https://file.aiquickdraw.com/mathbot/file/1732536448000fchppb17.jpg

To solve the given problem systematically, we will analyze the velocity vectors one by one to determine if the flow is incompressible or irrotational, and find the streamlines.

### Step 1: Key Definitions
1. **Incompressible Flow**: The divergence of the velocity field \(\nabla \cdot \mathbf{v} = 0\).
2. **Irrotational Flow**: The curl of the velocity field \(\nabla \times \mathbf{v} = 0\).
3. **Streamlines**: Defined by the differential equation \(\frac{dx}{v_x} = \frac{dy}{v_y} = \frac{dz}{v_z}\).

Now, let us examine each velocity field:

---

### Case (a): \(\mathbf{v} = [0, 3z^2, 0]\)

1. **Incompressibility**:
   Compute \(\nabla \cdot \mathbf{v}\):
   \[
   \nabla \cdot \mathbf{v} = \frac{\partial v_x}{\partial x} + \frac{\partial v_y}{\partial y} + \frac{\partial v_z}{\partial z} = 0 + 0 + 0 = 0
   \]
   The flow is **incompressible**.

2. **Irrotationality**:
   Compute \(\nabla \times \mathbf{v}\):
   \[
   \nabla \times \mathbf{v} = \begin{vmatrix}
   \mathbf{i} & \mathbf{j} & \mathbf{k} \\
   \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\
   0 & 3z^2 & 0
   \end{vmatrix} = \mathbf{0}
   \]
   The flow is **irrotational**.

3. **Streamlines**:
   \[
   \frac{dx}{0} = \frac{dy}{3z^2} = \frac{dz}{0}
   \]
   The streamline condition implies \(z = \text{constant}\) and \(y = y_0 + 3z^2 t\).

---

### Case (b): \(\mathbf{v} = [xy, yz, xz]\)

1. **Incompressibility**:
   Compute \(\nabla \cdot \mathbf{v}\):
   \[
   \nabla \cdot \mathbf{v} = \frac{\partial (xy)}{\partial x} + \frac{\partial (yz)}{\partial y} + \frac{\partial (xz)}{\partial z} = y + z + x
   \]
   The flow is **not incompressible**.

2. **Irrotationality**:
   Compute \(\nabla \times \mathbf{v}\):
   \[
   \nabla \times \mathbf{v} \neq \mathbf{0}
   \]
   The flow is **not irrotational**.

3. **Streamlines**:
   Solve \(\frac{dx}{xy} = \frac{dy}{yz} = \frac{dz}{xz}\).

---

### Case (c): \(\mathbf{v} = [cx, cy, 0]\)

1. **Incompressibility**:
   Compute \(\nabla \cdot \mathbf{v}\):
   \[
   \nabla \cdot \mathbf{v} = \frac{\partial (cx)}{\partial x} + \frac{\partial (cy)}{\partial y} + \frac{\partial (0)}{\partial z} = c + c + 0 = 2c
   \]
   The flow is incompressible only if \(c = 0\).

2. **Irrotationality**:
   Compute \(\nabla \times \mathbf{v}\):
   \[
   \nabla \times \mathbf{v} = 0
   \]
   The flow is irrotational.

---

### Case (d): \(\mathbf{v} = [-y, x, 0]\)

1. **Incompressibility**:
   Compute \(\nabla \cdot \mathbf{v}\):
   \[
   \nabla \cdot \mathbf{v} = \frac{\partial (-y)}{\partial x} + \frac{\partial (x)}{\partial y} + \frac{\partial (0)}{\partial z} = 0
   \]
   The flow is incompressible.

2. **Irrotationality**:
   Compute \(\nabla \times \mathbf{v}\):
   \[
   \nabla \times \mathbf{v} = 2
   \]
   The flow is **not irrotational**.

---

Would you like me to perform a specific calculation, or generate the GeoGebra illustration? Let me know! 

### Follow-Up Questions:
1. What tools can we use to visualize these flows geometrically?
2. How does incompressibility affect the physical behavior of the flow?
3. Can a flow be irrotational but compressible? Why?
4. How would you solve the streamline equation for case (b)?
5. What does the curl of a velocity field physically represent?

### Tip: Use computational tools like GeoGebra or Python to easily compute and visualize divergence, curl, and streamlines.

Let v be the velocity vector of a steady fluid flow. Is the flow irrotational? Incompressible? Find the streamlines for the given velocity vectors: (a) [0, 3z^2, 0], (b) [xy, yz, xz], (c) [cx, cy, 0], and (d) [-y, x, 0]. Using GeoGebra, demonstrate geometrically irrotational flow and incompressible flow, and also plot the streamlines.

Discover how to analyze steady fluid flow by determining whether the flow is incompressible or irrotational. Learn about key vector calculus principles, such as divergence and curl, and how to compute streamlines for various velocity fields. Includes practical demonstrations using GeoGebra.

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