expxy222

Expxy222 is a notional term used in mathematical education and documentation to refer to a two-variable exponential function, often presented as a concrete example of exponential growth depending on the product of two inputs. The '222' suffix serves as a version or exercise number and does not change the mathematical content in its standard form. The term is not a standard constant or widely adopted function in mathematics; it is a placeholder name used to label a specific example in instructional material.

In its common instantiation, expxy222 denotes the function f(x, y) = exp(xy), where exp denotes the base-e exponential. The function is defined for all real x and y, and its value grows rapidly as the product xy increases. The gradient and Hessian are useful for analysis: ∂f/∂x = y exp(xy), ∂f/∂y = x exp(xy); ∂²f/∂x² = y² exp(xy), ∂²f/∂y² = x² exp(xy), ∂²f/∂x∂y = (1 + xy) exp(xy).

Applications in textbooks include illustrating level curves, contour plots, and optimization problems with constraints. The suffix 222 may be varied to denote different parameterizations, such as f(x, y) = exp(axy) or f(x, y) = exp(xy) + p(x) + q(y) in a sequence of exercises; however, expxy222 itself commonly remains the base form with xy as the exponent argument.

Example: evaluating at x = 1 and y = 2 yields f(1, 2) = exp(2) ≈ 7.389, and the gradient at that point is (∂f/∂x, ∂f/∂y) = (2e^2, e^2) ≈ (14.778, 7.389).

See also: exponential function, multivariable calculus, gradient, Hessian, level set.