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Example: Find the gradient of f(x,y,z)=x^2+yz, the divergence of F(x,y,z)=(x, y^2, z^3), and the curl of G(x,y,z)=(yz, xz, xy). Step 1: Gradient of f ∇f = (∂f/∂x, ∂f/∂y, ∂f/∂z) ∂f/∂x = 2x, ∂f/∂y = z, ∂f/∂z = y So ∇f = (2x, z, y). Step 2: Divergence of F ∇·F = ∂/∂x(x) + ∂/∂y(y^2) + ∂/∂z(z^3) = 1 + 2y + 3z^2. Step 3: Curl of G ∇×G = (∂/∂y(Gz)-∂/∂z(Gy), ∂/∂z(Gx)-∂/∂x(Gz), ∂/∂x(Gy)-∂/∂y(Gx)) = (∂/∂y(xy)-∂/∂z(xz), ∂/∂z(yz)-∂/∂x(yz), ∂/∂x(xz)-∂/∂y(yz)) = (x - x, z - y, z - y) = (0, z - y, z - y).