The electric field ina region is given as
`E= (1)/(epsilon_(0))[( 2y^(2) + z) hat(i)+ 4 xyhat(j)+ xhat(k)] V//m`
Find volume charge density at ( - 1, 0,3) .

Answer

To find the volume charge density at the point (-1, 0, 3) given the electric field \( E \), we will use the differential form of Gauss's law, which relates the divergence of the electric field to the volume charge density.

### Step-by-Step Solution:

1. **Write the Electric Field**:
The electric field is given as:
\[ E = \frac{1}{\epsilon_0} \left( (2y^2 + z) \hat{i} + 4xy \hat{j} + x \hat{k} \right) \text{ V/m} \]

2. **Use the Divergence of the Electric Field**:
According to Gauss's law in differential form:
\[ \nabla \cdot E = \frac{\rho}{\epsilon_0} \]
where \( \rho \) is the volume charge density.

3. **Calculate the Divergence**:
The divergence operator in Cartesian coordinates is:
\[ \nabla \cdot E = \frac{\partial E_x}{\partial x} + \frac{\partial E_y}{\partial y} + \frac{\partial E_z}{\partial z} \]
Here, \( E_x = \frac{1}{\epsilon_0}(2y^2 + z) \), \( E_y = \frac{1}{\epsilon_0}(4xy) \), and \( E_z = \frac{1}{\epsilon_0}(x) \).

4. **Differentiate Each Component**:
- For \( E_x \):
\[ \frac{\partial E_x}{\partial x} = \frac{\partial}{\partial x} \left( \frac{1}{\epsilon_0}(2y^2 + z) \right) = 0 \]
- For \( E_y \):
\[ \frac{\partial E_y}{\partial y} = \frac{\partial}{\partial y} \left( \frac{1}{\epsilon_0}(4xy) \right) = \frac{4x}{\epsilon_0} \]
- For \( E_z \):
\[ \frac{\partial E_z}{\partial z} = \frac{\partial}{\partial z} \left( \frac{1}{\epsilon_0}(x) \right) = 0 \]

5. **Combine the Results**:
Now, we can combine these results to find the divergence:
\[ \nabla \cdot E = 0 + \frac{4x}{\epsilon_0} + 0 = \frac{4x}{\epsilon_0} \]

6. **Relate to Charge Density**:
From Gauss's law, we have:
\[ \frac{4x}{\epsilon_0} = \frac{\rho}{\epsilon_0} \]
Therefore, the volume charge density \( \rho \) is:
\[ \rho = 4x \]

7. **Substitute the Point (-1, 0, 3)**:
Now, we need to find \( \rho \) at the point (-1, 0, 3):
\[ \rho = 4(-1) = -4 \text{ C/m}^3 \]

### Final Answer:
The volume charge density at the point (-1, 0, 3) is:
\[ \rho = -4 \text{ C/m}^3 \]