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Tangent Plane

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Let $ S$ be a surface implicitely given by the equation

$\displaystyle f\left(x_1,\ldots,x_n\right) = c . $

Suppose that $ f$ is differentiable. Then the tangent plane to $ S$ at $ (p_1,\ldots,p_n)$ has the equation

$\displaystyle E :\quad \left(\operatorname{grad}f(p)\right)^{\operatorname t}\,(x-p) = 0 \,, $

provided $ \left\vert \operatorname{grad}f(p)\neq 0\right\vert$. Note that the normal vector is parallel to $ \operatorname{grad}f$.
\includegraphics[width=0.6\linewidth]{a_tangentialebene2}
In particular for the graph of a function $ y = g\left( x_1 \,, \ldots \,, x_{n-1} \right)$ the tangent plane at $ \left( q_i \,, \ldots \,, q_{n-1} \,,g(q)\right)^{\operatorname{t}}$ has the equation

$\displaystyle E : \quad y-g(q) = \sum_{i=1}^{n-1} \partial_i g(q) \left( x_i-q_i \right) \ . $

The partial derivative $ \partial_i g$ gives the slope of the tangent plane into the direction of the $ i$-th coordinate axis.
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Examples:

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  automatically generated 8/ 4/2008