The gravitational potential due to a mass distribution is `V = 3X^(2) Y+ Y^(3) Z`. Find the gravitational filed.

Explanation

To find the gravitational field from the given gravitational potential \( V = 3X^2 Y + Y^3 Z \), we will follow these steps:

### Step 1: Understand the relationship between gravitational field and potential
The gravitational field \( \mathbf{E} \) is related to the gravitational potential \( V \) by the equation:
\[ \mathbf{E} = -\nabla V \]
where \( \nabla V \) is the gradient of the potential.

### Step 2: Write the gradient in three dimensions
In three-dimensional Cartesian coordinates, the gradient of a scalar function \( V(x, y, z) \) is given by:
\[ \nabla V = \left( \frac{\partial V}{\partial x} \hat{i} + \frac{\partial V}{\partial y} \hat{j} + \frac{\partial V}{\partial z} \hat{k} \right) \]
Thus, the gravitational field can be expressed as:
\[ \mathbf{E} = -\left( \frac{\partial V}{\partial x} \hat{i} + \frac{\partial V}{\partial y} \hat{j} + \frac{\partial V}{\partial z} \hat{k} \right) \]

### Step 3: Calculate the partial derivatives of \( V \)
Given \( V = 3X^2 Y + Y^3 Z \), we will compute the partial derivatives:

1. **Partial derivative with respect to \( x \)**:
\[ \frac{\partial V}{\partial x} = \frac{\partial}{\partial x}(3X^2 Y + Y^3 Z) = 6XY \]

2. **Partial derivative with respect to \( y \)**:
\[ \frac{\partial V}{\partial y} = \frac{\partial}{\partial y}(3X^2 Y + Y^3 Z) = 3X^2 + 3Y^2 Z \]

3. **Partial derivative with respect to \( z \)**:
\[ \frac{\partial V}{\partial z} = \frac{\partial}{\partial z}(3X^2 Y + Y^3 Z) = Y^3 \]

### Step 4: Substitute the derivatives into the field equation
Now we can substitute these derivatives back into the equation for the gravitational field:
\[ \mathbf{E} = -\left( 6XY \hat{i} + (3X^2 + 3Y^2 Z) \hat{j} + Y^3 \hat{k} \right) \]

### Step 5: Write the final expression for the gravitational field
Thus, the gravitational field \( \mathbf{E} \) is:
\[ \mathbf{E} = -6XY \hat{i} - (3X^2 + 3Y^2 Z) \hat{j} - Y^3 \hat{k} \]

### Final Answer:
\[ \mathbf{E} = -6XY \hat{i} - (3X^2 + 3Y^2 Z) \hat{j} - Y^3 \hat{k} \]

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