Tangent Planes to Surfaces

# Tangent Planes to Surfaces

When we were dealing with single variable real-valued functions \$y = f(x)\$, we noted that the derivative \$f'(a)\$ (provided \$f\$ was differentiable) gave us the slope of the tangent line corresponding to the point \$(a, f(a))\$ on the graph of \$f\$.

Let \$z = f(x, y)\$ be two variable real-valued function that generates the surface \$S\$. Recall from the Partial Derivatives page that for a fixed value of \$y = b\$, the partial derivative \$f_x (x, b)\$ will give us a curve \$C_1\$. This curve is the intersection of the surface \$S\$ with the plane \$y = b\$ (in blue). For any defined value \$x = a\$, the partial derivative \$f_x (x, b)\$ evaluated at \$x = a\$, in other words, \$f_x (a, b)\$ gives us the slope of the tangent line of \$C_1\$ when \$x = a\$. Similarly, for a fixed value of \$x = a\$, the partial derivative \$f_y (a, y)\$ will also give us a curve \$C_2\$. This curve is the intersection of the surface \$S\$ with the plane \$x = a\$ (in pink). For any defined value \$y = b\$, the partial derivative \$f_x (a, y)\$ evaluated at \$y = b\$, in other words, \$f_y (a, b)\$ gives us the slope of the tangent line of \$C_2\$ when \$y = b\$. These two tangent lines are important. The plane that passes through these two tangent lines is known as the tangent plane at the point \$(a, b, f(a,b))\$. Definition: Let \$z = f(x, y)\$ be a two variable real-valued function that generates the surface \$S\$. Let \$C_1\$ be the curve obtained by intersecting the plane \$y = b\$ with \$S\$, and let \$C_2\$ be the curve obtained by intersecting the plane \$x = a\$ with \$S\$. Let \$T_1\$ and \$T_2\$ be the respective tangent lines to these curves at \$(a, b, f(a,b))\$. Then the Tangent Plane on \$S\$ to the point \$(a, b, f(a,b))\$ is the plane that contains both \$T_1\$ and \$T_2\$. Like tangent lines for curves; the tangent plane on the surface \$S\$ at \$(a, b, f(a,b))\$ best approximates \$S\$ at \$(a, b, f(a,b))\$. Another important property of the tangent plane at a general point \$(a, b, f(a,b))\$ is that any curve \$C\$ that lies on \$S\$ and passes through \$(a, b, f(a,b))\$ will have its tangent lie on the tangent plane.