Question

Determine whether or not the vector field is conservative. $$ \mathbf{F}(x, y, z)=\sin y z \mathbf{i}+x z \cos y z \mathbf{j}+x y \sin y z \mathbf{k} $$

Short answer

The vector field \(\mathbf{F}(x, y, z)\) is conservative if and only if its curl, i.e., \( \nabla \times \mathbf{F} = 0 \)

Step-by-step solution

Identify the Components of the Vector Field

The given vector field \(\mathbf{F}(x, y, z) = \sin y z \mathbf{i} + x z \cos y z \mathbf{j} + x y \sin y z \mathbf{k} \). Therefore its components are \( F_1(x, y, z) = \sin yz \), \( F_2(x, y, z) = xz\cos yz \), and \( F_3(x, y, z) = xy\sin yz \).

Compute the Curl of the Vector Field

The curl of the vector field \(\mathbf{F}\) can be given by the determinant of a matrix that includes the unit vectors on the first row, the derivatives on the second row, and the vector field components on the third row. This will give us a new vector, say \( \mathbf{G}(x, y, z) = \nabla \times \mathbf{F} \). Therefore, let us compute \( \mathbf{G}(x, y, z) \).

Check if the Curl of the Vector Field is Zero

If the vector formed by taking the curl of the vector field is the zero vector, i.e. its components are all zero, then the original vector field is conservative. Here, it should be determined if \( \mathbf{G}(x, y, z) = 0 \).