Let S be the surface defined by p(x, y, z) = 0 where z = f(x,y); i.e. z is a function of the variables x and y. (i) Let P be an arbitrary point on S with position vector p. Show that the equation of the tangent plane to the surface S at p is given by Vo(x-p) = 0 where x = xi+yj+zk is the position vector of an arbitrary point in R³. Hint: Start with the usual equation of the tangent plane at the point P = (a, b, f(a, b)) on the surface z = f(x, y): : - f(a, b) = = af -(a, b) (x − a) + ?x af ду -(a, b) (y — b)
Let S be the surface defined by p(x, y, z) = 0 where z = f(x,y); i.e. z is a function of the variables x and y. (i) Let P be an arbitrary point on S with position vector p. Show that the equation of the tangent plane to the surface S at p is given by Vo(x-p) = 0 where x = xi+yj+zk is the position vector of an arbitrary point in R³. Hint: Start with the usual equation of the tangent plane at the point P = (a, b, f(a, b)) on the surface z = f(x, y): : - f(a, b) = = af -(a, b) (x − a) + ?x af ду -(a, b) (y — b)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:(a) Let S be the surface defined by p(x, y, z)
function of the variables x and y.
(i) Let P be an arbitrary point on S with position vector p. Show that
the equation of the tangent plane to the surface S at p is given by
=
0 where z = f(x, y); i.e. z is a
z — f(a, b)
-
Vo. (x − p)
=
= 0
where x = xi+yj+zk is the position vector of an arbitrary point in
R³.
=
Hint: Start with the usual equation of the tangent plane at the point
P = (a, b, ƒ (a, b)) on the surface z = f(x, y):
af
af
(a, b) (x − a) + (a, b) (y - b)
ду
əx
(ii) Lety: R→→→→ R³; be a curve in R³ that lies on the surface S and passes
through the point P on S, i.e. there exists a to such that y(to) = P.
Show that the tangent vector y'(to) to the curve at P lies in the tan-
gent plane to the surface S at the point P.
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