Prerequisites

### The Multivariable Differential

The tangent line to the graph of $y=f(x)$ at the point ($x_0$,$y_0$) is given by $$y-y_0 = m \left( x-x_0 \right)$$ where the slope $m$ is of course just the derivative $\frac{df}{dx} \big|_{x=x_0}$. It is tempting to rewrite the equation of the tangent line as

Figure 1: Linear approximation using the tangent line.

$$\Delta y = \frac{df}{dx} \Delta x$$ which is also used for linear approximation in the form $$\Delta f = f(x+\Delta x)-f(x) \approx \frac{df}{dx} \Delta x$$ as shown in Figure 1. Regarding $\Delta x$ as small, this also leads to the interpretation of the differential $$df = \frac{df}{dx} \,dx$$ as the corresponding small change in $f$.

What if $f$ is a function of more than one variable? Consider the tangent plane to the graph of $z=f(x,y)$ at the point ($x_0$,$y_0$,$z_0$). What is the height difference $\Delta z$ between two points on the tangent plane? Hold a piece of paper at an arbitrary angle in front of you, and imagine moving on it first to the right, then directly forwards. How high did you go? The sum of the height differences in each step. How big are these height differences? As above, they are precisely the horizontal distance traveled multiplied by the appropriate slope.

In other words, the equation of the tangent plane is given by $$z-z_0 = m_x \left( x-x_0 \right) + m_y \left( y-y_0 \right)$$ where $m_x$ and $m_y$ are the slopes in the $x$ and $y$ directions. But these slopes are (by definition) the partial derivatives of $f$ with respect to $x$ and $y$ (at the given point).

In the $xy$-plane, the partial derivative of $f$ with respect to $x$ is written as $\frac{\partial f}{\partial x}$, and means “the derivative of $f$ with respect to $x$ with $y$ held constant”. In some contexts, it is necessary to state explicitly what variables are being held constant, in which case this partial derivative may be written as $\left(\frac{\partial f}{\partial x}\right)_y$. Make sure you distinguish between $d$ and $\partial$; writing the letter “d” so that it looks like a $\partial$ is not correct mathematics.

Returning to our graph, we can therefore write the equation of the tangent plane as $$\Delta z = \frac{\partial f}{\partial x} \Delta x + \frac{\partial f}{\partial y} \Delta y$$ which leads to the following expression for the differential of $f$ $$df = \frac{\partial f}{\partial x} \,dx + \frac{\partial f}{\partial y} \,dy \label{multidiff}$$ which again can be thought of as the small change in $f$ corresponding to small changes in $x$ and $y$. Similar expressions hold for functions of more than two variables.

Notice that there is nothing special about the variable names $x$ and $y$. If $f$ is a function of any two parameters $\alpha$ and $\beta$, then we have a formula equivalent to ($\ref{multidiff}$), namely $$df = \frac{\partial f}{\partial \alpha} \,d\alpha + \frac{\partial f}{\partial \beta} \,d\beta$$ In particular, it is not necessary for $\alpha$ and $\beta$ to have the same dimensions.