exprD

exprD is a symbolic operator used in some mathematical and computer algebra contexts to denote the total differential of a symbolic expression with respect to a set of independent variables. While not a universally standardized symbol, exprD is commonly interpreted as producing a differential form that encodes how the expression changes in all directions of the variable space.

Definition and notation. Let f be a smooth function of variables x1, x2, ..., xn. The action of exprD on f is the total differential: exprD f = sum_{i=1}^n (∂f/∂xi) dxi. Here dxi represents the differential of xi, and the expression exprD f is the differential form df. If the variables appear through intermediary functions, the chain rule is applied in the usual multivariable form: if yj = yj(x1, ..., xn), then exprD f(y1, ..., ym) = sum_j (∂f/∂yj) exprD yj, with exprD yj = sum_i (∂yj/∂xi) dxi.

Properties. The operator exprD is linear: exprD(af + bg) = a exprD f + b exprD g. It satisfies the Leibniz rule: exprD(fg) = f exprD g + g exprD f. It aligns with the standard total differential in multivariable calculus and is compatible with the chain rule when expressed through intermediate variables.

Examples. For f(x, y) = x^2 y, exprD f = 2xy dx + x^2 dy. If y is a function of x, say y = y(x), then exprD f becomes (∂f/∂x) dx + (∂f/∂y) dy/dx dx, which simplifies to df in terms of dx and dy.

Applications and context. exprD appears in discussions of differential geometry, perturbation analysis, and symbolic differentiation where emphasis is on the differential change of an expression rather than a numeric derivative. Its exact usage and notation may vary between texts and software environments, and it is not universally adopted as a standard operator.