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The problem involves determining if the given vector fields \( \mathbf{F}(x, y) \) are conservative (i.e., whether they are gradients of some scalar function \( f(x, y) \)), and if so, finding \( f(x, y) \). Let’s break down the solutions shown:

### (a) \( \mathbf{F}(x, y) = 5x \mathbf{i} + 5y \mathbf{j} \)
- **Check for conservativeness**: Since \( \frac{\partial}{\partial y}(5x) = 0 \) and \( \frac{\partial}{\partial x}(5y) = 0 \), the curl condition \( \frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x} \) is satisfied.
- **Scalar potential**:
  \[
  f(x, y) = \int 5x \, dx + \int 5y \, dy = \frac{5}{2}x^2 + \frac{5}{2}y^2 + C.
  \]

### (b) \( \mathbf{F}(x, y) = 7xy \mathbf{i} + 7xy \mathbf{j} \)
- **Check for conservativeness**: For \( P = 7xy \) and \( Q = 7xy \),
  \[
  \frac{\partial P}{\partial y} = 7x, \quad \frac{\partial Q}{\partial x} = 7y.
  \]
  Since \( \frac{\partial P}{\partial y} \neq \frac{\partial Q}{\partial x} \), the vector field is not conservative, and \( f(x, y) \) does not exist.

### (c) \( \mathbf{F}(x, y) = (4x^2 + 4y^2) \mathbf{i} + 8xy \mathbf{j} \)
- **Check for conservativeness**: For \( P = 4x^2 + 4y^2 \) and \( Q = 8xy \),
  \[
  \frac{\partial P}{\partial y} = 8y, \quad \frac{\partial Q}{\partial x} = 8y.
  \]
  Since \( \frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x} \), the vector field is conservative.
- **Scalar potential**:
  \[
  f(x, y) = \int (4x^2 + 4y^2) \, dx + \int 8xy \, dy = \frac{4}{3}x^3 + 4xy^2 + C.
  \]

Let me know if you want detailed derivations for any of these cases!

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### Relative Questions
1. What are the conditions for a vector field to be conservative in \( \mathbb{R}^2 \)?
2. How do you derive the scalar potential \( f(x, y) \) from a given conservative vector field \( \mathbf{F} \)?
3. Why does the curl condition fail for part (b)?
4. How does the scalar potential \( f(x, y) \) relate to the physical concept of potential energy?
5. Can a vector field be conservative if defined on a domain with holes or discontinuities?

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### Tip
Always check the domain of the vector field when determining if it is conservative; some fields may fail to be conservative due to a restricted or non-simply connected domain.

Determine which of the following vector fields F in the plane is the gradient of a scalar function f. If such an f exists, find it. (If an answer does not exist, enter DNE.)

Explore solutions to determining if vector fields are gradients of scalar functions, with detailed explanations of conservativeness checks and scalar potential derivation using exact differential equations.

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