Beauty of Mathematics

The Laplace transform is a simple way of converting functions in one domain to functions of another domain.

Here's an example :

Suppose we have a function of time, such as cos(w*t). With the Laplace transform, we can convert this to a function of frequency, which is

cos(w*t) ----L{}-----> w / (s^2 + w^2)

This is useful for a very simple reason: it makes solving differential equations much easier.

The development of the logarithm was considered the most important development in studying astronomy. In much the same way, the Laplace transform makes it much easier to solve differential equations.

Since the Laplace transform of a derivative becomes a multiple of the domain variable, the Laplace transform turns a complicated n-th order differential equation to a corresponding nth degree polynomial. Since polynomials are much easier to solve, we would rather deal with them. This occurs all the time.

In brief, the Laplace transform is really just a shortcut for complex calculations. It may seem troublesome, but it bypasses some of the most difficult mathematics.

Laplace transform is a technique mainly utilized in engineering purposes for system modeling in which a large differential equation must be solved.

The Laplace transform can also be used to solve differential equations and is used extensively in electrical engineering.

Laplace Transform is used in electrical circuits for the analysis of linear time-invariant systems

Conformal Mapping

Geometric interpretation of a complex function.

If D is the domain of real-valued functions and u(x,y) and v(x,y) then the system of equations u = u(x,y) and v = v(x,y) describes a transformation (or mapping) from the x y - plane into the u v -plane, also called the w-plane.

Therefore, we consider the function w= f(z) = u(x,y) + i v (x,y)

to be a transformation (or mapping) from the set D in the z-plane onto the range R in the w-plane.

Conformal Mapping:

A function f: C → C is conformal at a point z₀ if and only if it is holomorphic and its derivative is everywhere non-zero on C.

i.e., if f is analytic at z₀ and f’(z₀) ≠ 0

Isogonal Mapping:

An isogonal mapping is a transformation w = f (z) that preserves the magnitudes of local angles, but not their orientation.

Standard Transformations:

• Translation

- Maps of the form z → z + k, where k є C

• Magnification and rotation

- Maps of the form z → k z , where k є C

• Inversion

- Maps of the form z → 1 / z

CIRCLE AND ITS RELATED TERMS

Circle is the set of all points in a plane that are equidistant from a given point in the plane known as the center of the circle.

Chord - is a segment whose endpoints lie on a circle

Secant - is a line that intersects the circle at two points

Diameter is the longest secant in a circle.

Tangent - is a line in the plane of the circle that intersects the circle at exactly one point

Sector -The part of a circle enclosed by two radii of a circle and an arc.

The cut piece of pizza is minor sector and the remaining is the major sector.

Segment -part of a circle bounded by a chord and an arc

Applications of Trigonometry in Real life

Trigonometry is commonly used in finding the height of towers and mountains.

It is used in navigation to find the distance of the shore from a point in the sea.

It is used in oceanography in calculating the height of tides in oceans

It is used in finding the distance between celestial bodies

The sine and cosine functions are fundamental to the theory of periodic functions such as those that describe sound and light waves.

Architects use trigonometry to calculate structural load, roof slopes, ground surfaces and many other aspects, including sun shading and light angles