Let the vector $\vec{v} = v_1\hat{i} + v_2\hat{j} + v_3\hat{k}$ be a differentiable vector function of Cartesian coordinates $(x, y, z)$. The curl of vector $\vec{v}$ is:
By definition, curl v⃗ = ∇ × v⃗ = det[[î,ĵ,k̂],[∂/∂x,∂/∂y,∂/∂z],[v₁,v₂,v₃]] = (∂v₃/∂y − ∂v₂/∂z)î + (∂v₁/∂z − ∂v₃/∂x)ĵ + (∂v₂/∂x − ∂v₁/∂y)k̂. This is exactly option (C), the standard determinant expansion of the curl operator.
Practice This in Real CBT Exam Interface
Test yourself with the official TCS-iON palette, virtual calculator, and immediate performance analytics.