Fourier Transform from Fourier Series

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see

Definite Integral of Multiplied 2 Trigonometric Functions

Taylor Series

Express Functions from Taylor Series

We can think f(t) is sum of wave functions and a constant.

if t=0, f(0) = a_0.

Then, let's find a_0.

We can find a_0 like process above.

Through

Definite Integral of Multiplied 2 Trigonometric Functions,

we can find a_n ( except n = 0 ) and b_n.

Use

Express Functions from Taylor Series,

we can develop the formula.

In the process, we can derive Fourier Transform from Fourier Series.

Definite Integral of Multiplied 2 Trigonometric Functions

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see

Integral Some Functions

Synthesis of Trigonometric Functions

Integral Some Functions

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see

e ( mathematical constant )

Differential of A Function : sin x, cos x

Relationship between Integral and Differential

Differential of A Function : sin x, cos x

Before you see this post,

see

A Property of Limit of A Function : f(x)/g(x)

A Property of Limit of A Function : f(x)g(x)

Synthesis of Trigonometric Functions

Relationship between Integral and Differential

Before you see this post,

see

Riemann Sum and Riemann Integral

and

Error Estimate of Riemann Sum

f(q) can be integrated by Riemann integral and is a continuous function

and

F(x) can be differentiated.

Integral is an inverse operation of Differential.

Error Estimate of Riemann Sum

Before you see this post,

see

Riemann Sum and Riemann Integral

f(x) can be integrated by Riemann integral.

If you want to know more,

see this report.

Riemann Sum and Riemann Integral

If the limit of Riemann sum is convergence,

f(x) can be integrated by Riemann integral.

If f(x) can be integrated by Riemann integral,

some properties in picture below are valid.

If you want to know error of Riemann Sum

see

Error Estimate of Riemann Sum

Differential Function : [f(x)]^r = r[f(x)]^(r-1)

Before you see this post,

see

Differential Function : ln x

and

Differential of A Function : f(x)g(x)

q isn't zero.

Differential of A Function : f(x) / g(x)

Before you see this post,

see

A Property of Limit of A Function : f(x)g(x)

f(x) and g(x) are continuous functions.

Differential of A Function : f(x)g(x)

Before you see this post,

see

A Property of Limit of A Function : f(x)g(x)

f(x) and g(x) are a continuous functions

Differential of A Function : f(x)+g(x)

Before you see this post,

see

A Property of Limit of A Function : f(x)+g(x)

f(x) and g(x) are a continuous functions.

A Property of Limit of A Function : cf(x)

f(x) is a continuous function.

A Property of Limit of A Function : constant

Uniqueness of Limit of A Function

f(x) is a continuous function

Derivative

Total differential

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see

Derivative

A Property of Limit of A Function : f(x)g(x)

[pic 1]

Δf can be notated like [pic 1].

[pic 2]

We can continuously notate the function like [pic 2].

[pic 3]

Δf can be notated like [pic 3].

[pic 4]

Limit of Δf can be written to df and notated like [pic 4].

[pic 5]

[pic 6]

Let's use definition of partial differential in [pic 4]

and

a property of limit of a function like [pic 6].

See

A Property of Limit of A Function : f(x)g(x)

[pic 7]

We use definition in [pic 5] and a property in [pic 6] at a formula in [pic 4],

we can make a formula in [pic 7].

That is the formula of total differential.

Green's Theorem

1. Setting

[pic 1]

Let a closed curve C be set on x,y plane.

Domain D is covered by closed curve C.

[pic 2]

Let's functions, M(x,y) and N(x,y) exist and they have partial derivatives on D.

The formula can be notated like it in [pic 2] on C.

[pic 3]

This formula can be divided like [pic 3] 

because M(x,y)dx and N(x,y)dy are on same closed curve C. 

2. Induction

2-1. Considering The Formula at X Coordinate

First, we consider the formula in [pic 2] at x coordinate.

[pic 4]

[pic 5]

We set a domain like a formula in [pic 5].

[pic 6]

So, result is like [pic 6] on x coordinate.

2-2. Considering The Formula at Y Coordinate

Next, we also consider the formula in [pic 2] at y coordinate.

[pic 7]

[pic 8]

We set a domain like a formula in [pic 8].

[pic 9]

So, result is like [pic 6] on y coordinate.

3. Conclusion

[pic 10]

We can combine same domain.

Therefore, we can prove Green's Theorem.