91Ó°ÊÓ

The curl of a vector field \(\mathbf{F} = \langle F_1, F_2, F_3 \rangle\) can be calculated using the following determinant involving partial derivatives: $$ \nabla \times \mathbf{F} = \left\langle \frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z}, \frac{\partial F_1}{\partial z} - \frac{\partial F_3}{\partial x}, \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y} \right\rangle $$ For our given vector field, \(\mathbf{F} = \langle y z, x z, x y \rangle\). Let's compute the curl for our vector field: $$ \nabla \times \mathbf{F} = \left\langle \frac{\partial (x y)}{\partial y} - \frac{\partial (x z)}{\partial z}, \frac{\partial (y z)}{\partial z} - \frac{\partial (x y)}{\partial x}, \frac{\partial (x z)}{\partial x} - \frac{\partial (y z)}{\partial y} \right\rangle = \left\langle x - x, y - y, z - z \right\rangle $$ So we have \(\nabla \times \mathbf{F} = \left\langle 0, 0, 0 \right\rangle\).

Since the curl of the vector field is zero, i.e., \(\nabla \times \mathbf{F} = \left\langle 0, 0, 0 \right\rangle\), the vector field is conservative on the specified region of \(\mathbb{R}^{3}\).

To find the potential function, we need to solve the following equations for a scalar function \(\phi(x, y, z)\): 1. \(\frac{\partial \phi}{\partial x} = yz\) 2. \(\frac{\partial \phi}{\partial y} = xz\) 3. \(\frac{\partial \phi}{\partial z} = xy\) Now, integrate each equation with respect to their respective variables. 1. Integrating equation (1) with respect to \(x\): \(\phi(x, y, z) = \int yz \, dx = xyz + g(y, z)\) 2. Integrating equation (2) with respect to \(y\): \(\phi(x, y, z) = \int xz \, dy = xyz + h(x, z)\) 3. Integrating equation (3) with respect to \(z\): \(\phi(x, y, z) = \int xy \, dz = xyz + k(x, y)\) Notice that we get the same term \(xyz\) when we integrate each equation. Let's combine these results: $$ \phi(x, y, z) = xyz + g(y, z) + h(x, z) + k(x, y) $$ From this equation, we see that \(g(y,z) = 0\), \(h(x,z)= 0\), and \(k(x,y) = 0\). Thus, the potential function is given by: $$ \phi(x, y, z) = xyz $$ We have determined that the vector field is conservative and the potential function is \(\phi(x, y, z) = xyz\).