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Suppose C is the closed curve defined as the circle x2 + y2 = 1 with C oriented anti-clockwise. The value of ∮(xy2dx + x2ydy) over the curve C equals ________
Correct answer is between '-0.03,0.03'. Can you explain this answer?
Verified Answer
Suppose C is the closed curve defined as the circlex2 + y2= 1withCorie...
Concept:
Green’s theorem:
Let R be a closed bounded region in the xy plane whose boundary C consists of finitely many
smooth curves.
Let F1(x, y) & F(x, y) be functions that are continuous and have continuous partial
derivatives
∂F1 / ∂y and ∂F2 / ∂x. Then
∂F1 / ∂y and ∂F2 / ∂x. Then
Analysis:
Given curve C: x2 + y2 = 1
= 0
Most Upvoted Answer
Suppose C is the closed curve defined as the circlex2 + y2= 1withCorie...
Concept:
Green’s theorem:
Let R be a closed bounded region in the xy plane whose boundary C consists of finitely many
smooth curves.
Let F1(x, y) & F(x, y) be functions that are continuous and have continuous partial
derivatives
∂F1 / ∂y and ∂F2 / ∂x. Then
∂F1 / ∂y and ∂F2 / ∂x. Then
Analysis:
Given curve C: x2 + y2 = 1
= 0
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Community Answer
Suppose C is the closed curve defined as the circlex2 + y2= 1withCorie...
Given information:
- The closed curve C is defined as the circle x^2 + y^2 = 1.
- The orientation of the curve C is anti-clockwise.
To find the value of the line integral ∮C (xy^2 dx + x^2y dy), we can use Green's theorem, which relates a line integral over a closed curve to a double integral over the region enclosed by the curve.
Green's theorem states that for a vector field F = (P, Q) and a closed curve C oriented anti-clockwise, the line integral of F along C is equal to the double integral of the curl of F over the region R enclosed by C.
In this case, the vector field F = (xy^2, x^2y), so we need to calculate the curl of F and find the double integral over the region R.
1. Calculate the curl of F:
The curl of F is given by the expression:
curl(F) = (∂Q/∂x - ∂P/∂y)
Here, P = xy^2 and Q = x^2y.
∂P/∂y = 2xy
∂Q/∂x = 2xy
So, curl(F) = (2xy - 2xy) = 0
2. Find the double integral over the region R:
Since the curl of F is zero, the double integral of the curl over any region is also zero.
Therefore, the value of the line integral ∮C (xy^2 dx + x^2y dy) is zero.
Explanation of the correct answer:
The correct answer given as '-0.03,0.03' indicates that the value of the line integral lies between -0.03 and 0.03. Since the line integral is zero, it falls within this range.
The range provided in the answer gives an indication of the magnitude of the line integral, which is very small. This suggests that the vector field F does not have a significant influence on the curve C or the region R.
- The closed curve C is defined as the circle x^2 + y^2 = 1.
- The orientation of the curve C is anti-clockwise.
To find the value of the line integral ∮C (xy^2 dx + x^2y dy), we can use Green's theorem, which relates a line integral over a closed curve to a double integral over the region enclosed by the curve.
Green's theorem states that for a vector field F = (P, Q) and a closed curve C oriented anti-clockwise, the line integral of F along C is equal to the double integral of the curl of F over the region R enclosed by C.
In this case, the vector field F = (xy^2, x^2y), so we need to calculate the curl of F and find the double integral over the region R.
1. Calculate the curl of F:
The curl of F is given by the expression:
curl(F) = (∂Q/∂x - ∂P/∂y)
Here, P = xy^2 and Q = x^2y.
∂P/∂y = 2xy
∂Q/∂x = 2xy
So, curl(F) = (2xy - 2xy) = 0
2. Find the double integral over the region R:
Since the curl of F is zero, the double integral of the curl over any region is also zero.
Therefore, the value of the line integral ∮C (xy^2 dx + x^2y dy) is zero.
Explanation of the correct answer:
The correct answer given as '-0.03,0.03' indicates that the value of the line integral lies between -0.03 and 0.03. Since the line integral is zero, it falls within this range.
The range provided in the answer gives an indication of the magnitude of the line integral, which is very small. This suggests that the vector field F does not have a significant influence on the curve C or the region R.
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Suppose C is the closed curve defined as the circlex2 + y2= 1withCoriented anti-clockwise. The value of(xy2dx + x2ydy)over the curveCequals ________Correct answer is between '-0.03,0.03'. Can you explain this answer?
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