UPSC Exam > UPSC Questions > Consider a closed triangular contour traverse...
Start Learning for Free
Consider a closed triangular contour traversed in counter-clockwise direction, as shown in the figure. The value of the integral I ⃗F · d⃗l evaluated along this contour, for a vector field ⃗F = yˆex−xˆey, is?
Most Upvoted Answer
Consider a closed triangular contour traversed in counter-clockwise di...
Understanding the Vector Field
The vector field given is F = y * e^x * i - x * e^y * j. To evaluate the line integral along a closed triangular contour, we can apply Green's Theorem.
Green's Theorem Overview
- Green's Theorem relates a line integral around a simple closed curve C to a double integral over the region D enclosed by C.
- It states: ∮C (P dx + Q dy) = ∬D (∂Q/∂x - ∂P/∂y) dA, where P and Q are components of the vector field F.
Identifying Components
- For our vector field, we identify:
- P = y * e^x
- Q = -x * e^y
Calculating Partial Derivatives
- We compute the necessary partial derivatives:
- ∂P/∂y = e^x
- ∂Q/∂x = -e^y
Applying Green's Theorem
- Now, applying Green’s Theorem:
- ∂Q/∂x - ∂P/∂y = -e^y - e^x
- The integral becomes:
- ∬D (-e^y - e^x) dA
Evaluating the Integral
- Since the contour is closed, if the enclosed area D is symmetric and the function is continuous, the integral can often evaluate to zero over symmetric regions.
- In many cases for polynomial or exponential functions over symmetric regions, the contributions tend to cancel out.
Final Result
- Thus, the value of the integral I = ∮C F · dl evaluated along the closed triangular contour is likely to be zero:
I = 0
This conclusion arises from the properties of the vector field and the application of Green’s Theorem.
The vector field given is F = y * e^x * i - x * e^y * j. To evaluate the line integral along a closed triangular contour, we can apply Green's Theorem.
Green's Theorem Overview
- Green's Theorem relates a line integral around a simple closed curve C to a double integral over the region D enclosed by C.
- It states: ∮C (P dx + Q dy) = ∬D (∂Q/∂x - ∂P/∂y) dA, where P and Q are components of the vector field F.
Identifying Components
- For our vector field, we identify:
- P = y * e^x
- Q = -x * e^y
Calculating Partial Derivatives
- We compute the necessary partial derivatives:
- ∂P/∂y = e^x
- ∂Q/∂x = -e^y
Applying Green's Theorem
- Now, applying Green’s Theorem:
- ∂Q/∂x - ∂P/∂y = -e^y - e^x
- The integral becomes:
- ∬D (-e^y - e^x) dA
Evaluating the Integral
- Since the contour is closed, if the enclosed area D is symmetric and the function is continuous, the integral can often evaluate to zero over symmetric regions.
- In many cases for polynomial or exponential functions over symmetric regions, the contributions tend to cancel out.
Final Result
- Thus, the value of the integral I = ∮C F · dl evaluated along the closed triangular contour is likely to be zero:
I = 0
This conclusion arises from the properties of the vector field and the application of Green’s Theorem.
|
Explore Courses for UPSC exam
|
|
Similar UPSC Doubts
Top Courses for UPSCView all
Consider a closed triangular contour traversed in counter-clockwise direction, as shown in the figure. The value of the integral I ⃗F · d⃗l evaluated along this contour, for a vector field ⃗F = yˆex−xˆey, is?
Question Description
Consider a closed triangular contour traversed in counter-clockwise direction, as shown in the figure. The value of the integral I ⃗F · d⃗l evaluated along this contour, for a vector field ⃗F = yˆex−xˆey, is? for UPSC 2025 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Consider a closed triangular contour traversed in counter-clockwise direction, as shown in the figure. The value of the integral I ⃗F · d⃗l evaluated along this contour, for a vector field ⃗F = yˆex−xˆey, is? covers all topics & solutions for UPSC 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider a closed triangular contour traversed in counter-clockwise direction, as shown in the figure. The value of the integral I ⃗F · d⃗l evaluated along this contour, for a vector field ⃗F = yˆex−xˆey, is?.
Consider a closed triangular contour traversed in counter-clockwise direction, as shown in the figure. The value of the integral I ⃗F · d⃗l evaluated along this contour, for a vector field ⃗F = yˆex−xˆey, is? for UPSC 2025 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Consider a closed triangular contour traversed in counter-clockwise direction, as shown in the figure. The value of the integral I ⃗F · d⃗l evaluated along this contour, for a vector field ⃗F = yˆex−xˆey, is? covers all topics & solutions for UPSC 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider a closed triangular contour traversed in counter-clockwise direction, as shown in the figure. The value of the integral I ⃗F · d⃗l evaluated along this contour, for a vector field ⃗F = yˆex−xˆey, is?.
Solutions for Consider a closed triangular contour traversed in counter-clockwise direction, as shown in the figure. The value of the integral I ⃗F · d⃗l evaluated along this contour, for a vector field ⃗F = yˆex−xˆey, is? in English & in Hindi are available as part of our courses for UPSC.
Download more important topics, notes, lectures and mock test series for UPSC Exam by signing up for free.
Here you can find the meaning of Consider a closed triangular contour traversed in counter-clockwise direction, as shown in the figure. The value of the integral I ⃗F · d⃗l evaluated along this contour, for a vector field ⃗F = yˆex−xˆey, is? defined & explained in the simplest way possible. Besides giving the explanation of
Consider a closed triangular contour traversed in counter-clockwise direction, as shown in the figure. The value of the integral I ⃗F · d⃗l evaluated along this contour, for a vector field ⃗F = yˆex−xˆey, is?, a detailed solution for Consider a closed triangular contour traversed in counter-clockwise direction, as shown in the figure. The value of the integral I ⃗F · d⃗l evaluated along this contour, for a vector field ⃗F = yˆex−xˆey, is? has been provided alongside types of Consider a closed triangular contour traversed in counter-clockwise direction, as shown in the figure. The value of the integral I ⃗F · d⃗l evaluated along this contour, for a vector field ⃗F = yˆex−xˆey, is? theory, EduRev gives you an
ample number of questions to practice Consider a closed triangular contour traversed in counter-clockwise direction, as shown in the figure. The value of the integral I ⃗F · d⃗l evaluated along this contour, for a vector field ⃗F = yˆex−xˆey, is? tests, examples and also practice UPSC tests.
|
|
Explore Courses for UPSC exam
|
|
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup