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 need to use the relationship between gravitational field \( \vec{E} \) and gravitational potential \( V \). The gravitational field is given by the negative gradient of the potential:

\[ \vec{E} = -\nabla V \]

Where \( \nabla V \) is the gradient of the potential \( V \).

### Step 1: Calculate the partial derivatives

The gradient in three dimensions is given by:

\[ \nabla V = \left( \frac{\partial V}{\partial x} \hat{i} + \frac{\partial V}{\partial y} \hat{j} + \frac{\partial V}{\partial z} \hat{k} \right) \]

We need to compute the partial derivatives of \( V \) with respect to \( x \), \( y \), and \( z \).

#### Partial derivative with respect to \( x \):

\[ \frac{\partial V}{\partial x} = \frac{\partial}{\partial x}(3X^2 Y + Y^3 Z) = 6XY \]

#### Partial derivative with respect to \( y \):

\[ \frac{\partial V}{\partial y} = \frac{\partial}{\partial y}(3X^2 Y + Y^3 Z) = 3X^2 + 3Y^2 Z \]

#### Partial derivative with respect to \( z \):

\[ \frac{\partial V}{\partial z} = \frac{\partial}{\partial z}(3X^2 Y + Y^3 Z) = Y^3 \]

### Step 2: Combine the results into the gradient

Now we can write the gradient:

\[ \nabla V = (6XY) \hat{i} + (3X^2 + 3Y^2 Z) \hat{j} + (Y^3) \hat{k} \]

### Step 3: Find the gravitational field

Now, we can find the gravitational field \( \vec{E} \):

\[ \vec{E} = -\nabla V = -\left(6XY \hat{i} + (3X^2 + 3Y^2 Z) \hat{j} + Y^3 \hat{k}\right) \]

Thus, the gravitational field is:

\[ \vec{E} = -6XY \hat{i} - (3X^2 + 3Y^2 Z) \hat{j} - Y^3 \hat{k} \]

### Final Answer:

\[ \vec{E} = -6XY \hat{i} - (3X^2 + 3Y^2 Z) \hat{j} - Y^3 \hat{k} \]

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