Functions of several variable are important in many applications. Extending the calculus to these is the subject of this section.
A simple example is a function that adds its arguments \(f(x,y) = x+y\). The function \(f\) has two arguments. Or the function can be thought of as taking an ordered pair as its argument. So we can also thing of functions whose values are ordered pairs. For example \(w(x,y) = (-y, x)\).
A function of multiple inputs assignes to each valid combination of input values an output value. For example \(f(x,y) = x+y\) or \( g(x,y) = xy \). More complicated examples are the electric field, which gives a vector for each point in space, at least within the volume being considered.
By holding all but one of the inputs constant, a function of multiple variables can be considered a function of a single variable. Using the example function \(f\), and thinking of \(y\) as a constant then \[ \frac{df(x,y)}{dx} = 1 \] and applying this to \(g\) gives \[ \frac{dg(x,y)}{dx} = \frac{d xy}{dx} = y \]
In order to emphasize that we are holding all but one of the variables constant, these derivatives are written using a slightly different notation: \[ \frac{\partial f(x,y)}{\partial x} = 1 \] and \[ \frac{\partial g(x,y)}{\partial x} = \frac{\partial \hspace{1 pt} xy}{\partial x} = y \]
The partial derivative is ...d/dx d/dy = d/dy d/dx -- the partials commute
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