To calculate the **divergence** of the vector field \( \mathbf{F}(x, y, z) = (f(y), 0, 0) \), we apply the divergence operator in 3D, which is defined as:

\[
\nabla \cdot \mathbf{F} = \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} + \frac{\partial F_3}{\partial z}
\]

Given:
\[
\mathbf{F}(x, y, z) = (f(y), 0, 0)
\]
- \( F_1 = f(y) \)
- \( F_2 = 0 \)
- \( F_3 = 0 \)

Now, applying the divergence formula:

\[
\nabla \cdot \mathbf{F} = \frac{\partial f(y)}{\partial x} + \frac{\partial 0}{\partial y} + \frac{\partial 0}{\partial z}
\]

Since \( f(y) \) depends only on \( y \), the partial derivative with respect to \( x \) is zero:

\[
\frac{\partial f(y)}{\partial x} = 0
\]

Similarly, the derivatives of the other components are also zero:

\[
\frac{\partial 0}{\partial y} = 0 \quad \text{and} \quad \frac{\partial 0}{\partial z} = 0
\]

Thus, the divergence of the vector field is:

\[
\nabla \cdot \mathbf{F} = 0
\]

### Conclusion:
The divergence of the vector field \( \mathbf{F}(x, y, z) = (f(y), 0, 0) \) is zero.

Would you like more details or have any questions?

### 5 Related Questions:
1. How does divergence relate to the physical interpretation of vector fields?
2. What is the divergence of a more general vector field \( (f(x, y), g(x, y), h(x, y)) \)?
3. Can a vector field with zero divergence still have a non-zero curl?
4. What are the implications of zero divergence in fluid dynamics?
5. How can divergence be applied in the context of electromagnetism?

### Tip:
For vector fields, a zero divergence often indicates that the field is **solenoidal**, meaning it has no sources or sinks.

Learn how to calculate the divergence of the vector field F(x, y, z) = (f(y), 0, 0) using the divergence operator in 3D. This solution explains the steps for computing partial derivatives and applying the divergence formula. Suitable for advanced high school or university-level students.

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