PROOF OF PROPERTIES OF LAPLACE TRANSFORM (part 1)

0) PROOF OF SOME ELEMENTARY FUNCTION

1) FIRST SHIFTING PROPERTIES 

2) LINEARITY PROPERTY

WHERE "a" IS A CONSTANT 

PROOF OF LINEARITY PROPERTY

3) MULTIPLY BY T  PROPERTY

4) DIVISION BY T PROPERTY

PROOF OF DIVISION BY T

5) LAPLACE TRANSFORM OF DERIVATIVE

Proof of laplace transfrom of first derivative

PROOF OF PROPERTIES OF LAPLACE TRANSFORM (part 2)

1) LAPLACE TRANSFORM OF INTEGRAL PROPERTY

PROOF

2) LAPLACE TRANSFORM UNIT STEP FUNCTION

L.T OF UNIT STEP FUNCTION

3) SECOND SHIFTING PROPERTY OF LAPLACE TRANSFORM

PROOF OF SECOND SHIFTING PROPERTY

EXAMPLE 1

EXAMPLE 2

EXAMPLE 3

EXAMPLE 4

matrices

Types of Matrices

Matrices are distinguished on the basis of their order, elements and certain other conditions. There are different types of matrices but the most commonly used are discussed below. Let’s find out the types of matrices in the field of mathematics.

Types of Matrices

Different types of Matrices and their forms are used for solving numerous problems. Some of them are as follows:

1) Row Matrix

A row matrix has only one row but any number of columns. A matrix is said to be a row matrix if it has only one row. For example,

A=[−1/2√523]

is a row matrix of order 1 × 4. In general, A = [aij]1 × n is a row matrix of order 1 × n.2) Column Matrix

A column matrix has only one column but any number of rows. A matrix is said to be a column matrix if it has only one column. For example,

A=⎡⎣⎢⎢⎢0√3−11/2⎤⎦⎥⎥⎥

is a column matrix of order 4 × 1. In general, B = [bij]m × 1 is a column matrix of order m × 1.3) Square Matrix

Types of Matrices

Different types of Matrices and their forms are used for solving numerous problems. Some of them are as follows:

1) Row Matrix

A row matrix has only one row but any number of columns. A matrix is said to be a row matrix if it has only one row. For example,

A=[−1/2√523]

is a row matrix of order 1 × 4. In general, A = [a_ij]_{1 × n} is a row matrix of order 1 × n.

2) Column Matrix

A column matrix has only one column but any number of rows. A matrix is said to be a column matrix if it has only one column. For example,

A=⎡⎣⎢⎢⎢0√3−11/2⎤⎦⎥⎥⎥

is a column matrix of order 4 × 1. In general, B = [b_ij]_{m × 1} is a column matrix of order m × 1.

3) Square Matrix

A square matrix has the number of columns equal to the number of rows. A matrix in which the number of rows is equal to the number of columns is said to be a square matrix. Thus an m × n matrix is said to be a square matrix if m = n and is known as a square matrix of order ‘n’. For example,

A=⎡⎣⎢33/24−1√3/2301−1⎤⎦⎥

 is a square matrix of order 3. In general, A = [a_ij] m × m is a square matrix of order m.

A square matrix has the number of columns equal to the number of rows. A matrix in which the number of rows is equal to the number of columns is said to be a square matrix. Thus an m × n matrix is said to be a square matrix if m = n and is known as a square matrix of order ‘n’. For example,

A=⎡⎣⎢33/24−1√3/2301−1⎤⎦⎥

 is a square matrix of order 3. In general, A = [aij] m × m is a square matrix of order m.

Differentiation

What is Differentiation in Maths

In Mathematics, Differentiation can be defined as a derivative of a function with respect to an independent variable. Differentiation, in calculus, can be applied to measure the function per unit change in the independent variable.

Let y = f(x) be a function of x. Then, the rate of change of “y” per unit change in “x” is given by:

dy / dx

If the function f(x) undergoes an infinitesimal change of ‘h’ near to any point ‘x’, then the derivative of the function is defined as

\(\begin{array}{l}\lim\limits_{h \to 0} \frac{f(x+h) – f(x)}{h}\end{array} \)

Derivative of Function As Limits

If we are given with real valued function (f) and x is a point in its domain of definition, then the derivative of function, f, is given by:

f'(a) = lim_h→0[f(x + h) – f(x)]/h

provided this limit exists.

Let us see an example here for better understanding.

Example: Find the derivative of f(x) = 2x, at x =3.

Solution: By using the above formulas, we can find,

f'(3) = lim_h→0 [f(3 + h) – f(3)]/h = lim_h→0[2(3 + h) – 2(3)]/h

f'(3) = lim_h→0 [6 + 2h – 6]/h

f'(3) = lim_h→0 2h/h

f'(3) = lim_h→0 2 = 2Notations

When a function is denoted as y = f(x), the derivative is indicated by the following notations.

D(y) or D[f(x)] is called Euler’s notation.

dy/dx is called Leibniz’s notation.

F’(x) is called Lagrange’s notation.

The meaning of differentiation is the process of determining the derivative of a function at any point.

Linear and Non-Linear Functions

Functions are generally classified into two categories under Calculus, namely:

(i) Linear functions

(ii) Non-linear functions

A linear function varies at a constant rate through its domain. Therefore, the overall rate of change of the function is the same as the rate of change of a function at any point.

However, the rate of change of function varies from point to point in the case of non-linear functions. The nature of variation is based on the nature of the function.

The rate of change of a function at a particular point is defined as a derivative of that particular function.

Differentiation Formulas

The important Differentiation formulas are given below in the table. Here, let us consider f(x) as a function and f'(x) is the derivative of the function.

If f(x) = tan (x), then f'(x) = sec2x

If f(x) = cos (x), then f'(x) = -sin x

If f(x) = sin (x), then f'(x) = cos x

If f(x) = ln(x), then f'(x) = 1/x

If f(x) = ex, then f'(x) = ex

If f(x) = xn, where n is any fraction or integer, then f'(x) = nxn-1

If f(x) = k, where k is a constant, then f'(x) = 0

Integration

Integration can be considered the reverse process of differentiation or called Inverse Differentiation. Integration is the process of finding a function with its derivative. Basic integration formulas on different functions are mentioned here. Apart from the basic integration formulas, classification of integral formulas and a few sample questions are also given here, which you can practice based on the integration formulas mentioned in this article. When we speak about integration by parts, it is about integrating the product of two functions, say y = uv. More integral calculus concepts are given, so keep learning integral formulas to solve problems accurately. Also, watch the video given below to clear your concept.

List of Integral Formulas

The list of basic integral formulas are

∫ 1 dx = x + C

∫ a dx = ax+ C

∫ xn dx = ((xn+1)/(n+1))+C ; n≠1

∫ sin x dx = – cos x + C

∫ cos x dx = sin x + C

∫ sec2x dx = tan x + C

∫ csc2x dx = -cot x + C

∫ sec x (tan x) dx = sec x + C

∫ csc x ( cot x) dx = – csc x + C

∫ (1/x) dx = ln |x| + C

∫ ex dx = ex+ C

∫ ax dx = (ax/ln a) + C ; a>0, a≠1

Trigonometry

trigonometry, the branch of mathematics concerned with specific functions of angles and their application to calculations. There are six functions of an angle commonly used in trigonometry. Their names and abbreviations are sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (csc). These six trigonometric functions in relation to a right triangle are displayed in the figure. For example, the triangle contains an angle A, and the ratio of the side opposite to A and the side opposite to the right angle (the hypotenuse) is called the sine of A, or sin A; the other trigonometry functions are defined similarly. These functions are properties of the angle A independent of the size of the triangle, and calculated values were tabulated for many angles before computers made trigonometry tables obsolete. Trigonometric functions are used in obtaining unknown angles and distances from known or measured angles in geometric figures.

ABOUT THE GROUP....

                      NESOS 

JOURNEY

THE GROUP 'NESOS' WAS FORMED ON  20 APRIL,2022 & STRIVES TO PROVIDE GOOD QUALITY ENGINEERING MATHEMATICS NOTES ACCESSIBLE TO STUDENTS SPECIFICALLY PEOPLE APPEARING IN MST 2, FINALS & EVEN PU-MEET.

THE MEMBERS:

• JANVI (CO22523)
• RIMA(CO22536)
• PIYUSH(CO22534)
• ISHWINDER(CO22522)
  

                                                                  

JANVI

SHE IS SHORT-TEMPERD, ADAMANT & INTELLIGENT PERIODTT..................

SHE ACTED OUT SWIFT BY INITIATING THE BLOGGING JOURNEY VIA SENDING OUT INVITES TO HER GROUPMATES.SHE TILL LAST DATE ACTED OUT AS A MEDIATOR OF HER GROUPMATES FOR DECIDING WHICH COLOR TEE TO WEAR👊AKA SHE LEARNT A BIT OF PATIENCE . SHE  CHOSE THE LAYOUT, HEADERS & WIDGETS FOR THE SITE .SHE EVEN SELECTED THE THEME FOR THE SITE . SHE IS INCHARGE OF REVIWING & GIVING THE FINAL WORD FOR THE BLOG.  ONLY DIFFICULTY IN HER JOURNEY WAS REPEATEDLY DEALING WITH PIYUSH & ISHWINDER TO WRITE BLOGS . SHE IS QUITE STRAIGHT-FORWARD & DOESN'T TOLERATE DISCRIPANCIES 

JANVI SUMMED UP THE WHOLE OF HER BLOGS IN A SINGLE DAY  WHICH SHOWS HER SPEED & DEDICATION  (MAYBE👀).

Rima

she is good adviser she gives good ideas and suggested.

an excellent topic for our blog and help Janvi to decorate our blogs interface......

she suggests good topics and subtopic for our blog as she is good in mathematics, she helps us if any one of the members faced any problems. she is always available for any discussion. she helps or suggest survey question and share it to everyone.

Rima completes her four entries in our blog in a one day, it reflects that how much she is interested and dedicated in this blog😃

PIYUSH

He is all known for his dedication  towards maths.

The idea of creating a blog on math was of his.

At starting he was criticized for giving the idea of creating a blog on 

math. But had he decided to do something, he would do that at any cause.

Then he promised his teamates that this idea would not let them down. At last they all get agreed and from there comes a birth of engineering mathematics. Not only he had given idea of making a blog on math but the unique name of this group was also given by him. He  only made his own blogs that too on the last day . Thus  what this guy has done for his group till last moment ,nobody can do that. This shows how helping  he is as a team member.

LAPLACE TRANSFORM

Introduction of laplace transform             Laplace transform, in mathematics, a particular integral transform invented by the French mathematician Pierre-Simon Laplace (1749–1827), and systematically developed by the British physicist Oliver Heaviside (1850–1925) Its pupose is to simplify the solution of many differential equations that describe physical processes                 DEFINATION OF LAPLACE TRANSFOM

APPLICATION OF LAPLACE TRANSFORM

Laplace transform is an integral transform method which is particularly useful in solving linear ordinary differential equations. It finds very wide applications in various areas of physics, electrical engineering, control engineering, optics, mathematics and signal processing. The Laplace transform can be interpreted as a transformation from the time domain where inputs and outputs are functions of time to the frequency domain where inputs and outputs are functions of complex angular frequency

BASIC FORMULAS OF LAPLACE TRANSFORM

Survey report

 The survey report whose Google form was designed by Janvi (Co22523)

Report by Janvi (Co22523)

1. Our first question of the survey was :

The responses were as follows:

A greater number of respondents agrred to the fact that they are frustrated 🥴 from studying engineering mathematics

2.  The 2nd question ❓& its responses

A staggering 31.8% of the survey respondents have lost a hope for even understanding 😭 the mathematics due to miscellaneous personal or any other reasons 💻 ,followed by 27.3% respondents having no interest in the subject . Then, a percentage of 22.7% people found their professors boring 🪴, in a way found them incapable. Then a meagre amount of people were the ones who didn't 🚫 have had their basics clear 

3. The following response show the application of mathematics according to different people

4.  The 4th question went as follows:

5.  The response of a simple question ❓🌞

Yes , it's main application ✋ is found in paper  📜 by the people followed by road curve 🛣️ 🪝 .We can draw conclusion that the mere application that the education system has planted in students head for such a daily basis life application 
6.  The sequence guess went as follows:

The sequence here is both Fibonacci & Golden ratio which is one and the same thing 😂 .However, the mind of students have been 😭 corrupted by the miserable 😖 teaching practice based only on cramming & not on understanding a topic so as to widen their mind & correlation power

7. The seventh question response & it's analysis

The most people got this question right 👍

8.  The eight question is as follows:

Again a basic question but 9.1% didn't had their basics clear due to lack of engineering 😬 material on internet.

9.  Application of Laplace transforms???

10.  Now , a brief through our respondents favourite maths topic 

11. ☠️🤡

A staggering percentage of 38.9% students Don't miss their previous professor.PREIODT..

12.   Maths ur friend 🍟???

44.4% students are confident & don't deter to solve numerical thinking out 🌧️ their inaccurate answers 😉 cause they are the masters of their field 🏑. Then comes a 33.3% maybe 🤔 confused af students about their relationship with  maths numericals. 22.2% students are simply not confident about solving numericals.

13.  The views ✌️🗜️

14.    The most important question aka the fruitfulness of  our handwork  😉

The respondents'  nod of approval is quite enough to convince our PC 😁 teacher to give us 50/50

Signing off

XoXo