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## Area of mathematics

In mathematics field many area are like that, linear algebra, abstract algebra, mechanics, partial differential equation, general relativity and cosmology and many types branch in existence in this field.

### Mechanics

In math ,it is part of science in which , A motion of body under the action of forces in which a body remains constant. This leads study of such topics as gravity, electricity, and magnetism according to the nature of the forces involved. The manner in which bodies move under the action of forces.

Mechanics divides in to three parts. In the first parts in which we study force acting on a body at rest this represents **statics**. Again in the second parts in which we can study the possible motion of a body or system of a body which is known as **kinematics**. and in the third parts in which we have study prediction of the motion this represents **kinetics **.

The principal of applied mechanics as three phenomenon given as below.

The motion of a body as a stars , planets and satellites can be prediction with great accuracy thousands of year.

### Abstract algebra

Abstract algebra in math may be defined as” A branch of mathematics in which we study the algebraic function . In algebraic function in which states that Groups, Rings, Fields, modules, Vector space, lattice and algebra.

Abstract algebra Algebraic structures use many area in mathematics like that set theory. In set theory , groups (it is follow all property like closure, associative, identity and inverse) and semi groups(it follow only two property of group). and monoid are holds in.

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#### Groups

A group is holds all property four property given below.

**Closure property**

The product of any two integers will be also an integers i. e if p and q are two integers then p q will also an integers,

for example. p=5 an integer

q=7 an integer

p*q=5*7=35 also an integer

2. **Associative property**

A property in which products of three integers also takes place an a integers

for example l=2 an integer

m=3 an integer

n=1 an integer

l*(m*n)=(l*m)*n

2*(3*1)=(2*3)*1=6 also an integer

3. **Identity element**

a property in which a integers multiplication into 1 then results is also an integers

for example p=6 an integer

p*1=p

6*1=6 also as integers

4.** Inverse element**

a property in which two integers are equal one

for example a= 4 an integers

a a^{-1} =1

4 4^{-1}=1

#### Semi group

If S is a non empty set and * (multiplication) be a binary operation on S then the algebraic function {s, *} be a semi group.

i.e if for any a, b, c € S

(a * b) * c = a * (b * c)

a=1 an integer

b=4 an integer

c=3 an integer

(a * b )* c= a* ( b * c )

(1 * 4 )* 3= 1*(4 * 3 )=12 also an integer.

### General relativity and cosmology in math

The general relativity generalized as special relativity and refines and Newton’s law o gravitation which is unified description of gravity . A geometric property of space and time of four dimensional spacetime . the curvature of spacetime is directly related to the energy and momentum of whatever matter and radiation are present . the relation specified by the Einstein field equations a system of partial differential equations.

The geometry of space the motion of bodies in free fall and the propagation of light. for example gravitational time dilation , gravitational lensing the gravitational redshift of light. General relativity may be understood as according to Newton ‘s second law of motion, which is states that , the net force acting on a body is equal to that body mass multiplied by its acceleration. The preferred inertial motion ‘s are related to the geometry space and time .

## Conclusion

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