# vector normal of a surface

Discussion in 'Math' started by kokkie_d, Dec 7, 2010.

1. ### kokkie_d Thread Starter Active Member

Jan 12, 2009
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Hi,

A quick question:
Why is the vector normal of a surface or a line important?
What does it signify?

Cheers,

2. ### Georacer Moderator

Nov 25, 2009
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1,289
Typically only surfaces have normal vectors, lines don't. Maybe in some degenerate (can I use this word in that case?) cases, but these are special.

It generally declares the surface's direction at that point in space and is very useful in physics (involved in flux equations), math (in stereometrical geometry), mechanics (for static loads and torque calculations) and many more fields.

You will typically find it as a factor in an exterior or inner product.

3. ### Papabravo Expert

Feb 24, 2006
11,783
2,496
I don't quite see how an inner product(dot product), which produces a scalar from a pair of vectors can give rise to a vector which is normal to a surface.

The outer product of two vectors produces a tensor and again I fail to see the relationship of the tensor to a normal vector.

The cross product of two vectors does produce a vector which is normal to both of the original vectors. So is this what you meant?

4. ### Georacer Moderator

Nov 25, 2009
5,181
1,289
Maybe we are a bit lost in translation and I don't vow for it this time. Let's take things from the beginning:
Is this what a normal vector is about?
http://en.wikipedia.org/wiki/Normal_vector
If the answer is yes, then I 'm in a good way.

Is this an example of an inner product?
$\Phi _m=A \cdot B$ - Magnetic flux through a surface
If yes I 'm still on a good track.

Is this an example of an exterior product?
$\hat{n} \times (\overline{E_2} - \overline{E_1})$ - identity for the electrical field on separating surface
If yes then I don't see any problem with my post.

If there's a no in the above, please elaborate and give me the correct translation.

5. ### kokkie_d Thread Starter Active Member

Jan 12, 2009
72
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mmm maybe I had to be a bit more specific.

I am reading a paper entitled: stability analysis of the continuous conduction mode buck converter via filippov's method

in this paper the vector normal of the hypersurface is defined as:

gradient of a function which is reached to via the partial derivation of the function (off course)

The normal vector is used in the saltation matrix.

I only have access to the paper and can not find enough background on the saltation matrix and the meaning of the normal vector.

Maybe "meaning" is not the right word. I have the same problem with the flux in induction motors; using the d-q notation one can find the flux in the stator and rotor and as a consequence optimally control the torque. For some reason I have difficulty understanding the use of the vector normal and the reasons for their existence?

Does this make sense?

I mean in math terms yeah I understand but its implications in the real world or translation in to the real world I find difficult.

Maybe someone has some helpful words for me or an new explanation that does make sense to me?

6. ### Papabravo Expert

Feb 24, 2006
11,783
2,496
We're good for the normal vector

We're good for the dot product or inner product except for the fact that it is a SCALAR quantity, not a VECTOR quantity. As a scalar quantity there is no direction and thus ther is no sense of direction -- normal parallel or otherwise.

Where we go off the tracks is the thing you call an exterior product matches what I call a cross product. A cross product is a vector and it is perpendicular (normal) to the two operands.
http://en.wikipedia.org/wiki/Cross_product

What I call an exterior product is something else again. The two operands are vectors, but the result is a TENSOR. TENSORS are sometimes represented as a matrix in three dimensions, but they also exist in higher dimensions as well.
http://en.wikipedia.org/wiki/Outer_product

7. ### Georacer Moderator

Nov 25, 2009
5,181
1,289
As for the dot product, sure, I didn't renounce that the dot product is scalar. I just said that the normal vector is involved in a dot multiplication as a factor, not as a product.

As for the other case, thanks for the correction. My english is usually very fluent, but sometimes when it comes to jargon, I fall short due to the lack of need for such vocabulary. I have confirmed through your link that "outer product" is what I meant to say.

8. ### krishna chaitanya New Member

Nov 8, 2010
5
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gradient of the surface gives the normal to the surface at that point....well u all know about it...
(cosa, cosb, cosc) be the unit vector..
what we have to see is the tangent plane to the surface there....
its xcosa+ycosb+zcosc= K(constant say... u will get this bye dot product condition)...
hope this will help..

9. ### Redbelly98 New Member

Jan 16, 2010
5
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The dot product of the normal vector (to a surface) with some vector field gives the flux of that vector field through the surface. Even though it is a scalar, it has a sign (+ or -). If it's positive then the flux is outward from the surface, and if negative then the flux is into the surface.

This thread is about why the surface normal is useful. One use is to calculate flux which, yes, is a scalar quantity.