Differentiation

If f[r,θ,z] is a function of three variables, then for each θ = θ₀ and z=z₀, the function f[r,θ₀,z₀] is a differentiable function of r, and its derivative is denoted f_r[r₀,θ₀,z₀], called the first partial derivative of f with respect to r.

Similarly the derivative of the r = r₀, z=z₀ slice curve f[r₀,θ,z₀] with respect to θ is denoted by f_θ[r₀,θ₀,z₀], called the first partial derivative of f with respect to θ and the derivative of the r = r₀, θ = θ₀ slice curve f(r₀,θ₀,z) with respect to z is denoted by f_z[r₀,θ₀,z₀], called the first partial derivative of f with respect to z.

If there is a tangent plane at such a point, then the tangent plane is horizontal.

The r-partial derivative of the function f is itself a function f_r defined over the same domain as f. The collection of points [r,θ,z] in the domain such that f_r[r,θ,z] = 0 will be the fold set in the r-direction, the fold set in the θ-direction is the collection of points for which f_θ[r,θ,z] = 0, and the fold set in the z-direction is the collection of points for which f_z[r,θ,z] = 0. The critical points occur at the intersection of the fold set in the r-direction and the fold set in the θ-direction and the fold set in the z-direction.