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The problem asks us to analyze the vector field \( F: \mathbb{R}^2 \setminus \{0\} \to \mathbb{R}^2 \) defined by
\[
F(x, y) := \left( \frac{y}{x^2 + y^2}, \frac{-x}{x^2 + y^2} \right).
\]
Specifically, part (d) involves checking whether the restrictions \( F|_{U_1} \) and \( F|_{U_2} \) are gradient fields for the sets \( U_1 := \{ (x, y) \in \mathbb{R}^2 : x > 0 \} \) and \( U_2 := \{ (x, y) \in \mathbb{R}^2 : y > 0 \} \). If they are gradient fields, we should determine the potential function \( \Phi_j \) for \( F|_{U_j} \) with \( j = 1, 2 \).

### Solution Outline

1. **Check if \( F \) is a gradient field**: For \( F \) to be a gradient field in each region \( U_1 \) and \( U_2 \), it must satisfy the condition \( \frac{\partial F_1}{\partial y} = \frac{\partial F_2}{\partial x} \) where \( F = (F_1, F_2) \).

2. **Potential Function**: If \( F \) is a gradient field in each region, we would find a scalar potential function \( \Phi \) such that \( F = \nabla \Phi \).

### Step-by-Step Solution

1. **Compute the Partial Derivatives**:
   - For \( F(x, y) = \left( \frac{y}{x^2 + y^2}, \frac{-x}{x^2 + y^2} \right) \):
     - \( F_1(x, y) = \frac{y}{x^2 + y^2} \)
     - \( F_2(x, y) = \frac{-x}{x^2 + y^2} \)
   - Calculate \( \frac{\partial F_1}{\partial y} \) and \( \frac{\partial F_2}{\partial x} \) to determine if they are equal.

2. **Determine Potential Function**: If \( F|_{U_1} \) and \( F|_{U_2} \) are gradient fields, find potential functions \( \Phi_1 \) and \( \Phi_2 \) for these regions.

Would you like the full calculations and steps for each part? 

### Further Exploration

1. What properties make a vector field conservative or a gradient field?
2. How can you determine if a vector field is path-independent?
3. Why are the regions \( U_1 \) and \( U_2 \) necessary to consider separately?
4. How does excluding the origin \( (0,0) \) impact the vector field?
5. Can the concept of potential functions apply to fields in more than two dimensions?

**Tip**: Always check if a vector field is conservative by verifying if it satisfies the equality of mixed partial derivatives (e.g., \( \frac{\partial F_1}{\partial y} = \frac{\partial F_2}{\partial x} \)).

Betrachten Sie das Vektorfeld F: R^2 \ {0} -> R^2 definiert durch F(x, y) := (y / (x^2 + y^2), -x / (x^2 + y^2)). Prüfen Sie, ob die Einschränkungen F|_{U_1} und F|_{U_2} Gradientfelder für U_1 := {(x, y) ∈ R^2 : x > 0} und U_2 := {(x, y) ∈ R^2 : y > 0} sind. Falls ja, bestimmen Sie ein Potential Φ_j für F|_{U_j} mit j = 1, 2.

Analyze the vector field F defined as (y / (x^2 + y^2), -x / (x^2 + y^2)) and determine if its restrictions F|_{U_1} and F|_{U_2} are gradient fields for regions U_1 (x > 0) and U_2 (y > 0). Learn how to compute potential functions Φ_j for F|_{U_j}.

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