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Let S be the surface of the cone z=√x² y² bounded by the planes z=0 and z=3. Further let C be the closed surface forming the boundary of the surface S. A vector field is such that curl F is -xi-yj. Find absolute value of line integral F.dr?
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Let S be the surface of the cone z=√x² y² bounded by the planes z=0 an...
Given information:
S is the surface of the cone z=√(x² + y²) bounded by the planes z=0 and z=3.
C is the closed surface forming the boundary of S.
The vector field F has curl F = -xi - yj.
To find:
The absolute value of the line integral F.dr.
Explanation:
To find the line integral of a vector field F along a curve C, we can use the line integral formula:
∫(C) F.dr = ∫(a to b) F(r(t)).r'(t)dt
Where r(t) is the parameterization of the curve C, and r'(t) is the derivative of r(t) with respect to t.
Parameterization of C:
Since C forms the boundary of S, we can parameterize C using the equation of the cone surface:
r(t) = (t, t, √(t² + t²)) = (t, t, √2t²) = (t, t, √2)t
Where t ranges from 0 to 2π.
Calculating F(r(t)):
To calculate F(r(t)), we substitute the values of r(t) into the vector field F:
F(r(t)) = -(t)i - (t)j
Calculating r'(t):
r'(t) = (1, 1, √2)
Calculating F.dr:
Now we can calculate F.dr using the line integral formula:
∫(C) F.dr = ∫(0 to 2π) F(r(t)).r'(t)dt
= ∫(0 to 2π) [-(t)i - (t)j].(1, 1, √2) dt
= ∫(0 to 2π) [-t - t√2] dt
= -∫(0 to 2π) t(1 + √2) dt
= -(1 + √2) ∫(0 to 2π) t dt
= -(1 + √2) [t²/2] (0 to 2π)
= -(1 + √2) [(4π² - 0)/2]
= -2π²(1 + √2)
Calculating the absolute value:
Since the question asks for the absolute value of the line integral, we take the magnitude:
|F.dr| = |-2π²(1 + √2)|
= 2π²(1 + √2)
Therefore, the absolute value of the line integral F.dr is 2π²(1 + √2).
S is the surface of the cone z=√(x² + y²) bounded by the planes z=0 and z=3.
C is the closed surface forming the boundary of S.
The vector field F has curl F = -xi - yj.
To find:
The absolute value of the line integral F.dr.
Explanation:
To find the line integral of a vector field F along a curve C, we can use the line integral formula:
∫(C) F.dr = ∫(a to b) F(r(t)).r'(t)dt
Where r(t) is the parameterization of the curve C, and r'(t) is the derivative of r(t) with respect to t.
Parameterization of C:
Since C forms the boundary of S, we can parameterize C using the equation of the cone surface:
r(t) = (t, t, √(t² + t²)) = (t, t, √2t²) = (t, t, √2)t
Where t ranges from 0 to 2π.
Calculating F(r(t)):
To calculate F(r(t)), we substitute the values of r(t) into the vector field F:
F(r(t)) = -(t)i - (t)j
Calculating r'(t):
r'(t) = (1, 1, √2)
Calculating F.dr:
Now we can calculate F.dr using the line integral formula:
∫(C) F.dr = ∫(0 to 2π) F(r(t)).r'(t)dt
= ∫(0 to 2π) [-(t)i - (t)j].(1, 1, √2) dt
= ∫(0 to 2π) [-t - t√2] dt
= -∫(0 to 2π) t(1 + √2) dt
= -(1 + √2) ∫(0 to 2π) t dt
= -(1 + √2) [t²/2] (0 to 2π)
= -(1 + √2) [(4π² - 0)/2]
= -2π²(1 + √2)
Calculating the absolute value:
Since the question asks for the absolute value of the line integral, we take the magnitude:
|F.dr| = |-2π²(1 + √2)|
= 2π²(1 + √2)
Therefore, the absolute value of the line integral F.dr is 2π²(1 + √2).
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Let S be the surface of the cone z=√x² y² bounded by the planes z=0 and z=3. Further let C be the closed surface forming the boundary of the surface S. A vector field is such that curl F is -xi-yj. Find absolute value of line integral F.dr?
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Let S be the surface of the cone z=√x² y² bounded by the planes z=0 and z=3. Further let C be the closed surface forming the boundary of the surface S. A vector field is such that curl F is -xi-yj. Find absolute value of line integral F.dr? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Let S be the surface of the cone z=√x² y² bounded by the planes z=0 and z=3. Further let C be the closed surface forming the boundary of the surface S. A vector field is such that curl F is -xi-yj. Find absolute value of line integral F.dr? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let S be the surface of the cone z=√x² y² bounded by the planes z=0 and z=3. Further let C be the closed surface forming the boundary of the surface S. A vector field is such that curl F is -xi-yj. Find absolute value of line integral F.dr?.
Let S be the surface of the cone z=√x² y² bounded by the planes z=0 and z=3. Further let C be the closed surface forming the boundary of the surface S. A vector field is such that curl F is -xi-yj. Find absolute value of line integral F.dr? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Let S be the surface of the cone z=√x² y² bounded by the planes z=0 and z=3. Further let C be the closed surface forming the boundary of the surface S. A vector field is such that curl F is -xi-yj. Find absolute value of line integral F.dr? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let S be the surface of the cone z=√x² y² bounded by the planes z=0 and z=3. Further let C be the closed surface forming the boundary of the surface S. A vector field is such that curl F is -xi-yj. Find absolute value of line integral F.dr?.
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