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?

Question

Answer

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

Analysis:

Given curve C: x² + y² = 1

= 0

Answer

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.

Answer

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

Analysis:

Given curve C: x² + y² = 1

= 0

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