arctangradient

Arctangradient refers to the vector field obtained by taking the gradient of the arctangent of a real-valued function. In mathematical terms, let f: R^n → R be differentiable. Then the scalar field arctan(f(x)) has a gradient given by the chain rule: ∇ arctan(f(x)) = (1 / (1 + f(x)^2)) ∇ f(x). In one dimension, this reduces to the familiar derivative dy/dx = f'(x) / (1 + f(x)^2).

Examples illustrate the formula. If f(x, y) = x, then ∇ arctan(f) = (1 / (1 + x^2)) ∇f = (1 / (1 + x^2), 0). If f(x, y) = x^2 + y^2, then ∇ arctan(f) = (2x, 2y) / (1 + (x^2 + y^2)^2).

Properties and remarks. The arctangradient is smooth wherever f is differentiable. Its magnitude is bounded by the magnitude of ∇f, since 1 + f(x)^2 ≥ 1. The factor 1/(1 + f(x)^2) dampens large values of f, which can be useful in smoothing or regularizing gradient fields derived from scalar potentials.

Applications. The concept is used in optimization and vector-field design to produce smooth gradient fields from scalar functions. By applying arctan to a scalar field before differentiating, one can control the influence of large function values on the resulting gradient, which can be advantageous in numerical methods and modeling.

Notes. The term arctangradient is not a standard, widely adopted term in all mathematical literature; it is typically described as the gradient of arctan f or ∇[arctan(f)]. See also arctan, gradient, chain rule.