Linear Approximation and Differential of a Function of Several Variables

Just as a differentiable one-variable function is well approximated near $x_0$ by its tangent line, a function of two (or three) variables is well approximated near a point by its tangent plane (respectively tangent hyperplane).

Definition 8.10 (Linear Approximation and Differential, two variables). Let $A=\{(x,y)\mid a<x<b,\,c<y<d\}\subset\mathbb R^2$, $F:A\to\mathbb R$, $(x_0,y_0)\in A$.

1. The linear approximation of $F$ at $(x_0,y_0)$ is

   $F(x,y) \approx F(x_0,y_0) + \frac{\partial F}{\partial x}\bigg|_{(x_0,y_0)}(x-x_0) + \frac{\partial F}{\partial y}\bigg|_{(x_0,y_0)}(y-y_0). \qquad(12)$

2. The differential of $F$ is

   $dF = \frac{\partial F}{\partial x}(x,y)\,dx + \frac{\partial F}{\partial y}(x,y)\,dy, \qquad \text{where } dx=\Delta x,\ dy=\Delta y. \qquad(13)$

   Definition 8.11 (Linear Approximation and Differential, three variables). For $A=\{(x,y,z)\mid a<x<b,\,c<y<d,\,e<z<f\}\subset\mathbb R^3$, $F:A\to\mathbb R$, $(x_0,y_0,z_0)\in A$:

   $F(x,y,z) \approx F(x_0,y_0,z_0) + F_x\big|_{(x_0,y_0,z_0)}(x-x_0) + F_y\big|_{(x_0,y_0,z_0)}(y-y_0) + F_z\big|_{(x_0,y_0,z_0)}(z-z_0), \qquad(14)$

   $dF = F_x(x,y,z)\,dx + F_y(x,y,z)\,dy + F_z(x,y,z)\,dz, \qquad dx=\Delta x,\ dy=\Delta y,\ dz=\Delta z. \qquad(15)$

   (The same pattern extends to any number of variables, though this course restricts itself to at most three.) Geometric meaning. For one variable, the linear approximation at $x_0$ is the tangent line to $y=f(x)$ at $x_0$. For two variables, the linear approximation at $(x_0,y_0)$ is the tangent plane to the surface $z=F(x,y)$ at $(x_0,y_0)$ — the natural one-dimension-higher analogue. Worked pattern (differential). For $w(x,y,z)=x^2y+y^2z+z^2x$: compute $w_x=2xy+z^2$, $w_y=2yz+x^2$, $w_z=2zx+y^2$, then by (15), $dw = (2xy+z^2)\,dx + (2yz+x^2)\,dy + (2zx+y^2)\,dz.$ …