Friday, July 31, 2020

Mathematical Symbols - Calculus and Analysis

Mathematical Symbols are very important to learn mathematics, on using symbols we can easily understand the concepts of topics. 

Generally, Mathematical Symbols can be categorized on their importance in the following ways:

  1. Mathematics symbols - Number
  2. Mathematics symbols Basic
  3. Mathematics symbols - Algebraic
  4. Mathematics symbols - Geometric
  5. Mathematics symbols - Trigonometry
  6. Mathematics symbols - Set Theory
  7. Mathematics symbols - Statistics & Probability
  8. Mathematics symbols - Calculus And Analysis
  9. Mathematics symbols - Logical
  10. Mathematics symbols - Roman
  11. Mathematics symbols Greek
  12. Mathematics symbols Miscellaneous
Now we shall learn /discuss the eighth topic of Mathematical symbols Calculus and Analysis on this post.

Mathematics symbols  

 These are very important Basic Tools frequently used by us not only in Calculus and Analysis but also in Physics,  Chemistry, Economics as well as in other branches of applied mathematical sciences, humanities, and engineering.

Table of important Mathematical Symbol of  Calculus and Analysis

Symbol
Name of Symbol 
Definition / Meaning
Illustration
(a, b)
open interval
(a, b) = {x: a < x < b} or
(a, b) = {x / a < x < b}
Terminal / End points are not included (Excluded).
[a, b]
closed interval
[a, b] = {x: a ≤ x ≤ b} or
[a, b] = {x / a ≤ x ≤ b}
Terminal / End points are also included (Not Excluded).
Limit
Limit value of any function f
If value of function is not determined then calculated like, Evaluation of  f(2), when f(x) = 1/(x-2)


LHL
Left Hand Limit


RHL
Left Hand Limit

f(a) or g(x)

Value of function
Value of function f(x) or g(x) when x = a.

Epsilon
It represents very small number, near to zero.

e
Euler’s Number
It lies between 2 and 3,
e = 2.718281828…

dy/dx
Derivative
Derivative- Leibnitz’s Symbol
d(x4)/dx= 4x3
Dxy or D
Derivative
Derivative- Euler’s General Notation/Symbol
D(x4) = 4x3
Dx2y or D
Derivative
Derivative of Derivative- Euler’s Notation/Symbol
D2(x4) = 4.3x2= 12x2
Dx3y or D
Derivative
Three times Derivative - Euler’s Notation/Symbol
D3(x4) = 4.3.2x= 24x
Dxny or D
Derivative
n times Derivative - Euler’s Notation/Symbol
D’n(x4) = 0
Y’
Derivative
Derivative- Lagrange’s Symbol
(x4)’ = 4x3
Y’’
Second Derivative
Derivative of Derivative Lagrange’s Symbol
(x4)’’ = 4.3x2= 12x2
Y’’’
Third Derivative
Three times Derivative- Lagrange’s Symbol
(x4)’’’ = 4.3.2x= 24x
Y(n)
nth Derivative
n times Derivative- Lagrange’s Symbol
(x4)’n = 0
integral
Inverse process of differentiation
∫1. dx= x
∫∫
Double integral
Integration of two variable functions
∫∫f (x, y) dxdy
Or
 ∫∫f (x, y) dydx
∫∫∫
Triple integral
Integration of three variable functions
∫∫∫f (x, y, z) dxdydz

Closed contour integral / line integral
∮f(x)dx
 or
∮f(y)dy

Closed surface integral
f (x, y) dxdy
Or
  f (x, y) dydx

Closed volume integral
f (x, y, z) dxdydz
z
Complex number
A number having real or imaginary parts
Z = a + i b, where a & b both are Real Numbers
i
iota
Imaginary unit
i ≡ √-1
or
 i2 = -1
or
i3 = -i  
or
 i4 = 1
R(z) or Re(z)
Real part of complex number
If z = a + i b then R(z) = a
If z = 3 +i4 then
 R(z) = 3
I(z) or Im(z)
Imaginary part of complex number
If z = a +ib then  I(z) = a
If z = 3 +i4 then
 I(z) = 4
Z*
Complex conjugate
If z = a +ib then Z* = a - ib  
If z = 3 +i4 then
Z* = 3- i4  
Z bar
Complex conjugate
If z = a +ib then Z(bar) = a - ib  
If z = 3 +i4 then
Z* = 3- i4  

|z|
Magnitude of a complex number
If z = a +ib then |Z|= √(a2+b2)
If z = 3 +i4 then
 |3 +4i| = √25 = 5

|z|
Absolute value of a complex number
If z = a +ib then |Z|= √(a2+b2)
If z = 3 +i4 then
|3 +4i| = √25 = 5
Arg(z)
Argument value of a complex number
The angle made by the radius in the Argand or complex plane.
If z= 1 + i then
 arg(z)= pie /4

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