# Tangent Plane

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## tangent plane

[′tan·jənt ′plān] (mathematics)

The tangent plane to a surface at a point is the plane having every line in it tangent to some curve on the surface at that point.

## Tangent Plane

The tangent plane to a surface *S* at a point *M* is the plane that passes through the point *M* and that is characterized by the property that the distance from this plane to the variable point *M’* on the surface *S* is infinitesimal in comparison with the distance *MM’* as *M* ’ approaches *M*. If a surface *S* has the equation *z* = *f*(*x, y*), then the equation of the tangent plane at the 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 the function *f*(*x, y*) has a total differential at thepoint (*x*_{0}, *y*_{0}). In this case, *A* and *B* are the values of the partialderivatives ∂*f*/∂*x* and ∂*f*/∂*x* at the point (*x*_{0}*y*_{0}) (*see*DIFFER-ENTIAL CALCULUS).