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| − | ''to a surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092180/t0921801.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092180/t0921802.png" />'' | + | {{TEX|done}} |
| | + | ''to a surface $S$ at a point $M$'' |
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| − | The plane passing through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092180/t0921803.png" /> characterized by the property that the distance from this plane to a variable point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092180/t0921804.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092180/t0921805.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092180/t0921806.png" /> approaches arbitrarily close to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092180/t0921807.png" /> is infinitesimally small as compared to the distance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092180/t0921808.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092180/t0921809.png" /> is given by an equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092180/t09218010.png" />, then the equation of the tangent plane at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092180/t09218011.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092180/t09218012.png" />, has the form | + | The plane passing through $M$ characterized by the property that the distance from this plane to a variable point $M_1$ of $S$ as $M_1$ approaches arbitrarily close to $M$ is infinitesimally small as compared to the distance $MM_1$. If $S$ is given by an equation $z=f(x,y)$, then the equation of the tangent plane at a point $(x_0,y_0,z_0)$, where $z_0=f(x_0,y_0)$, has the form |
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| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092180/t09218013.png" /></td> </tr></table>
| + | $$z-z_0=A(x-x_0)+B(y-y_0)$$ |
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| − | if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092180/t09218014.png" /> has a total differential at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092180/t09218015.png" />. In this case, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092180/t09218016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092180/t09218017.png" /> are the values of the partial derivatives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092180/t09218018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092180/t09218019.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092180/t09218020.png" />. | + | if and only if $f(x,y)$ has a total differential at the point $(x_0,y_0)$. In this case, $A$ and $B$ are the values of the partial derivatives $\partial f/\partial x$ and $\partial f/\partial y$ at $(x_0,y_0)$. |
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Latest revision as of 10:22, 12 April 2014
to a surface $S$ at a point $M$
The plane passing through $M$ characterized by the property that the distance from this plane to a variable point $M_1$ of $S$ as $M_1$ approaches arbitrarily close to $M$ is infinitesimally small as compared to the distance $MM_1$. If $S$ is given by an equation $z=f(x,y)$, then the equation of the tangent plane at a point $(x_0,y_0,z_0)$, where $z_0=f(x_0,y_0)$, has the form
$$z-z_0=A(x-x_0)+B(y-y_0)$$
if and only if $f(x,y)$ has a total differential at the point $(x_0,y_0)$. In this case, $A$ and $B$ are the values of the partial derivatives $\partial f/\partial x$ and $\partial f/\partial y$ at $(x_0,y_0)$.
For references see Tangent line.
How to Cite This Entry:
Tangent plane. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tangent_plane&oldid=14114
This article was adapted from an original article by BSE-3 (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article