Gradient Vector Calculator

This calculator finds the gradient vector ∇f = (∂f/∂x, ∂f/∂y) of the quadratic surface f(x,y) = ax² + by² + cxy at a single (x, y) point you choose, then reports its magnitude and direction. The two partial derivatives are computed directly — ∂f/∂x = 2ax + cy and ∂f/∂y = 2by + cx (lines 14-15) — so each is driven by exactly two of the five inputs. With the cross-term coefficient c left at its default of 0, cy drops out of ∂f/∂x and cx drops out of ∂f/∂y entirely, decoupling the two partial derivatives: a (Coefficient of x²) and the x Point move only ∂f/∂x, while b (Coefficient of y²) and the y Point move only ∂f/∂y.

Because each partial derivative is a plain product of two inputs (2·a·x, for instance), a proportional nudge to either factor moves it by the same proportional amount — a and x pull exactly evenly on ∂f/∂x, with neither one driving it more than the other. Magnitude and Direction combine both partials, so both x-side and y-side inputs affect them. The calculator does compute f(x,y) itself internally (fValue, line 24), but it never displays that value or the level curve through the point as an output, even though it derives the level-curve slope too (levelSlope, line 27) — and it is restricted to the quadratic form ax²+by²+cxy — a cubic or higher-order surface is outside its model.