If f(x , y ) is a differentiable function of the two independent variables x and

y, and we set

zf (x, y),


dz @f

@x dx


@y dy

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

fdzgx @f



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