Laplace Transform - First Shifting Theorem with Example | By GP Sir

Dr.Gajendra Purohit

25 Jun 201810:14

Summary

TLDRIn this educational video, the instructor introduces the 'First Shifting Theorem' in the context of Laplace Transforms, focusing on its application for both direct and inverse transformations. The video simplifies the concept with straightforward examples, emphasizing the importance of understanding the theorem's implications rather than its proof. The instructor demonstrates how to apply the theorem to find the Laplace Transform of given functions and then invert them, using additional examples for clarity. The lecture aims to ensure comprehension and encourages students to engage with any lingering questions in the comment section. The video concludes with a call to action for viewers to share and subscribe to the channel.

Takeaways

📚 The video discusses the 'First Shifting Theorem' in the context of Laplace Transforms.
🔍 The theorem is used to find the Laplace Transform and its inverse.
📝 The presenter provides simple statements of the theorem without focusing on the proof.
📈 Three examples are given to demonstrate how to find the Laplace Transform using the theorem.
🔄 The 'First Shifting Theorem' is also applicable for inverse Laplace Transforms.
📑 The video includes examples to illustrate finding the inverse Laplace Transform.
🤔 The presenter assures that the concept is simple and that proofs will be taught when necessary.
👨‍🏫 The lecture aims to ensure understanding of the first shifting theorem for both Laplace and inverse Laplace Transforms.
💬 Students are encouraged to ask questions in the comment section if they have doubts.
📢 The presenter invites viewers to share and subscribe to the channel for more educational content.

Q & A

What is the 'First Shifting Theorem' in the context of Laplace Transform?
-The 'First Shifting Theorem' is a principle in Laplace Transform that allows for the simplification of finding the Laplace Transform of functions that are shifted in time. It is a simple concept that is crucial for solving problems involving time delays or advanced functions.
How does the 'First Shifting Theorem' simplify the process of finding Laplace Transforms?
-The 'First Shifting Theorem' simplifies the process by providing a direct formula to calculate the Laplace Transform of a function that is time-shifted. This avoids the need for complex integrations and makes it easier to handle functions with time delays.
What are the implications of the 'First Shifting Theorem' that students need to remember?
-Students need to remember that the 'First Shifting Theorem' allows them to easily find the Laplace Transform of functions that are shifted in time by a certain amount. The theorem provides a straightforward way to account for this time shift in the transform.
Is it necessary to understand the proof of the 'First Shifting Theorem' to use it effectively?
-While understanding the proof can be beneficial, it is not necessary for practical use. The focus should be on the application of the theorem to solve problems, and the proof can be studied when needed for a deeper understanding.
Can you provide an example of how the 'First Shifting Theorem' is applied to find the Laplace Transform?
-Sure, if you have a function f(t-a)u(t-a) where u(t) is the unit step function, the Laplace Transform using the 'First Shifting Theorem' would be e^{-as}F(s) where F(s) is the Laplace Transform of f(t).
What is the inverse process of the 'First Shifting Theorem'?
-The inverse process of the 'First Shifting Theorem' is used to find the inverse Laplace Transform of a function. It helps in determining the original time function when given its Laplace Transform, especially when the function has been shifted in the s-domain.
How does the 'First Shifting Theorem' assist in finding the inverse Laplace Transform?
-The 'First Shifting Theorem' assists in finding the inverse Laplace Transform by providing a method to shift the function back in time. It allows for the reconstruction of the original time-domain function from its Laplace Transform representation.
Can you give an example of using the 'First Shifting Theorem' for inverse Laplace Transform?
-If the Laplace Transform of a function is given as e^{-as}F(s), using the 'First Shifting Theorem' for inverse Laplace Transform, the original function can be found as f(t-a)u(t-a) where F(s) is the Laplace Transform of f(t).
What are the key takeaways from the lecture on the 'First Shifting Theorem'?
-The key takeaways include understanding what the 'First Shifting Theorem' is, how to use it to find Laplace Transforms of time-shifted functions, and how to apply it to find the inverse Laplace Transform. The lecture also emphasized the importance of knowing the implications of the theorem without necessarily focusing on the proof.
How can students ensure they understand the 'First Shifting Theorem' and its applications?
-Students can ensure their understanding by practicing with various examples, reviewing the theorem's implications, and asking questions in the comment section if they have doubts. Engaging with the material through practice and discussion is key to solidifying the concepts.

Outlines

00:00

📚 Introduction to First Shifting Theorem

The speaker begins by addressing the students and introducing the topic of the video, which is the 'First Shifting Theorem' in Laplace Transform. They mention that they will cover how to use this theorem for both direct and inverse Laplace Transforms. The speaker reassures the audience that although there is a proof for the theorem, it is not necessary to focus on it for this lesson. Instead, the emphasis is on understanding the implications of the theorem. The speaker provides examples to demonstrate how to find the Laplace Transform using the First Shifting Theorem and promises to teach the proof when needed.

05:01

🔍 Applying First Shifting Theorem in Practice

In this paragraph, the speaker delves into practical applications of the First Shifting Theorem by providing examples. They illustrate how to solve problems using the theorem and then move on to teach how to find the inverse Laplace Transform using the inverse shifting theorem. The speaker takes the audience through multiple examples to solidify their understanding of applying the First Shifting Theorem to both direct and inverse Laplace Transforms.

10:02

📢 Conclusion and Call to Action

The speaker concludes the lecture by summarizing the key points covered in the video. They have taught the concept of the First Shifting Theorem, how to use it to find the Laplace Transform, and subsequently, how to find the inverse Laplace Transform. The speaker expresses hope that the students have understood the material and encourages them to ask questions in the comment section if they have any doubts. They also invite the audience to share the video, subscribe to the channel, and engage with the content.

Mindmap

Keywords

💡Laplace Transform

Laplace Transform is a mathematical technique used to convert a function of a real variable, often time, into a function of a complex variable, known as the 's-plane'. It is widely used in the analysis of linear time-invariant systems and differential equations. In the video, the concept is central as the instructor discusses how to use the 'First Shifting theorem' to find the Laplace Transform of functions, illustrating its application with examples.

💡First Shifting Theorem

The 'First Shifting Theorem' in the context of Laplace Transforms is a principle that allows for the simplification of the transform process by shifting the function in the time domain. This theorem is crucial in the video as it is the main tool used to demonstrate how to find the Laplace Transform. The instructor mentions that it's a simple concept that students need to remember and applies it to various examples.

💡Inverse Laplace Transform

Inverse Laplace Transform is the process of converting a function in the 's-plane' back to its original function in the time domain. It is the reverse operation of the Laplace Transform and is essential for solving differential equations. The video script discusses how to find the inverse Laplace Transform using the 'First Shifting theorem', showing its importance in recovering the original function from its transformed version.

💡Time Domain

The time domain refers to the representation of a signal or function as it varies over time. It is one of the two primary domains used in signal processing, the other being the frequency domain. In the video, the instructor uses the time domain to explain the initial forms of functions before applying the Laplace Transform and then shifting them using the 'First Shifting theorem'.

💡s-plane

The s-plane, also known as the complex plane, is a graphical representation of complex numbers where the horizontal axis represents the real part and the vertical axis represents the imaginary part. It is used in the context of Laplace Transforms to represent the transformed functions. The video mentions the s-plane as the domain where the Laplace Transform of time-domain functions is represented.

💡Linear Time-Invariant Systems

Linear Time-Invariant (LTI) systems are systems that are both linear and do not change over time. They are characterized by their response to inputs being proportional and their behavior being consistent regardless of when the input is applied. The video's theme indirectly relates to LTI systems as Laplace Transforms are often used to analyze and solve problems related to such systems.

💡Differential Equations

Differential equations are equations that relate a function to its derivatives, describing how the rate of change of a quantity depends on other quantities. They are fundamental in various fields of science and engineering. The video's focus on Laplace Transforms is directly related to solving differential equations, as the Transform is a powerful tool for this purpose.

💡Proof

A proof in mathematics is a logical demonstration that a statement is true. In the video, the instructor mentions that while the 'First Shifting theorem' has a proof, the focus is on its implications rather than the proof itself. This suggests that the video aims to teach practical application over theoretical underpinnings.

💡Comment Section

The comment section is a feature of online platforms where users can post messages, questions, or feedback related to the content. In the video script, the instructor encourages students to use the comment section to ask questions if they have doubts, indicating an interactive approach to learning and addressing student queries.

💡Subscription

A subscription, in the context of online content, refers to the act of signing up to receive regular updates or access to premium content. The instructor ends the video by encouraging viewers to subscribe to the channel, which implies a series of related educational videos and a community of learners interested in the subject matter.

Highlights

Introduction to the use of 'First Shifting theorem' in Laplace Transform.

Explanation of how to use 'First Shifting theorem' in inverse Laplace Transform.

Guidance on finding the Laplace transform by using the first shifting theorem.

Emphasis on the simplicity of the 'First Shifting theorem' concept.

Clarification that the proof of the theorem is not a focus, but its implications are.

Illustration of the theorem's application through two examples.

Introduction of the concept of inverse first shifting theorem.

Demonstration of how to find the inverse Laplace of given functions.

Explanation of the 'First Shifting theorem' for inverse Laplace Transform.

Presentation of three examples to find Laplace using the first shifting theorem.

Instruction on finding the inverse Laplace using the inverse shifting theorem.

Discussion on solving questions using the first shifting theorem.

Presentation of additional examples to solidify understanding.

Summary of the lecture's content on the first shifting theorem.

Encouragement for students to ask questions in the comment section if they have doubts.

Request for viewers to share and subscribe to the channel for more content.