Question

ques

Answer 1

Consider a surface given by
\[\phi(x,y,z)\]
\[x^2+y^2+z^2=9\]
Now the normal to the surface is given by
\[\vec \nabla\phi=2x\hat i+2y\hat j+2z\hat k\]
Now my question is where did the 9 go, I know if u apply the operator to it will become 0 but shouldn't it be something like
\[2x\hat i+2y\hat j+2z\hat k=\vec 0\]

Answer 2

We may think of the given surface \(x^2+y^2+z^2=9\) as a specific level surface for the level set of function : \( \phi(x,y,z)=x^2+y^2+z^2\), namely, \(\phi=9\).

Answer 3

Next we use the fact the gradient of a function is perpendicular to the level surface to find a normal vector of given surface

Answer 4

"gradient" operates on a function

doesn't make much sense to say "gradient of an equation"

Answer 5

Yes, but shouldn't u also apply operator on the right side ??

Answer 6

we're simply using the fact that \(\nabla \phi \perp (\text{level surface}) \) to get a normal vector

Answer 7

Let me think about it lol

Answer 8

I think your equation is also correct as the gradient of \(g(x,y,z)=0\) is the 0 vector

Answer 9

In case you want to see it geometrically, may be consider a function of two variables, sketch the level curves, compute the gradient and verify...

Answer 10

gradient is only perpendicular at any point of the surface if it's a level surface right?

Answer 11

Lets say, the function f(x,y) is defined for all real values of ordered pairs (x,y)

then \(\nabla f(a,b)\) is perpendicular to the specific level curve that passes through the point \((a,b)\)

Answer 12

I think I understand it better now,
you can only find gradient of a function,
the function is
\[x^2+y^2+z^2\]
Doesn't make sense to find gradient of an equation
We're really just using the fact
that gradient is perpendicular to the level surface....

Answer 13

as long as the curve is level

Answer 14

Nice! It will be more illuminating to see these geometrically...

Answer 15

But I don't understand the point of level surface.
Suppose a point P on a level surface, if the value of the function at the point P is the same as the value of any other point on the surface, it's a level surface, but of course it will be same, because all points on the surface satisfy the equation of the surface

Answer 16

Right, as you move on a specific level surface, the value of function remains constant.

The gradient vector changes accordingly to remain perpendicular to the surface.

Answer 17

can u give an example of a non level surface??
even if we take
\[\phi(x,y,z)=x+c\]
where c is a constant
\[\phi(x,y,z)-x=c\]
This in itself defines a new function which makes it a level surface
\[\psi(x,y,z)=c\]
Now the value of the function at all points will be same and equal to c because they all lie on the surface
it seems as if there is nothing like a "non level surface"

Answer 18

you get level surface by fixing the value of the function, so every intersection of function and a horizontal plane is a level surface

Answer 19

maybe i shouldn't say that, because, that makes sense only for a function of 2 variables..

Answer 20

You're right, any equation in "n" variables can be thought of as a "level surface" of a function of "n" variables. The function itself requires "n+1" axes to visualize, but the level surface requires only "n" axes.

Answer 21

What i'm saying is that for a level surface we have the form
\[\phi(x,y,z)=c\]
even if we have an equation such that
both left and right side are entirely made of variables we can always rearrange to have all variables on one side and a constant on the right side
\[x^2+y^2+z^2=xyz\]\[\frac{x^2+y^2+z^2}{xyz}=1\]
of the form
\[\phi(x,y,z)=c\]
So it seems like no matter what we can't have a non level surface ??

Answer 22

RIght, you can always define a function such that your surface belongs to its level set

Answer 23

Thats how we use gradient to compute a normal vector of ANY surface

Answer 24

oh ok I understand now, so it's a level surface for
\[\psi(x,y,z)=\frac{x^2+y^2+z^2}{xyz}\]
but not for
\[\phi(x,y,z)=x^2+y^2+z^2\]

Answer 25

Yep! the connection between gradient and normal vector is with level surfaces

Answer 26

It depends on our function, if it's level or not !!

Answer 27

Hmm that sounds okay I guess!
but honestly i never heard that phrase "if its level or not" before haha!

Answer 28

thanks:)

Answer 29

np:) I think you will like this video...
http://ocw.mit.edu/courses/mathematics/18-02-multivariable-calculus-fall-2007/video-lectures/lecture-12-gradient/

when you're free try watching..

Answer 30

Yeh actually my syllabus is really messed up, they are teaching us vector calculus without teaching us about multivariable functions or partial derivatives

For example
\[ d(f(x,y,z))=\frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy+\frac{\partial f}{\partial z}dz\]
I didn't even knew what this meant and we were using it, I had to look it up separately to understand it

Answer 31

Ahh then I think those lectures really supplement your course of vector calculus. Except for the french accent, Prof. Denis Auroux is really good !

Answer 32

thx ill check it out

Answer 33

.

Answer 34

i spent ages chasing my tail on this so i hope this might be helpful - as i did find some peace of mind, eventually. [at the very least, i hope it makes some sense.]

strictly speaking, you do not apply the del operator to the RHS, only the LHS, ie only to \(\phi\)

because you have defined \(\phi(x,y,z)\) to be constant, you can say that \(d \Phi = 0\) **provided** you remain within the constraint \(\phi(x,y,z) = c\) , that is you limit yourself to x,y,z 's on the so-called level surface. that is why you created the level surface in the first place.

so, for the paraboloid \(z = x^2 + y^2\), you create the level surface \(\phi = z - x^2 - y^2 = 0\) knowing that \(d \Phi = 0\) without needing to do any further analysis provided x,y,z are related by \(z = x^2 + y^2\)

[which, BTW, is completely different from the scalar field \(\phi = z - x^2 - y^2 \) where there is no constraint on x,y,z values you choose, and where \(\vec \nabla \phi\) just gives you the direction/scale of greatest change ]

then, because \(d \Phi = \phi_x dx + \phi_y dy + \phi_z dz\)

it follows that \(\phi_x dx + \phi_y dy + \phi_z dz = 0\))

\(\implies <\phi _x,\phi _y, \phi _z > \bullet <dx,dy,dz> = 0\)

and again because of the constraint \(\phi(x,y,z) = c\), you have \(\phi(x+dx,y+dy,z+dz) = c\) so <dx,dy,dz> runs along the surface and is the tangent vector, so \(\vec \nabla \phi \) must be the normal vector.

this might sound trivial, but if you moved the paraboloid 5 units up the z axis as \(z = x^2 + y^2 + 5\), you could say

\(\phi = z - x^2 - y^2 = 5\)

or

\(\phi = z - x^2 - y^2 -5 = 0\)

and by definition, because of the constraint, \(d \phi = 0\)

and then apply the operator to the LHS

with the same result


At least that's how i have come to see it...