Re:

In article <43E76B27.9090803@xxxxxxxxx>,
JEMebius <jemebius@xxxxxxxxx> wrote:
Rob Johnson wrote:

In article <43E6028C.7010802@xxxxxxxxx>,
JEMebius <jemebius@xxxxxxxxx> wrote:
In article <43E6028C.7010802@xxxxxxxxx>,
JEMebius <jemebius@xxxxxxxxx> wrote:

iredshift@xxxxxxxxx wrote:

I am learning the method of characteristics for solving first order
pdes and I am trying to remember the formula for a surface normal to
the surface F = F(x1,x2,x3). I know that for 3d surfaces, the surface
normal to F = F(x1, x2) is given by:

( d/dx1 F, d/dx2 F, -1) = ( Fx1, Fx2, -1)

Does this formula extend to higher dimensions? I.e. Is the surface
normal to F = F(x1,x2,x3) given by

( Fx1, Fx2, Fx3, -1)?

Yes:
In 3D you give the explicit function z = f(x, y) to represent the
surface.
Now one has dz = df/dx . dx + df/dy . dy or as a dot product of vectors,
(df/dx, df/dy, -1).(dx, dy, dz) = 0

Exactly the same formula and reasoning hold - with due changes, of
course - in any number of dimensions.

Mr Rob Johnson says in his earlier reply (see below) essentially the
same in terms of implicit functions. In your problem the implicit
function is G(x, y, z) = f(x, y) - z = 0.
Strictly speaking, the gradient lies along the normal to a
(hyper)surface - it is a vector, it is not the normal itself.


============================================================================

I have been thinking about this and I am confused by what you say
here. The normal to a surface is just a vector that is perpindicular
to the surface at a given point. A normal does not need to be a unit
vector nor does it need to point inward or outward, unless specified.
Why is the gradient not "the" normal itself?

No wonder that one gets confused by the terminology.

Often the adjective "normal" is used without substantive, but rather as
a substantive itself. This word is then no longer unequivocal, and
indeed has different meanings depending on the context.

I did a little research into the concepts of normal line, normal vector,
unit normal vector.
[extensive research snipped]
I guess that with "normal" one means very often "normal vector" because
in practice normal vectors are more important than the normal line itself.

Perhaps this short article helps in putting away the confusion!

Yes, when I see normal in the context of differential geometry, I
usually think of normal vector instead of normal line, but I can see
how an unqualified reference to one could be interpreted as the other.

The line has an aesthetic appeal as there is only one normal line, but
there is a whole family of normal vectors. Furthermore, the vectors
bear no reference back to the point on the surface, whereas the normal
line at least intersects the surface at the point of normalcy.

Rob Johnson <rob@xxxxxxxxxxxxxx>
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