Inverse Matrix

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see

Identity Matrix

Determinant of Matrix

Identity Matrix

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see

Determinant of Matrix

Determinant of Matrix

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see

Permutation, Symmetric Group and Signature

Cyclic Permutation

Matrix Operation

Cyclic Permutation

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see

Permutation, Symmetric Group and Signature

Let's calculate cyclic permutations.

Permutation, Symmetric Group and Signature

Variance and Standard Deviation

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see

Possibility and Mean

If we find variance like upper picture,

variance has no meaning.

So, variance is calculated like bottom picture.

Standard deviation is a positive square root of variance. 

Possibility and Mean

Example

Differential of A Function : sin x, cos x

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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

Synthesis of Trigonometric Functions

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see

Distance between A Point and A line

and

Sum and Products by Unit Vector

Distance between A Point and A line

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see

A Property of Perpendicular Lines

correction in picture : parpendicular -> perpendicular

A Property of Perpendicular Lines

Relationship between Integral and Differential

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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.

Complex Number

Magnitude of a complex number z

 $\sqrt { z \bar{z}} = \sqrt { a^2 + b^2 }$

 = a-bi

 is a complex conjugate of z.

Real Number

National Number ⊂ Integer ⊂ Rational Number ⊂ Real Number

Irrational Number ⊂ Real Number

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