If f(x , y ) is a differentiable function of the two independent variables x and
y, and we set
zf (x, y),
is called the total differential of f (x , y ). To understand its meaning, let us
consider a fixed point (x0, y0) in the (x , y )-plane and start by holding y
constant at the value y/y0, so f (x , y0) is then a function only of x . If we now change x from x0 to x0/dx , where dx is a differential change in x , it follows that
is the corresponding differential change in z , produced by this change in x .
Similarly, if we hold x constant at x/x0 and change y from y0/dy, where dy is a differential change in y, it follows that