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7. Consider a vector field ⃗F = yˆi+xz3ˆj−zyˆk. Let C be the circle x2+y2 = 4 on the plane z = 2, oriented counter-clockwise. The value of the contour integral?
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7. Consider a vector field ⃗F = yˆi+xz3ˆj−zyˆk. Let C be the circle x2...
Understanding the Vector Field
The vector field is given by F = y i + x z^3 j - z y k. We need to evaluate the contour integral of this field over the circle C defined by x² + y² = 4 in the plane z = 2.
Setting Up the Circle
- The circle C can be parameterized as:
- x = 2 cos(t)
- y = 2 sin(t)
- z = 2
- The parameter t varies from 0 to 2π.
Calculating the Vector Field on the Path
- Substitute the parameterization into F:
- F = (2 sin(t)) i + (2 cos(t))(2^3) j - (2)(2 sin(t)) k
- This simplifies to F = 2 sin(t) i + 16 cos(t) j - 4 sin(t) k.
Finding the Differential Element
- The differential element along the curve is given by:
- dr = (dx, dy, dz) = (-2 sin(t) dt, 2 cos(t) dt, 0).
Computing the Line Integral
- The line integral is calculated as:
- ∫C F · dr = ∫(F · dr) dt from t = 0 to 2π.
- This involves computing the dot product of F and dr, followed by integration over the interval.
Using Green’s Theorem
- Since the region is simply connected, Green's theorem can be applied:
- The contour integral translates to a double integral over the region enclosed by C.
- However, we must compute the curl of F, since the line integral around a closed curve yields zero if the curl is zero.
Conclusion
- After performing the calculations, it turns out that the integral evaluates to zero, indicating that the vector field has no net circulation around the closed path C. Thus, the value of the contour integral is zero.
The vector field is given by F = y i + x z^3 j - z y k. We need to evaluate the contour integral of this field over the circle C defined by x² + y² = 4 in the plane z = 2.
Setting Up the Circle
- The circle C can be parameterized as:
- x = 2 cos(t)
- y = 2 sin(t)
- z = 2
- The parameter t varies from 0 to 2π.
Calculating the Vector Field on the Path
- Substitute the parameterization into F:
- F = (2 sin(t)) i + (2 cos(t))(2^3) j - (2)(2 sin(t)) k
- This simplifies to F = 2 sin(t) i + 16 cos(t) j - 4 sin(t) k.
Finding the Differential Element
- The differential element along the curve is given by:
- dr = (dx, dy, dz) = (-2 sin(t) dt, 2 cos(t) dt, 0).
Computing the Line Integral
- The line integral is calculated as:
- ∫C F · dr = ∫(F · dr) dt from t = 0 to 2π.
- This involves computing the dot product of F and dr, followed by integration over the interval.
Using Green’s Theorem
- Since the region is simply connected, Green's theorem can be applied:
- The contour integral translates to a double integral over the region enclosed by C.
- However, we must compute the curl of F, since the line integral around a closed curve yields zero if the curl is zero.
Conclusion
- After performing the calculations, it turns out that the integral evaluates to zero, indicating that the vector field has no net circulation around the closed path C. Thus, the value of the contour integral is zero.
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7. Consider a vector field ⃗F = yˆi+xz3ˆj−zyˆk. Let C be the circle x2+y2 = 4 on the plane z = 2, oriented counter-clockwise. The value of the contour integral?
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7. Consider a vector field ⃗F = yˆi+xz3ˆj−zyˆk. Let C be the circle x2+y2 = 4 on the plane z = 2, oriented counter-clockwise. The value of the contour integral? for UPSC 2025 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about 7. Consider a vector field ⃗F = yˆi+xz3ˆj−zyˆk. Let C be the circle x2+y2 = 4 on the plane z = 2, oriented counter-clockwise. The value of the contour integral? covers all topics & solutions for UPSC 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for 7. Consider a vector field ⃗F = yˆi+xz3ˆj−zyˆk. Let C be the circle x2+y2 = 4 on the plane z = 2, oriented counter-clockwise. The value of the contour integral?.
7. Consider a vector field ⃗F = yˆi+xz3ˆj−zyˆk. Let C be the circle x2+y2 = 4 on the plane z = 2, oriented counter-clockwise. The value of the contour integral? for UPSC 2025 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about 7. Consider a vector field ⃗F = yˆi+xz3ˆj−zyˆk. Let C be the circle x2+y2 = 4 on the plane z = 2, oriented counter-clockwise. The value of the contour integral? covers all topics & solutions for UPSC 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for 7. Consider a vector field ⃗F = yˆi+xz3ˆj−zyˆk. Let C be the circle x2+y2 = 4 on the plane z = 2, oriented counter-clockwise. The value of the contour integral?.
Solutions for 7. Consider a vector field ⃗F = yˆi+xz3ˆj−zyˆk. Let C be the circle x2+y2 = 4 on the plane z = 2, oriented counter-clockwise. The value of the contour integral? in English & in Hindi are available as part of our courses for UPSC.
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