TITLE: Definition of a Tangent Plane
# We all have and intuitive idea of what the tangent plane to a point
on a given surface should be. It is the plane that goes through
that point and that it is tangent to the surface at that point.
Suppose

z = f(x,y)
is a smooth surface above the xy-plane (see the picture below).
The tangent plane at the point x=a, y=b is the plane that
contains all the tangent vectors (at the point) of curves
on the surface that pass through the point (a,b,f(a,b)).
!tplane.gif
# Another way to define it is to say that this is the plane
through the point (a,b,f(a,b)) which is perpendicular to
the NORMAL to the surface at that point. The NORMAL
to the surface at a point is just defined as the direction
perpendicular to all tangent vectors to the surface at
that point. Therefore all we need to do to find the equation
of the tangent plane is to find a normal to the surface at
that point. To find the normal notice that the position
vector of points on the surface are given by:
> r := (x,y) -> x*i+y*j+f(x,y)*k;
r := (x,y) -> x i + y j + f(x, y) k
# The tangent vectors to the coordinate curves r(x,b) and
r(a,y) (these are the dashed lines on the surface above)
when x=a and y=b are,
> 'fx(a,b)' = subs(x=a,diff(r(x,b),x));
/ d \
fx(a, b) = i + |---- f(a, b)| k
\ dx /
> 'fy(a,b)' = subs(y=b,diff(r(a,y),y));
/ d \
fy(a, b) = j + |---- f(a, b)| k
\ dy /
# These are the arrows tangent to the surface displayed in the above
picture. A normal to the surface must be perpendicular to these
two vectors so we can take the crossproduct of them to find it,
> N := crossprod([1,0,D1f(a,b)],[0,1,D2f(a,b)]);
N := [ - D1f(a, b), - D2f(a, b), 1 ]
# where we have denoted by D1f(a,b) the partial derivative of f
with respect to x evaluated at the point x=a, y=b. D2f(a,b)
is the partial derivative of f w.r.t. y at (a,b). These
are just the coefficients of "k" in the expressions for
fx(a,b) and fy(a,b) resp.
So we are done. The tangent plane must go through the point
and have N as a normal. We know from
our previous study of planes in 3D
that the equation of the plane is given by:
> TPlane := innerprod(N,evalm([x,y,z]-[a,b,f(a,b)])) = 0;
TPlane :=
- D1f(a, b) x + D1f(a, b) a - D2f(a, b) y + D2f(a, b) b + z - f(a, b) = 0
# The next three maple commands are just for displaying the above equation
in the usual form. Notice how you can move things around in an expression
with the commands, "isolate", "collect", "sort".
> readlib(isolate): isolate(TPlane,z);
z = D1f(a, b) x - D1f(a, b) a + D2f(a, b) y - D2f(a, b) b + f(a, b)
> collect(",[D1f(a,b),D2f(a,b)]);
z = D1f(a, b) (x - a) + D2f(a, b) (y - b) + f(a, b)
> sort(",f(a,b));
z = f(a, b) + D1f(a, b) (x - a) + D2f(a, b) (y - b)
# This is the equation of the tangent plane to the surface z=f(x,y)
at the point x=a,y=b.
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