Final Answer:
The curl of the vector field F(x, y, z) = (ln x, ln xy, ln xyz) is the vector (0, 1/x, 1/y).
Step-by-step explanation:
Components of Curl: The curl of a vector field F(x, y, z) is given by:
curl(F) = ∂Q/∂x - ∂P/∂y + (∂R/∂y - ∂Q/∂z) i + (∂P/∂z - ∂R/∂x) j + (∂Q/∂x - ∂P/∂y) k
where P, Q, and R are the x, y, and z components of F, respectively.
Calculation: Applying the formula to F(x, y, z), we get:
curl(F) = (0 - 0) i + (1/x - 0) j + (1/y - 1/x) k
= 0 i + 1/x j + (1/y - 1/x) k
Simplified Result: Therefore, the curl of F is the vector (0, 1/x, 1/y).