Dx) + (Y - y) (Df / Dy) = 0      and 

        (X - x) (Df / Dy) - (Y - y)(Df / Dx) = 0

Parametric Cartesian Equations  :- 

At the pont  of the curve    x = f (t) , y = F(t)  ;
       where we have  f'(t)   is not equivalent to   0    ;
we have ;

    dy/dx = (dy/dt) (dt/dx) = F' (t)/f' (t)  

Hence the equations of the tangents and the normal at any point    t    of the curve    x=f(t) , y=F(t)   are ;

   [X-f(t)]F'(t)-[Y-F(t)]f'(t)=0
   [X-f(t)]f'(t)+[Y-F(t)]F'(t)=0

respectively .

Sunday, 22 December 2013

Maxima and Minima

 

 Greatest  and Least Value :-

In this section we shall be concerned with the application of Calculus to determining the values of a function which are greatest or least in their immediate neighborhood technically known as  Maximum and Minimum Values .

It will be assumed that    f (x)   possesses continuous derivatives of every order that come in equation .

Maximum Value of a Function :-

Let     be any interior points of the interval of definition of a function   f (x)    , if it is the greatest of all its value for values of   x   lying in some neighborhood of   c   . To b more definite and to avoid the vague words   "Some Neighborhood" we say that    f (c)    is a maximum value of function if there exists some interval     ( c - D , c + D )       around     c   such that   .

                   f (c) > f (x) 

for all values of     other then      c   lying in the interval

So that ,      f (c)    is  maximum value of      f (x)     if .

f (c) > f ( c+ h )     i.e.       f ( c + h ) - f (c) < 0 

where value of    h   lying between   -D   and   D  .

Minimum Value of a Function :-

f (c)   is said to be a minimum value of    f (x)    , if it is the least of all its values for values of   x    lying in some neighborhood of     c    .

This is equivalent to saying that     f (c)    is a minimum value of    f (x)   ,if there exist a positive     D     such that ;

f (c) < f ( c + h ) ,   i.e.      f ( c + h ) - f (c) > 0

for value of    h    lying between    -D   and   .

For values of   h     sufficiently small in numerical value  .

The term Extreme Value is used both for a maximum as well as for a minimum value , so that    f (c)   is an extreme value if    f ( c+ h ) - f (c)     keeps on invariable sign for value of     h    sufficiently small numerically .

Friday, 20 December 2013

Velocity and Acceleration

 

Velocity and Acceleration of a Moving Particles :-

If a particle is moving along a straight line , and if at any instant     t     the position        of the particle be given by the distance   s    measured along the path from a suitable fixed point    on it , then      denoting the velocity and   f   the acceleration of the particle at the instant ,

     We have;

                  v = rate of displacement

                    = rate of change of   s   with respect to time

                    = ds/dt ;

and ,           f = rate of change of velocity with respect to time

                   = dv/dt

                   = ( d/dt ) ( ds/dt )

If instead of moving in a straight line , the particle be moving in any manner in a plane , the position of a particle at any instant   t   being given by the Cartesian Co-ordinates    x , y     referred to a fixed  set of axes the components of velocity and acceleration parallel to those axes will similarly be given by

       vx = rate of displacement parallel to    x-axis  = dx/dt

       vy = rate of displacement parallel to    y-axis   = dy/dt

       fx = rate of chang of     vx = ( d/dt ) ( dx/dt )

       fy = rate of change of    vy = ( d/dt ) ( dy/dt )


Thursday, 19 December 2013

Differential Equations

 

In this post we will discuss about "Formation of Ordinary Differential Equations"

Let ;     f ( x , y , c1 ) = 0  ---------------------(1)

be an equation containing   x ,  y  an  on arbitrary constant   c1   .

Differentiating  (1)  we get ;

                              ( Df/Dx ) + ( Df/Dx ) (dy/dx ) = 0 ------(2) 

equation  (2)  will in general contains   c1   . If   c1  be eliminated between  (1)  and  (2)  , we shall get a relation involving   x,  y  and   dy/dx  which will evidently be a differential equation  of the first order .

Similarly , if we have an equation

                             f ( x , y , c1 , c2 ) = 0 ----------------(3)

containing two arbitrary constant  c1  and  c2  , then by differentiating this twice , we shall get two equations . Now between these two equations and given equations , in all three equations , if the two arbitrary constant   c1  and   c2  be eliminated , we shall evidently get a differential equation  of the second order .

in general , if we have an equation ;

                         f ( x , y , c1 , c2 ,  .......cn ) = 0 ----------(4)

containing  n  arbitrary constants  c1 , c2 , .....cn      then by differentiating this   n  times , we shall get    equations . Now between these    n   equations  and the given equation in all   ( n+1 )   equations , if the   n   arbitrary constants   c1 , c2 , ...cn   be eliminated , we shall evidently get a differential equation for  n th    order , for there being    n   differentiations , the resulting equation must contain a derivative of the   n th order .



Homogeneous Equations

      
We have to understand about Homogeneous Equation in Calculus .

If  and  N  of the equation   Mdx + Ndy = 0   are both of the same degree in  x  and  y   , and are homogeneous , the equation is said to be homogeneous  . Such an equation can be put in the form

                                          dy/dx = f ( y/x )

Every homogeneous equation of the above type can be easily solved by putting     y = vx       
where   v  is a function of , and consequently

                                          dy/dx = v + x ( dv/dx )  

whereby it reduced to the form     v + xdv/dx = f ( v ) 

                                      i.e.        dx/x = dv/ [ f(v) - v ] 

in which the variables are separated

A Special Form :-

The equation of the form

                         dy/dx = [ ( a1x +b1y +c1 ) / ( a2x + b2y +c2 ) ] 

                         where ,  [ a1/a2 is not equal to b1/b2 ] --------------(1)

can be easily solved by putting
                                               x = x' + h 
                           and              y = y' + k    
                           where   and are constant

So that ,              dx = dx'    and      dy = dy' 
                         and choosing   h  and  k  in such a way that


         a1h + b1k + c1 = 0    and    a2h + b2k + c2 =0        -------(2)

For now the equation reduces to the form

                      dy'/dx' = ( a1x' +b1y' ) / ( a2x' + b2y' )

which is homogeneous in   x'  and  y'  and hence solved .



Monday, 16 December 2013

Clairaut's Equation

  

An equation of the form
         
                         y = px + f (p) , where p = dy/dx

is called Clairaut's Equation .

Differentiating both sides of the equation with respect to   x   , we have  ,

p = p + xdp/dx + f' (p) dp/dx  or   dp/dx {x + f' (p)} = 0

therefore , either ,  dp/dx = 0 ------------ (1)

                 or ,     x + f' (p) =0 ------------(2)

                from (1) ,    p = C --------------(3)

Now if    p    be eliminated between  (3)  and the original equation , we get   y = Cx + f(C)  as the general or complete solution of the equation .

Again , if  p  be eliminated between  (2)  an the original equation , we shall obtain a relation between   x  and  y  which also satisfies the differential equation , and as such can be called a solution  of the given equation . Since this solution does not contain any arbitrary constant nor can it be derived from the complete solution by giving any particular value to the arbitrary constant , it is called the Singular Solution of the differential equation .

Thus we see that the Equation of Clairaut's form has two kinds of solution .

a)   The complete solution ( linear in  x  and  y  ) containing one arbitrary constant .

b)   The singular solution containing no arbitrary constant .

Now , to eliminate  p  between

                             y = px + f(p)  and   0 = x + f'(x)

is the same as to eliminate   between ,

                            y = Cx + f(C)   and   0 = x + f'(C) 

i.e. , the same as the process of finding the envelop of the line  y = Cx + f(C)  for different values of  .

Thus , the singular solution represents the envelope of the family of straight lines represented by the complete solution .



Thursday, 12 December 2013

Method of Isoclines

It is only in the limited number of cases that a differential equation may be solve analytically be the preceding methods , and in many practical cases where the solution of a differential equation is needed under given initial conditions and the above methods  fail , a graphical method , the method of isoclines is sometimes adopted . We proceed to explain below this method in case of simple differential equation of the first order .

Let us consider an equation of the type

dy/dx = f (x ,y ) ------------(1)

As already explained before , the general solution of this equation involves one arbitrary constant of integration . and hence represents a family of curves and in general , one member of the family passes through a given point  ( x , y ) .

Now if in  (1)  we replace   dy/dx  by   m   we get an equation   f( x , y) = m    , which for any particular numerical value of   m  represents a curve , at every point of which the value of  dy/dx  i.e.  the slope of the tangent line to the family of curves represented by the general solution of   (1)  is the same as the numerical value of   . This curve   f(x , y) = m   is called an Isoclinal or Isocline .

Which may be graphically constructed on a graph paper



   

Through different points on anyone isocline , short parallel lines are drawn having their common slope equal to the particular value of  m  for that Isocline .

Tuesday, 10 December 2013

Simpson's Rule

Approximate evaluation of a definite integral :
Simpson's Rule .

In many cases , a definite integral can not be obtained either because the quantity to be integrated can not be expressed as a mathematical function , or because the indefinite integral of the unction itself can not be determined directly . In such cases formula of approximation are used . One such important formula is Simpson's Rule . By this rule the definite integral of any function is expressed in terms of the individual values of any number of ordinates within the interval , by assuming that the function within each of the small ranges into which the whole interval may be divided can be represented to a sufficient degree of approximation by a parabolic function .
 


In numerical analysis, Simpson's rule is a method for numerical integration, the numerical approximation of definite integrals. Specifically, it is the following approximation:
 \int_{a}^{b} f(x) \, dx \approx \frac{b-a}{6}\left[f(a) + 4f\left(\frac{a+b}{2}\right)+f(b)\right].
Simpson's rule also corresponds to the three-point Newton-Cotes quadrature rule.



Suppose that the interval [a, b] is split up in n subintervals, with n an even number. Then, the composite Simpson's rule is given by
\int_a^b f(x) \, dx\approx 
\frac{h}{3}\bigg[f(x_0)+2\sum_{j=1}^{n/2-1}f(x_{2j})+
4\sum_{j=1}^{n/2}f(x_{2j-1})+f(x_n)
\bigg],
where x_j=a+jh for j=0, 1, ..., n-1, n with h=(b-a)/n; in particular, x_0=a and x_n=b. The above formula can also be written as
\int_a^b f(x) \, dx\approx\frac{h}{3}\bigg[f(x_0)+4f(x_1)+2f(x_2)+4f(x_3)+2f(x_4)+\cdots+4f(x_{n-1})].

The error committed by the composite Simpson's rule is bounded (in absolute value) by
\frac{h^4}{180}(b-a) \max_{\xi\in[a,b]} |f^{(4)}(\xi)|,
where h is the "step length", given by h=(b-a)/n.     

In other words this Simpson's Rule can be written as  :

h/3 [sum of the extreme ordinates  +  2.sum of the remaining odd ordinates   +  4.sum of the even ordinates] 





Saturday, 7 December 2013

Areas of Plane Curves

 

Areas in Cartesian Co - ordinate :-

Suppose we want to determine the area A1 bounded by the curve   y = f(x)   , the x - axis an two fixed ordinates   x = a   and   x = b    . The function   f(x)  , is supposed to be single - valued , finite and continuous in the interval  (a , b ) .

The process of finding the area , bounded by any defined contour line is called Quadrature , "the term meaning the investigation of the size of a square which shall have the same area as that  of the region under consideration" .


Here are the example of some Plain Curves :-


Name Implicit equation Parametric equation As a function graph
Straight line a x+b y=c (x_0 + \alpha t,y_0+\beta t) y=m x+c Gerade.svg
Circle x^2+y^2=r^2 (r \cos t, r \sin t)
framless
Parabola y-x^2=0 (t,t^2) y=x^2 Parabola.svg
Ellipse \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a \cos t, b \sin t)
framless
Hyperbola \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a \cosh t, b \sinh t)







Hyperbola.svg 

Thursday, 5 December 2013

Standared Methods of Integration

 

The different methods of Integration will aim at reducing a given Integral to one of the Fundamental or known Integrals . As a matter of facts , there are two principal processes :

1)   The method of substitution , i.e. the change of the independent variable .

Integration by substitution can be derived from the fundamental theorem of calculus as follows. Let ƒ and ϕ be two functions satisfying the above hypothesis that ƒ is continuous on I and ϕ is continuous on the closed interval [a,b]. Then the function ƒ(ϕ(t))ϕ(t) is also continuous on [a,b]. Hence the integrals

\int_{\phi(a)}^{\phi(b)} f(x)\,dx
and

\int_a^b f(\phi(t))\phi'(t)\,dt
in fact exist, and it remains to show that they are equal.
Since ƒ is continuous, it possesses an antiderivative F. The composite function Fϕ is then defined. Since F and ϕ are differentiable, the chain rule gives

(F \circ \phi)'(t) = F'(\phi(t))\phi'(t) = f(\phi(t))\phi'(t).
Applying the fundamental theorem of calculus twice gives

\begin{align}
\int_a^b f(\phi(t))\phi'(t)\,dt & {} = (F \circ \phi)(b) - (F \circ \phi)(a) \\
& {} = F(\phi(b)) - F(\phi(a)) \\
& {} = \int_{\phi(a)}^{\phi(b)} f(x)\,dx,
\end{align}
which is the substitution rule.

2) Integration by Parts ;


Integrating the product rule for three multiplied functions, u(x), v(x), w(x), gives a similar result:
\int_a^b u v \, dw = u v w - \int_a^b u w \, dv - \int_a^b v w \, du.
In general for n factors
\frac{d}{dx} \left(\prod_{i=1}^n u_i(x) \right)= \sum_{j=1}^n \prod_{i\neq j}^n u_i(x) \frac{du_j(x)}{dx},
which leads to
 \Bigl[ \prod_{i=1}^n u_i(x) \Bigr]_a^b = \sum_{j=1}^n \int_a^b \prod_{i\neq j}^n u_i(x) \, du_j(x),
where the product is of all functions except for the one differentiated in the same term.

It may be noted that classes of integrals which are reducible to one or other of the  fundamental forms by the above processes are very limited , and that the large majority of the expressions, under proper restrictions , can only be integrated by the aid of infinite series , and in some cases when the integrand involves expression under a radical sign containing powers of   x   beyond the second , the investigation of such integrals has necessitated the introduction of higher classes of transcendental function such as elliptic functions etc .

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