梯度和散度

Consider in Cartesian coordinates the three-dimensional vector field 𝑢 = u1(x, y, z)𝑖 + u2(x, y, z)𝑗 + u3(x, y, z)𝑘. The curl of 𝑢, denoted as ∇ × 𝑢 and pronounced “del-cross-u”, is defined as the vector field given by ∇ × 𝑢 . Here, the cross product is used between a vector differential operator and a vector field. The curl measures how much a vector field rotates, or curls, around a point. A more math-based description will be given later. Example: Show that the curl of a gradient is zero, that is, ∇ × (∇f) = 0. We have ∇ × (∇f) , using the equality of mixed partials. Example: Show that the divergence of a curl is zero, that is, ∇ · (∇ × 𝑢) = 0. We have ∇ · (∇ × 𝑢) , again using the equality of mixed partials.