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It appears that you have uploaded a problem that involves Green's theorem and the evaluation of an integral. Here's a breakdown of the questions in the image:

1. **(a)** The question asks you to state Green's theorem.
2. **(b)** You're given a positive orientation of a unit circle in \(\mathbb{R}^2\) and asked to compute an integral over that circle:
   \[
   \int_{\Gamma} \left( \frac{-y}{(x^2 + y^2)^a} \, dx + \frac{x}{(x^2 + y^2)^a} \, dy \right)
   \]
3. **(c)** For which values of the parameter \(\alpha \in \mathbb{R}\) can Green's theorem be applied to evaluate the integral in part (b)?

Let's start with **Green's theorem**, which is:

\[
\oint_{\Gamma} P \, dx + Q \, dy = \iint_{D} \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \, dA
\]
where \(\Gamma\) is the positively oriented, simple closed curve enclosing a region \(D\), and \(P(x, y)\) and \(Q(x, y)\) are functions of \(x\) and \(y\).

For **part (b)**, you're looking to evaluate the integral over the unit circle \(\Gamma\), with the vector field:
\[
P(x, y) = \frac{-y}{(x^2 + y^2)^a}, \quad Q(x, y) = \frac{x}{(x^2 + y^2)^a}
\]
To proceed, we will need to analyze the behavior of this vector field and check the conditions under which Green's theorem can be applied.

For **part (c)**, we'll examine the conditions under which the integral in part (b) is well-defined and Green's theorem holds. Specifically, we'll need to check whether the integral encounters any singularities, particularly when \( (x^2 + y^2) = 0 \), which occurs at the origin. Green's theorem can generally be applied when the vector field is smooth and well-behaved in the region of integration, so we need to determine whether this condition holds for the given \(a\).

Would you like me to proceed with solving each part in detail, or do you have any specific questions about the problem?

State Green's theorem. Given a positively oriented unit circle in R^2, determine for which parameters α ∈ R Green's theorem can be applied to evaluate the given integral. Then, compute the integral for the suitable α.

Learn how to apply Green's theorem to evaluate a given line integral over a positively oriented unit circle. This problem explores the conditions under which Green's theorem is valid and provides a step-by-step solution. Suitable for undergraduate mathematics students.

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