Let \(S\) be a level surface of a differentiable function \(F\) having the equation: \[F(x,y,z)=c.\] Consider a curve \(C\) on \(S\) that passes through \(\mathbf{x}_0=(x_0,y_0,z_0)\) (Fig. 1).

Assume \(C\) is given parametrically by a differentiable vector-valued function \(\mathbf{r}(t)=(x(t),y(t),z(t))\) and \(\mathbf{r}(t_0)=(x_0,y_0,z_0)\). Because \(C\) is on \(S\), \[F(\mathbf{r}(t))=F(x(t),y(t),z(t))=c.\] If \(\phi(t)=F(x(t),y(t),z(t))\), the chain rule states (also see (\(\dagger\)) on page ) that:

\[\phi'(t)=\overrightarrow{\nabla} F(\mathbf{r}(t))\bullet\mathbf{r}'(t).\]

Because \(\phi(t)=c\) is constant, we have \(\phi'(t)=0\). In particular, \(\phi'(t_0)=0\) and therefore:

\[\overrightarrow{\nabla} F(\underbrace{x_0,y_0,z_0}_{=\mathbf{r}(t_0)})\cdot\mathbf{r}'(t_0)=0.\]

This means the gradient of \(f\) at \((x_0,y_0,z_0)\) is normal to the tangent vector \(\mathbf{r}'(t)\). Consider all curves on \(S\) passing through \((x_0,y_0,z_0)\). A plane that contains the tangent vectors of all these curves is called the tangent plane.^{1} If \(\overrightarrow{\nabla} F(x_0,y_0,z_0)\neq (0,0,0)\), because \(\overrightarrow{\nabla} F(x_0,y_0,z_0)\) is normal to the tangent vectors at \((x_0,y_0,z_0)\), the (Recall that a plane through \(\mathbf{x}_0=(x_0,y_0,z_0)\) with normal \(\mathbf{n}=(n_1,n_2,n_3)\) consists of all points \(\mathbf{x}=(x,y,z)\) that satisfy: \(\mathbf{n}\cdot (\mathbf{x}-\mathbf{x}_0)=0\) )equation of tangent plane; is:

\[\overrightarrow{\nabla} F(\mathbf{x}_0)\bullet (\mathbf{x}-\mathbf{x}_0)=0,\]

\[\text{or}\quad \frac{\partial F}{\partial x}(x_0,y_0,z_0)(x-x_0)+\frac{\partial F}{\partial y}(x_0,y_0,z_0)(y-y_0)+\frac{\partial F}{\partial z}(x_0,y_0,z_0)(z-z_0)=0.\]

**Theorem 1.**Let \(F:U\subseteq\mathbb{R}^3\to \mathbb{R}\) be a differentiable function. If \((x_0,y_0,z_0)\) lies on the level surface \(S\) defined by \(F(x_0,y_0,z_0)=c\) and \(\overrightarrow{\nabla} F(x_0,y_0,z_0)\neq \mathbf{0}\), then the equation of the tangent plane at \((x_0,y_0,z_0)\) is:

\[\frac{\partial F}{\partial x}(x_0,y_0,z_0)(x-x_0)+\frac{\partial F}{\partial y}(x_0,y_0,z_0)(y-y_0)+\frac{\partial F}{\partial z}(x_0,y_0,z_0)(z-z_0)=0.\]

The equations of the normal line to the level surface \(F(x,y,z)=0\) at \((x_0,y_0,z_0)\) are:

\[\frac{x-x_0}{\frac{\partial F}{\partial x}(x_0,y_0,z_0)}=\frac{y-y_0}{\frac{\partial F}{\partial y}(x_0,y_0,z_0)}=\frac{z-z_0}{\frac{\partial F}{\partial z}(x_0,y_0,z_0)}\]

(The equation of the ellipsoid has the standard form\(\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1.\) The points \((a,0,0)\), \((0,b,0)\) and \((0,0,c)\) lie on the surface of the ellipsoid.);

The arguments are the same if we consider the level curves of \(F(x,y)=c\). In the previous section we saw that \(\overrightarrow{\nabla} F\) is normal to its level curves. The equation of the line tangent to the level curve at \((x_0,y_0)\) becomes: \[\overrightarrow{\nabla} F(x_0,y_0)\boldsymbol{\cdot} (x-x_0,y-y_0)=0\] or \[F_x(x_0,y_0)(x-x_0)+F_y(x_0,y_0)(y-y_0)=0. \tag{*}\] Note that (*) can also be written as: \[y-y_0=-\frac{F_x(x_0,y_0)}{F_y(x_0,y_0)}(x-x_0).\]

^{1 }In Section 3.8 we defined the tangent plane as a plane that contains the tangent vectors in the x and y directions. When F is differentiable, the surface is smooth enough and the tangent plane contains the tangent vectors in all directions.↩