Problems in Mathematical Analysis.pdf - pwp.net.ipl.pt

Sec. 11]_The Tangent Plane and the Normal to a Surface_217
1980. Transform the equation
putting u = x+y, v = x y, w = xyz, where w=w(u, v).
Sec. 11. The Tangent
Plane and the Normal to a Surface
1. The equations of a tangent plane and a normal for the case of explicit
representation of a surface. The tangent plane to a surface at a point M
(point of tangency) is a plane in which lie all the tangents at the point M to
various curves drawn on the surface through this point.
The normal to the surface is the perpendicular to the tangent plane
at the
point of tangency
If the equation of a surface, in a rectangular coordinate system, is given
in explicit form, z f (x, y), where f (x, y) is a differentiate function, then
the equation of the tangent plane at the point M (x , f/ , z ) of the surface is
z-*o=/i(* .
0o)(X-*o)
+ /i(*o, )0 r
-0o). (i)
Here, z f (x , t/ ) and X, K, Z are the current coordinates of the point of
the tangent plane.
The equations of the normal are of the form
where .Y, F, Z are the current coordinates of the point of the normal.
Example 1. Write the equations of the tangent plane and the normal to
the surface z = y 2 at the point M (2, 1,1).
Solution. Let us lind the partial derivatives of the given function and
their values at the point M