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To determine if the vector field \( \mathbf{F} = (y \sin z - \sin x) \, \mathbf{i} + (x \sin z + 2 y z) \, \mathbf{j} + (x y \cos z + y^2) \, \mathbf{k} \) is irrotational and to find the scalar potential \( \phi \) such that \( \mathbf{F} = \nabla \phi \), we proceed with the following steps:

### Step 1: Check if \( \mathbf{F} \) is Irrotational
A vector field \( \mathbf{F} \) is irrotational if its curl is zero, i.e., \( \nabla \times \mathbf{F} = 0 \).

Given \( \mathbf{F} = F_x \, \mathbf{i} + F_y \, \mathbf{j} + F_z \, \mathbf{k} \) where:
- \( F_x = y \sin z - \sin x \)
- \( F_y = x \sin z + 2 y z \)
- \( F_z = x y \cos z + y^2 \)

We compute \( \nabla \times \mathbf{F} \) using the determinant formula for the curl:
\[
\nabla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ F_x & F_y & F_z \end{vmatrix}
\]
Expanding this, we calculate each component of \( \nabla \times \mathbf{F} \) and verify if it equals zero.

### Step 2: Find the Scalar Potential \( \phi \)
If \( \nabla \times \mathbf{F} = 0 \), then \( \mathbf{F} \) is a conservative field, and we can find a scalar function \( \phi \) such that \( \mathbf{F} = \nabla \phi \). For this:
1. Integrate \( F_x = \frac{\partial \phi}{\partial x} \) with respect to \( x \) to find \( \phi(x, y, z) \).
2. Differentiate the resulting expression with respect to \( y \) and set it equal to \( F_y = \frac{\partial \phi}{\partial y} \).
3. Differentiate the expression with respect to \( z \) and set it equal to \( F_z = \frac{\partial \phi}{\partial z} \).

By solving these equations, we obtain \( \phi(x, y, z) \), if it exists.

Would you like me to go through the calculations in detail for each step? 

Here are some related questions:
1. What does it mean for a vector field to be irrotational?
2. How do we use the curl to determine if a vector field is conservative?
3. What is the significance of the scalar potential in a vector field?
4. How does the existence of a potential function simplify computations in vector fields?
5. What are the applications of conservative vector fields in physics?

**Tip:** Always start by verifying if the curl of a vector field is zero when searching for a potential function. This simplifies your calculations and ensures the field is conservative.

Determine if the vector field \( \mathbf{F} = (y \sin z - \sin x) \, \mathbf{i} + (x \sin z + 2 y z) \, \mathbf{j} + (x y \cos z + y^2) \, \mathbf{k} \) is irrotational, and find the scalar function \( \phi \) such that \( \mathbf{F} = \nabla \phi \).

Learn how to determine if a vector field is irrotational and find the scalar potential function for \( \mathbf{F} = (y \sin z - \sin x) \, \mathbf{i} + (x \sin z + 2 y z) \, \mathbf{j} + (x y \cos z + y^2) \, \mathbf{k} \). This involves computing the curl of \( \mathbf{F} \) and verifying if it equals zero, indicating that the field is conservative.

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