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!

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?

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.