Total Differential

If $z = f(x,y)$ is a function of two independent variables $x$ and $y$, the total differential of $z$, denoted $dz$ or $df$, is defined as:

$$ dz = \frac{\partial f}{\partial x}dx + \frac{\partial f}{\partial y}dy $$

Here, $dx$ and $dy$ represent small changes in $x$ and $y$ respectively. $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ are the partial derivatives of $f$ with respect to $x$ and $y$.

Geometrically, $dz$ represents the change in height along the tangent plane to the surface $z=f(x,y)$ at a point $(x,y)$, when $x$ changes by $dx$ and $y$ changes by $dy$. This is an approximation of the actual change $\Delta z$ in the function $f(x,y)$.

For a function of three variables, $w = f(x,y,z)$, the total differential is:

$$ dw = \frac{\partial f}{\partial x}dx + \frac{\partial f}{\partial y}dy + \frac{\partial f}{\partial z}dz $$

The total differential is fundamental in understanding how small changes in independent variables affect the dependent variable, for error analysis, and for deriving rates of change in multivariable contexts.

If $z = f(x,y)$ and both $x$ and $y$ are functions of a single variable $t$ (e.g., time), then $z$ is also implicitly a function of $t$. The rate of change of $z$ with respect to $t$, $\frac{dz}{dt}$, can be found using the chain rule for multivariable functions:

$$ \frac{dz}{dt} = \frac{\partial f}{\partial x} \cdot \frac{dx}{dt} + \frac{\partial f}{\partial y} \cdot \frac{dy}{dt} $$

This formula allows us to determine how $z$ changes as $t$ changes, considering the contributions from the changes in both $x$ and $y$.

For example, if the dimensions of a rectangle $x(t)$ and $y(t)$ are changing with time, this formula can be used to find the rate of change of its area $A(x,y) = xy$.

The total differential provides a linear approximation for the change in a function $z=f(x,y)$ when its independent variables $x$ and $y$ undergo small changes $\Delta x$ and $\Delta y$. The actual change is $\Delta z = f(x+\Delta x, y+\Delta y) - f(x,y)$.

The approximate change (or error) is given by the total differential $dz$:

$$ \Delta z \approx dz = \frac{\partial f}{\partial x}\Delta x + \frac{\partial f}{\partial y}\Delta y $$

This is particularly useful in experimental sciences and engineering where measurements have inherent errors. If we know the errors in measuring $x$ and $y$ (i.e., $\Delta x$ and $\Delta y$), we can estimate the resulting error in a quantity calculated from $x$ and $y$.