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## Linear Algebra

In math way Linear Algebra is a branch of mathematics in which we solve linear equation such as A_{1 }x_{1 }+ a_{2 }x_{2 }+ a_{3 }x_{3 }+ …………….. a_{n }x_{n} = b . In this type of equation solve in linear function, linear Algebra is fundamental of latest version of geometry of basic object such as lines planes and rotations. It is using in sciences fields and engineering fields.

### History of Linear algebra

In Math way the first searching linear systems by Leibniz in 1693. Then again Gabriel Cramer used for giving explicit solution in linear algebra in this time called Cramer rule. After some time Gauss further described method of elimination. Which is initially listed as geodesy.

In 1844 Hermann Grassmann published in his “Theory of extension” which is included as fundamental new topics which is called Linear Algebra. In 1848 James Joseph Sylvester introduced the term of matrix which is known as Latin for Womb.

Liner algebra grew with ideas noted in complex plane .Two numbers w and z in C have a difference w-z, and the line segments wz and 0(w-z) are of the same length and direction. The segments are equipollent . The four – dimensional system H of quaternions was started in 1843.

Arthur Cayley is gives matrix multiplication and inverse matrix in 1856 making possible the general linear group. The mechanism of group describing complex and hypercomplex numbers. He also realized the connection between matrices and determinants and wrote “there would be many things to say that theory of matrix.

### Algebra Formula

This formula is use in liner algebra for better understanding. all types of formula in linear algebra

- If a=b then a + c= b + c a + c = b + c for any c
- I.e a=b then a−c=b−c a − c = b − c for any c
- If a=b then ac=bc a c = b c for any c
- If a=b then ac=bc a c = b c for any non-zero c
- a² – b² = (a-b)(a+b)
- (a+b)² = a² + 2ab + b²
- (a-b)² = a² – 2ab + b²
- a² + b² = (a-b)² +2ab
- (a+b+c)² = a²+b²+c²+2ab+2ac+2bc
- (a-b-c)² = a²+b²+c²-2ab-2ac+2bc
- a³-b³ = (a-b) (a² + ab + b²)
- a³+b³ = (a+b) (a² – ab + b²)
- (a+b)³ = a³+ 3a²b + 3ab² + b³
- (a-b)³ = a³- 3a²b + 3ab² – b³
- “n” is a natural number, a
^{n}– b^{n}= (a-b) (a^{n-1}+ a^{n-2}b +….b^{n-2}a + b^{n-1}) - “n” is a even number, a
^{n}+ b^{n}= (a+b) (a^{n-1 }– a^{n-2}b +….+ b^{n-2}a – b^{n-1}) - “n” is an odd number a
^{n}+ b^{n}= (a-b) (a^{n-1}– a^{n-2}b +…. – b^{n-2}a + b^{n-1}) - (a
^{m})(a^{n}) = a^{m+n}(ab)^{m}= a^{mn}

### Trigonometric Function

Trigonometric function are real function which relates angle of right angle triangle is ratio of two side length. The trigonometric function mostly use in sine, cosine series and we are also use in reciprocal form.

sinx⋅cosy=sin(x+y)+sin(x−y)/2

cosx⋅cosy=cos(x+y)+cos(x−y)/2

sinx⋅siny=cos(x+y)−cos(x−y)/2

sinx+siny=2sin(x+y)/2 cos(x−y)/2

sinx−siny=2cos(x+y)/2 sin(x−y)/2

cosx+cosy=2cos(x+y)/2 cos(x−y)/2

cosx−cosy=−2sin(x+y)/2 sin(x−y)/2

sin 2x = 2 sin (x) cos(x)

tan2x=sin2x/(1−2sin^{2}x)

## Calculus

In calculus mathematics originally called infinitesimal calculus. In the same way calculus was developed in 17 century by Isaac Newton and Gottfried Wilhelm Leibniz.

Calculus uses in science and economics field generally. It has two major branches **differential calculus** and **integral calculus**.

### Differential calculus

In differential calculus. We can define as “the rates of at which quantity change”. The objects of study is known as **derivative of a function**. The derivative of a function describes as “the rate of change of function that input value. The process of finding derivative is called differentiation.

in geometry function, the derivative of point is in slope of the tangent line to the graph of function .The derivative of a function of a real variable is linear approximation at function of the point. It is use frequently to find **maxima** and **minima** of the function . Equation involving derivatives of function are called differential equation and fundamental is called **natural phenomena**.

**History of differential calculus**

The modern development of calculus is credited by Isaac Newton in 1643 to 1727 and **Gottfried Wilhelm Leibniz **further changes in 1646 to 1716 .Who provided independent and unified approaches to differentiation and integration.

For the ideas of derivative Newton and Leibniz built on significant earlier work on the concept. Isaac Borrow is credit goes to development of derivative. Newton and Leibniz further given history o differentiation. **Bernhard Riemann** and **Karl Weierstrass** in 1814 to 1898 was also generalize to **euclidean space **and **complex plane**.

**Application of differential function**

Differential function are used in different ways.

**Differential equation form** **of math way**

A differential function is a relation between collection of function and their derivative. An ordinary differential equation is differential equation relates that function of variable and their derivative. A partial differential equation is equation relates function of more than one variables in their partial derivative.

For Newton s second law, the relation between acceleration and force as a differential form

F(t) = m d^{2}x/dt^{2}

And the heat equation in one space variable describes as diffuse through a straight rod is in partial differential equation form.

**Mean value theorem** **in math way**

The mean value theorem gives a relationship between values of the derivative and values of the original function . If f(x) is real valued function and a and b are numbers with a<b, then the mean value theorem says that , the slope between the two points (a, f(a)) and (b, f(b)) is equal to the slope and tangent line to f at some point c between a and b.

In other words, f'(c) = (f(b)-f(a))/b-a

All those slopes are zero so an line form one point on the another graph to another point then slopes are zero.

**Implicit function theorem in math way**

The implicit function theorem converts relates as f(x, y)=0 into the function. it states that if f is continuously differentiable then around most points the zero set of f looks graph of function. The graphs of two function .In neighborhood of every point on the circle (-1,0)and (1,0) one of these two graph of the circle.

The implicit function is related as inverse function theorem ,which states that when a function on the graph its invertible function.

#### Formula of differentiation

- dr
^{2}/dx=nx (n−1) - d(fg)/dx=fg
^{1}+gf^{1} - d/dx(fg)= (gf
^{1}−fg^{1})/g^{2} - df(g(x))/dx=f
^{1}(g(x))g^{1}(x) - d(sinx)/dx=cosx
- d(cosx)/dx=−sinx
- d(tanx)/dx=−sec
^{2}x - d(cotx)/dx=cosec
^{2}x - d(secx)/dx=secx.tanx
- d(cosec x)/dx=−cosecx.cotx
- d(e
^{x})/dx=e^{x} - d(a
^{x})/dx=a^{x}.lna - d/dxlnx=1/x
- d(arc sinx)/dx=1/√1–x
^{2} - d(arc sinx)/dx=1/√1+x
^{2}

### Integral calculus

In The math way Integral calculus are the value of the function found the process of integration. The process in which we get f(x)form into f'(x) form called as integration. f(x) is called as anti derivative or Newtons Leibnitz integral or primitive function of an interval.

i. e **F'(x)=f(x)** for every value of x in i.

Integral is the representation of the area of region under a curve .a definite integral of a function represents area of region bounded by graphs on given function between two points on the line .

The integral of a function represents family of curve both derivative and integral in fundamental of integration.

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### Type of integral

Integral calculus are most of the two type.

- Definite integral

- Indefinite integral

**Definite integral**

The definite integral have a pre existing value of limit ,and making the final value of an integral. if f(x) is a function of the curve then.

** f(x)d(x)= f(b)-f(a)**

**Indefinite integral**

The Indefinite integral do not have a pre existing value of the limits making a final value of integral. if f(x) is a function of the curve then

** g'(x)dx=g(x)+c**

### Properties of integral calculus

1. The derivative of an integral is the integrand itself. **f(x)d(x)= f(x)+c**

2. Two indefinite integrals with the same derivative leads two the same family of curves and they are equivalent .

** ʃ [ f(x)d(x)-g(x)d(x)]=0**

3. The integral of the difference of finite numbers is equal to sum of difference of finite integral in individual function.

** ʃ [ f(x)d(x)+g(x)d(x)]= ʃ [ f(x)d(x)] + ʃ [g(x)d(x)]**

4. The constant integral is given by

** ʃ k f(x)d(x)= k ʃ f(x)d(x)**

5. The previous two properties is in the integral is given by

** ʃ **(x)d(x)=k_{1}** ʃ **f_{1}(x)d(x)+k_{2} **ʃ** f_{2}(x)d(x)+k_{3}** ʃ **f_{3}(x)d(x)……………………………..k_{n} ** ʃ** f_{n}(x)d(x)

#### Formula of integration

- ∫a dx=ax+C
- ∫x
^{n}dx = x^{n-1}+ C - ∫1/x dx=ln|x|+C
- ∫a
^{x}dx=a^{x}/lna+C - ∫lnx dx=xlnx−x+C
- ∫sinx dx=−cosx+C
- ∫cosx dx=sinx+C
- ∫tanx dx+ln|secx|+C=−ln|cosx|+C
- ∫cotx dx=ln|sinx|+C
- ∫secx dx=ln|secx+tanx|+C
- ∫cosecx dx=ln|cosecx–cotx|+C
- ∫sec
^{2}x dx=tanx+C - ∫secx.tanx dx=secx+C
- ∫cosec
^{2}x dx=−cotx+C - ∫tan
^{2}x dx=tanx−x+C - ∫frac dx √a
^{2}−x^{2}= arc sinx/a+C - ∫frac dx √a
^{2}+x^{2}=1/a arc sinx/a+C

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