Application

Introduction:

•  The Laplace transform is named in honor of mathematician and astronomer Pierre-Simon Laplace, who used the transform in his work on probability theory.

• Like the Fourier transform, the Laplace transform is used for solving differential and integral equations.

• Laplace transform is a widely used integral transform.

• Laplace transform is just a shortcut for complex calculations.

 Real Life Applications:

•    The Laplace transform turns a complicated nth order differential equation to a corresponding nth degree polynomial.
  
  
• In physics and engineering, it is used for analysis of linear time-invariant systems such as electrical circuits, harmonic oscillators, optical devices, and mechanical systems.

• The Laplace transform is one of the most important equations in digital signal processing and electronics.

•  In Nuclear physics, Laplace transform is used to get the correct form for radioactive decay.

• It has also been applied to the economic and managerial problems, and most recently, to Materials Requirement Planning (MRP) 

• The Laplace transform reduces a linear differential equation to an algebraic equation, which can then be solved by the formal rules of algebra

maths in nature

MATHS AND NATURE

"The laws of nature are but the mathematical thoughts of God"

                                                                                      - Euclid

Mathematics is everywhere in this universe. We seldom note it. We enjoy nature and are not interested in going deep about what mathematical idea is in it. Here are a very few properties of mathematics that are depicted  in nature.

SYMMETRY

Symmetry is everywhere you look in nature .

Symmetry is when a figure has two sides that are mirror images of one another. It would then be possible to draw a line through a picture of the object and along either side the image would look exactly the same. This line would be called a line of symmetry.

There are two kinds of symmetry.

One is bilateral symmetry in which an object has two sides that are mirror images of each other.

The human body would be an excellent example of a living being that has bilateral symmetry.




Few more pictures in nature showing bilateral symmetry.

The other kind of symmetry is radial symmetry. This is where there is a center point and numerous lines of symmetry could be drawn.

The most obvious geometric example would be a circle.



Few more pictures in nature showing radial symmetry.

SHAPES

Geometry is the branch of mathematics  that describes shapes.

Sphere:

A sphere  is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball.

The shape of the Earth is very close to that of an oblate spheroid, a sphere flattened along the axis from pole to pole such that there is a bulge around the equator.

Hexagons:

Hexagons are six-sided polygons, closed, 2-dimensional, many-sided figures with straight edges.

 
For a beehive, close packing is important to maximise the use of space. Hexagons fit most closely together without any gaps; so hexagonal wax cells are what bees create to store their eggs and larvae.

Cones:

A cone is a three-dimensional geometric shape that tapers smoothly from a flat, usually circular base to a point called the apex or vertex.

Volcanoes form cones, the steepness and height of which depends on the runniness (viscosity) of the lava. Fast, runny lava forms flatter cones; thick, viscous lava forms steep-sided cones.

Few more cones in nature:

Parallel lines:

In mathematics, parallel lines stretch to infinity, neither converging nor diverging.

These parallel dunes in the Australian desert aren't perfect - the physical world rarely is.

Fibonacci spiral:

If you construct a series of squares with lengths equal to the Fibonacci numbers (1,1,2,3,5, etc) and trace a line through the diagonals of each square, it forms a Fibonacci spiral.

Many examples of the Fibonacci spiral can be seen in nature, including in the chambers of a nautilus shell.

 
 
 
 
 
 
 
 
 
 
 
 

SOMU

Locus
The set of all points that share a property. This usually results in a curve or surface. Example: A Circle is "the locus of points on a plane that are a certain distance from a central point". As shown below, just a few points start to look like a circle, but if you collect ALL the points, you will actually have a circle.

Basic

What is an Equation

An equation says that two things are equal. It will have an equals sign "=" like this:

x

+

2

=

6

That equations says: what is on the left (x + 2) is equal to what is on the right (6)
So an equation is like a statement "this equals that"

Parts of an Equation

So people can talk about equations, there are names for different parts (better than saying "that thingy there"!)
Here we have an equation that says 4x-7 equals 5, and all its parts:

	A Variable is a symbol for a number we don't know yet. It is usually a letter like x or y. A number on its own is called a Constant. A Coefficient is a number used to multiply a variable (4x means 4 times x, so 4 is a coefficient) An Operator is a symbol (such as +, ×, etc) that represents an operation (ie you want to do something with the values).
	A Term is either a single number or a variable, or numbers and variables multiplied together. An Expression is a group of terms (the terms are separated by + or - signs)

So, now we can say things like "that expression has only two terms", or "the second term is a constant", or even "are you sure the coefficient is really 4?"

Exponents

The exponent (such as the 2 in x2) says how many times to use the value in a multiplication. Examples: 82 = 8 × 8 = 64 y3 = y × y × y y2z = y × y × z

Exponents make it easier to write and use many multiplications

Example: y⁴z² is easier than y × y × y × y × z × z, or even yyyyzz

Polynomial

Example of a Polynomial: 3x² + x - 2

A polynomial can have constants, variables and the exponents 0,1,2,3,...
And they can be combined using addition, subtraction and multiplication, ... but not division!

Monomial, Binomial, Trinomial

There are special names for polynomials with 1, 2 or 3 terms:

Like Terms

Like Terms are terms whose variables (and their exponents such as the 2 in x²) are the same.
In other words, terms that are "like" each other. (Note: the coefficients can be different)

Example:

(1/3)xy2

-2xy2

6xy2

Are all like terms because the variables are all xy²

Polynomial

Definition of a polynomial

Before giving you the definition of a polynomial, it is important to provide the definition of a monomial

Definition of a monomial:

A monomial is a variable, a real number, or a multiplication of one or more variables and a real number with whole-number exponents

Examples of monomials and non-monomials

Monomials	9	x	9x	6xy	0.60x4y
Not monomials	y - 6	x-1 or 1/x	√(x) or x1/2	6 + x	a/x

Polynomial definition:

A polynomial is a monomial or the sum or difference of monomials. Each monomial is called a term of the poynomial

Important!:Terms are seperated by addition signs and subtraction signs, but never by multiplication signs

A polynomial with one term is called a monomial

A polynomial with two terms is called a binomial

A polynomial with three terms is called a trinomial


Examples of polynomials:

Polynomial	Number of terms	Some examples
Monomial	1	2, x, 5x3
Binomial	2	2x + 5, x2 - x, x - 5
Trinomial	3	x2 + 5x + 6, x5 - 3x + 8

Difference between a monomial and a polynomial:

A polynomial may have more than one variable.

For example, x + y and x² + 5y + 6 are still polynomials although they have two different variables x and y

By the same token, a monomial can have more than one variable. For example, 2 × x × y × z is a monomial



Exercices

For all expressions below, look for all expressions that are polynomials.

For those that are polynomials, state whether the polynomial is a monomial, a binomial, or a trinomial

1) 3.4 + 3.4x

2) z² + 5z^-1 + 6

3) -8

4) 2c² + 5b + 6

5) 14 + x

6) 5x - 2^-1

7) 4 b² - 2 b^-2

8) f² + 5f + 6

Answer: 1), 3), 4), 5), 6), and 8) are polynomials. 1), 5), and 6) are binomials. 3) is a monomial. 4) and 8) are trinomials

2) and 7) are not because they have negative exponents

Notice that 6) is still a polynomial although it has a negative exponent. It is because it is the exponent of a real number, not a variable

In fact, 5x - 2^-1 = 5x + 1/2 = 5x + 0.5

It is subtle, but if you have any questions about the definition of a polynomial, feel free to contact me