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The line integral of a vector field
where r²=x²+y² is taken
around a square (see figure) of side of unit length and
centered at (xₒ , yₒ) with │xₒ│>1/2 & │yₒ│>1/2. If the
value of the integral is L, then
(JAM 2014)
(a)
(b)
(c)
(d)
L depends on (xₒ , yₒ)
L is independent of (xₒ , yₒ) and its value is -1
L is independent of (xₒ , yₒ) and its value is 0
L is independent of (xₒ , yₒ) and its value is 2?
where r²=x²+y² is taken
around a square (see figure) of side of unit length and
centered at (xₒ , yₒ) with │xₒ│>1/2 & │yₒ│>1/2. If the
value of the integral is L, then
(JAM 2014)
(a)
(b)
(c)
(d)
L depends on (xₒ , yₒ)
L is independent of (xₒ , yₒ) and its value is -1
L is independent of (xₒ , yₒ) and its value is 0
L is independent of (xₒ , yₒ) and its value is 2?
Most Upvoted Answer
The line integral of a vector field where r²=x²+y² is taken around a s...
Analysis of the Line Integral:
- The line integral of a vector field around a closed curve is given by the circulation of the vector field along the curve.
- In this case, the vector field is defined by r²=x²+y², which represents the magnitude of the vector at any point (x, y).
- The square with side length 1 centered at (xₒ, yₒ) is a closed curve along which the line integral is to be calculated.
Determination of the Line Integral:
- Since the vector field is radial (dependent only on the distance from the origin), the line integral around a closed curve will be zero.
- This is because the positive and negative contributions along opposite sides of the square will cancel out due to the symmetry of the vector field.
- Therefore, the value of the line integral is independent of the position of the square and is always zero.
Conclusion:
- The value of the line integral is independent of the position of the square and is always zero.
- Hence, the correct option is:
(c) L is independent of (xₒ, yₒ) and its value is 0.
- The line integral of a vector field around a closed curve is given by the circulation of the vector field along the curve.
- In this case, the vector field is defined by r²=x²+y², which represents the magnitude of the vector at any point (x, y).
- The square with side length 1 centered at (xₒ, yₒ) is a closed curve along which the line integral is to be calculated.
Determination of the Line Integral:
- Since the vector field is radial (dependent only on the distance from the origin), the line integral around a closed curve will be zero.
- This is because the positive and negative contributions along opposite sides of the square will cancel out due to the symmetry of the vector field.
- Therefore, the value of the line integral is independent of the position of the square and is always zero.
Conclusion:
- The value of the line integral is independent of the position of the square and is always zero.
- Hence, the correct option is:
(c) L is independent of (xₒ, yₒ) and its value is 0.
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The line integral of a vector field where r²=x²+y² is taken around a square (see figure) of side of unit length and centered at (xₒ , yₒ) with │xₒ│>1/2 & │yₒ│>1/2. If the value of the integral is L, then (JAM 2014) (a) (b) (c) (d) L depends on (xₒ , yₒ) L is independent of (xₒ , yₒ) and its value is -1 L is independent of (xₒ , yₒ) and its value is 0 L is independent of (xₒ , yₒ) and its value is 2?
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The line integral of a vector field where r²=x²+y² is taken around a square (see figure) of side of unit length and centered at (xₒ , yₒ) with │xₒ│>1/2 & │yₒ│>1/2. If the value of the integral is L, then (JAM 2014) (a) (b) (c) (d) L depends on (xₒ , yₒ) L is independent of (xₒ , yₒ) and its value is -1 L is independent of (xₒ , yₒ) and its value is 0 L is independent of (xₒ , yₒ) and its value is 2? for UPSC 2025 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about The line integral of a vector field where r²=x²+y² is taken around a square (see figure) of side of unit length and centered at (xₒ , yₒ) with │xₒ│>1/2 & │yₒ│>1/2. If the value of the integral is L, then (JAM 2014) (a) (b) (c) (d) L depends on (xₒ , yₒ) L is independent of (xₒ , yₒ) and its value is -1 L is independent of (xₒ , yₒ) and its value is 0 L is independent of (xₒ , yₒ) and its value is 2? covers all topics & solutions for UPSC 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The line integral of a vector field where r²=x²+y² is taken around a square (see figure) of side of unit length and centered at (xₒ , yₒ) with │xₒ│>1/2 & │yₒ│>1/2. If the value of the integral is L, then (JAM 2014) (a) (b) (c) (d) L depends on (xₒ , yₒ) L is independent of (xₒ , yₒ) and its value is -1 L is independent of (xₒ , yₒ) and its value is 0 L is independent of (xₒ , yₒ) and its value is 2?.
The line integral of a vector field where r²=x²+y² is taken around a square (see figure) of side of unit length and centered at (xₒ , yₒ) with │xₒ│>1/2 & │yₒ│>1/2. If the value of the integral is L, then (JAM 2014) (a) (b) (c) (d) L depends on (xₒ , yₒ) L is independent of (xₒ , yₒ) and its value is -1 L is independent of (xₒ , yₒ) and its value is 0 L is independent of (xₒ , yₒ) and its value is 2? for UPSC 2025 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about The line integral of a vector field where r²=x²+y² is taken around a square (see figure) of side of unit length and centered at (xₒ , yₒ) with │xₒ│>1/2 & │yₒ│>1/2. If the value of the integral is L, then (JAM 2014) (a) (b) (c) (d) L depends on (xₒ , yₒ) L is independent of (xₒ , yₒ) and its value is -1 L is independent of (xₒ , yₒ) and its value is 0 L is independent of (xₒ , yₒ) and its value is 2? covers all topics & solutions for UPSC 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The line integral of a vector field where r²=x²+y² is taken around a square (see figure) of side of unit length and centered at (xₒ , yₒ) with │xₒ│>1/2 & │yₒ│>1/2. If the value of the integral is L, then (JAM 2014) (a) (b) (c) (d) L depends on (xₒ , yₒ) L is independent of (xₒ , yₒ) and its value is -1 L is independent of (xₒ , yₒ) and its value is 0 L is independent of (xₒ , yₒ) and its value is 2?.
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The line integral of a vector field where r²=x²+y² is taken around a square (see figure) of side of unit length and centered at (xₒ , yₒ) with │xₒ│>1/2 & │yₒ│>1/2. If the value of the integral is L, then (JAM 2014) (a) (b) (c) (d) L depends on (xₒ , yₒ) L is independent of (xₒ , yₒ) and its value is -1 L is independent of (xₒ , yₒ) and its value is 0 L is independent of (xₒ , yₒ) and its value is 2?, a detailed solution for The line integral of a vector field where r²=x²+y² is taken around a square (see figure) of side of unit length and centered at (xₒ , yₒ) with │xₒ│>1/2 & │yₒ│>1/2. If the value of the integral is L, then (JAM 2014) (a) (b) (c) (d) L depends on (xₒ , yₒ) L is independent of (xₒ , yₒ) and its value is -1 L is independent of (xₒ , yₒ) and its value is 0 L is independent of (xₒ , yₒ) and its value is 2? has been provided alongside types of The line integral of a vector field where r²=x²+y² is taken around a square (see figure) of side of unit length and centered at (xₒ , yₒ) with │xₒ│>1/2 & │yₒ│>1/2. If the value of the integral is L, then (JAM 2014) (a) (b) (c) (d) L depends on (xₒ , yₒ) L is independent of (xₒ , yₒ) and its value is -1 L is independent of (xₒ , yₒ) and its value is 0 L is independent of (xₒ , yₒ) and its value is 2? theory, EduRev gives you an
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