Advanced Mathematical Research Author Md Tauseef Ibrahim/Abraham Malik

Dealing in the area of Mathematics,Algorithms,Physics, chemistry,Bio chemistry, Biology.Providing multiple solutions for one problem with the help of domain of mathematics like Algebra,Vector Calculus,Calculus,Coordinate Geometry,Trigonometry etc .Providing the basics and Advanced concept and creating Alternative solution paths for problems with the help of different domain of mathematics.Thus providing different views and interpretation for same phenomenon.

Saturday, January 24, 2015

Domain of a function and its analysis.

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Calculus :It is the study of rate of change of a function with respect to variation of domain of a function.

Since function can always be drawn in x-y Cartesian plane,the study of function reduces to the study of curves in the x-y Cartesian plane and hence calculus comes into picture to study the rate of change of range with respect to domain.First we will analyses some function and use calculus to study it.

GRAPHS OF FUNCTIONS
If f is a real-valued function of a real variable, then the graph of f in the xy-plane is
defined to be the graph of the equation y = f(x). For example, the graph of the function
f(x) = x is the graph of the equation y = x, shown in Figure 0.1.4. That figure also shows
the graphs of some other basic functions that may already be familiar to you.

Graphs can provide valuable visual information about a function. For example, since
the graph of a function f in the xy-plane is the graph of the equation y = f(x), the points
on the graph of f are of the form (x, f(x)); that is, the y-coordinate of a point on the graph
of f is the value of f at the corresponding x-coordinate

The y-coordinate of a point on the graph of y = f(x) is the value of f at the corresponding x-coordinate.

The values of x for which f(x) = 0 are the x-coordinates of the points where the graph of f intersects the
x-axis (Figure 0.1.6). These values are called the zeros of f , the roots of f(x) = 0, or the x-intercepts of the graph of y = f(x).

Zeros of a function is the domain where the function intersect the x axis.

So looking at the above figure we can say x=x1, x=0,x=x2 and x=x3 are the zeros of the function means

f(x1)=f(0)=f(x2)=f(x3)=0.

and x1,0,x2,x3 are called solution of the equation y=f(x)=0.

The graph of the equation x2 + y2 = 25 is a circle of radius 5 centered at the origin which is not a function because a function for unique domain should have unique range but here for one x, y1=y2=y.

THE ABSOLUTE VALUE FUNCTION
Recall that the absolute value or magnitude of a real number x is defined by

|x| = x, x ≥

−x, x < 0 , its corresponding graph is as shown in the following figure.

many to one function and symmetrical about x axis which property is called even function ie

f(x)=f(-x) , even function.

Another example of even function is cosine function which is symmetrical about y-axis such that f(x)=f(-x).

Example:Sketch the graph of the function defined piecewise by the formula

The formula for f changes at the points x = −1 and x = 1. (We call these the breakpoints for the formula.) Agood procedure for graphing functions defined piecewise is to graph the function separately over the open intervals determined by the breakpoints, and then graph f at the breakpoints themselves. For the function f in this example the graph is the horizontal ray y = 0 on the interval (−infinity ,−1], it is the semicircle y = √1 − x2 on the interval (−1, 1), and it is the ray y = x on the interval [1,+infinity ). The formula for f specifies that the equation y = 0 applies at the breakpoint −1 [so y = f(−1) = 0], and it specifies that the equation y = x applies at the breakpoint 1 [so y = f(1) = 1]. The graph of f is shown in

the function is discontinuous at x=1.

DOMAIN AND RANGE

Find the domain of this function.

By observing the graph of arccos(x) function where -1 <=x<=1.

Slope of a function.

https://www.youtube.com/watch?v=dL5lBxZ68WE
https://www.youtube.com/watch?v=StCpKfsetLM

This is an example of increasing function whose slope =Tan(theta)>0.so we call it an increasing function.

Sunday, January 18, 2015

Bernoulli’s equation P+KE+U=constant ,continuity equation for incompressible fluid, A1v1 = A2v2 .

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We are going to analyses the flow of an incompressible,non-viscous and non-turbulent ie streamed lined flow of fluid.

The Continuity Equation

The mass of a moving fluid doesn’t change as it flows. This leads to an important quantitative relationship called the continuity equation. Consider a portion of a flow tube between two stationary cross sections with areas A1 and A2.The fluid speeds at these sections are v1 and v2.respectively.During a small time interval dt, the fluid at A1moves a distance v1dt,so a cylinder of fluid with height v1dt and volume dV1 = A1v1dt flows into the tube across A1.During this same interval, a cylinder of volume dV2 = A2v2dt flows out of the tube across A2.Let’s first consider the case of an incompressible fluid so that the density has the same value at all points.The mass dm1 flowing into the tube across A1in time dt is dm1 = rA1v1 dt.Similarly, the mass dm2 that flows out across A2 in the same time is dm2 = rA2v2 dt. In steady flow the total mass in the tube is constant so
So dm1=dm2.   and

                                                              rA1v1 dt = rA2v2 dt    or

                                                                    A1v1 = A2v2    (continuity equation, incompressible fluid)
The figure illustrating the above derivation

Deriving Bernoulli’s Equation

To derive Bernoulli’s equation, we apply the work–energy theorem to the fluid in a section of a flow tube.And the figure that will assist us in deriving this relation is as following:

Let’s compute the work done on this fluid element during dt. We assume that there is negligible internal friction in the fluid (i.e., no viscosity), so the only nongravitational forces that do work on the fluid element are due to the pressure of the surrounding fluid. The pressures at the two ends are p1 and p2.the force on cross section at a is p1A1 and the force at c is p2A2.The net work dW done on the element by the surrounding fluid during this displacement is therefore

The work dW is due to forces other than the conservative force of gravity, so it equals the change in the total mechanical energy (kinetic energy plus gravitational potential energy) associated with the fluid element. The mechanical energy for the fluid between sections b and c does not change.At the beginning of dt the fluid between a and b has volume A1 ds1, mass rA1 ds1 and kinetic energy