The total differential

The tangent plane provides the first-order local approximation of the surface at the reference point. It is defined by the function's value and its two partial derivatives at that point.

Conditions: The surface is differentiable at the reference point $(x_0, y_0)$.; The plane is constructed using the value and partial derivatives at $(x_0, y_0)$.

Differentiability requires that the linear approximation (the tangent plane) becomes arbitrarily accurate as the input displacement approaches zero. Specifically, the remaining error between the actual increment and the total differential, divided by the input displacement length, must tend to zero.

Conditions: The function is evaluated at a reference point $(x_0, y_0)$.; The input is displaced by $(\Delta x, \Delta y)$ with length $\rho = \sqrt{\Delta x^2 + \Delta y^2}$.

The total differential $dz$ is computed by evaluating both partial derivatives at the reference point $(x_0, y_0)$ and taking their linear combination with the input displacements $\Delta x$ and $\Delta y$. The formula is $dz = f_x(x_0, y_0)\Delta x + f_y(x_0, y_0)\Delta y$.

Conditions: The function is differentiable at the reference point $(x_0, y_0)$.; The input is moved by a displacement $(\Delta x, \Delta y)$.

The actual increment $\Delta z$ is the difference between the new function value and the reference function value after the input is displaced. It is calculated as $\Delta z = f(x_0+\Delta x, y_0+\Delta y) - f(x_0, y_0)$.

Conditions: The function is evaluated at a reference point $(x_0, y_0)$.; The input is moved by a displacement $(\Delta x, \Delta y)$.

The actual increment $\Delta z$ measures the true change in height on the curved surface, calculated as the new function value minus the reference value. The total differential $dz$ measures the height change on the tangent plane for the same input displacement.

Conditions: The input is displaced by $(\Delta x, \Delta y)$ from a reference point $(x_0, y_0)$.; The function is differentiable at the reference point.