Let's analyze each vector field to determine if it's conservative or not.

(a) Vector Field:

$F = \left( \frac{x^4}{x^4 + y^4}, \frac{y^4}{x^4 + y^4} \right)$

To determine if this vector field is conservative, we need to check if the field is path-independent, meaning it should have a scalar potential function $f$ such that $\nabla f = F$. Additionally, in 2D, a necessary and sufficient condition for conservativeness is that $\frac{\partial F_y}{\partial x} = \frac{\partial F_x}{\partial y}$.

1. Compute $F_x = \frac{x^4}{x^4 + y^4}$ and $F_y = \frac{y^4}{x^4 + y^4}$.
2. Differentiate:
   • $\frac{\partial F_y}{\partial x}$ and $\frac{\partial F_x}{\partial y}$.
3. Check if $\frac{\partial F_y}{\partial x} = \frac{\partial F_x}{\partial y}$.

Calculations yield that $\frac{\partial F_y}{\partial x} \neq \frac{\partial F_x}{\partial y}$, meaning this vector field is not conservative.

(b) Vector Field:

$F = y^2 \, i + (2xy + e^z) \, j + y e^z \, k$

To determine if this field is conservative, we need to check if $\nabla \times F = 0$. For a 3D vector field to be conservative, its curl must vanish everywhere.

The components of $F$ are:

• $F_x = y^2$
• $F_y = 2xy + e^z$
• $F_z = y e^z$

Compute the curl: $\nabla \times F = \left( \frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z} \right) i - \left( \frac{\partial F_z}{\partial x} - \frac{\partial F_x}{\partial z} \right) j + \left( \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} \right) k$

Evaluating each component:

• $\frac{\partial F_z}{\partial y} = e^z$ and $\frac{\partial F_y}{\partial z} = e^z$, so $\frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z} = 0$.
• $\frac{\partial F_z}{\partial x} = 0$ and $\frac{\partial F_x}{\partial z} = 0$, so $\frac{\partial F_z}{\partial x} - \frac{\partial F_x}{\partial z} = 0$.
• $\frac{\partial F_y}{\partial x} = 2y$ and $\frac{\partial F_x}{\partial y} = 2y$, so $\frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} = 0$.

Since $\nabla \times F = 0$, the field $F$ is conservative.

To find a potential function $f$ such that $\nabla f = F$:

1. Integrate $F_x = y^2$ with respect to $x$, yielding $f(x, y, z) = x y^2 + g(y, z)$.
2. Differentiate with respect to $y$ and compare with $F_y$.
3. Integrate with respect to $z$ and match terms with $F_z$.

Following these steps, we find that a potential function is: $f(x, y, z) = x y^2 + y e^z$

Summary:

• (a) Not conservative.
• (b) Conservative with potential function $f(x, y, z) = x y^2 + y e^z$.

Would you like further details or have any questions?

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1. What is the difference between conservative and non-conservative vector fields?
2. How do you verify if a vector field is conservative in 2D versus 3D?
3. What physical phenomena are described by conservative vector fields?
4. How does path independence relate to conservative fields?
5. Can a vector field be conservative in a limited region but not in the entire space?

Tip: For a vector field to be conservative in 3D, checking the curl (if it vanishes everywhere) is often the quickest method.