## gradient, unit normal in vector calculus

1. The problem statement, all variables and given/known data

there is a surface xy3z2=4. What is the unit normal to this surface at a pt in the surface (-1,-1,2)??

2. Relevant equations
what is a unit normal to a scalar region? how can it be calculated?

3. The attempt at a solution
i calculated the gradient (del operator) of this surface at the given point to be 3i-12j+4k . this is in the direction of the unit normal along which max. rate of increase occurs. So far so good. But how dowe get the unit normal from this ? i dont know how to go further.

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 here, all points in the surface has a constant value =4, irrespective of the location of the points in the surface. does this have something to do with the calculation of the unit normal? it does sound absurd..but i'm equally blank..so just throwing all that's on my mind.
 by the way, i'm writing down the gradient of the surface as i calculated it, so that any mistakes i've made can be corrected too. $\nablaS$ = y3z2i+3xy2z2j+2xy3zk. now,$\nablaS$ at the point (-1,-1,2) = 3i-12j+4k the unit normal as i got is 3i-12j+4k ----------------------------- 13 but my textbook says it is (-1/rt 11, -3/rt 11, 1/rt 11) i've no idea how to arrive at this one .. any help is appreciated

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## gradient, unit normal in vector calculus

There's a theorem that says whenever you have a surface defined by f(x,y,z) = c, where c is a constant, the normal to the surface is given by grad f. This is exactly the case you have here, so taking the gradient will indeed give you the normal.

The reason you're not getting the right answer is that you simply calculated the numbers wrong. The x-component, for example, should be (-1)322=-4, not 3. (I didn't check the rest.)

 Recognitions: Homework Help Hi msslowlearner! When I calculate your gradient I get a different result. Your gradient formula is correct, but filling in (-1, -1, 2) should give a different result. Furthermore, your problem statement asks for a unit normal. This means your vector should have length 1.

 Quote by I like Serena Hi msslowlearner! Furthermore, your problem statement asks for a unit normal. This means your vector should have length 1.
i dont think it should have length 1. its the vector/mod vector, which is not 1. its the smallest multipliable unit of the vector, when multipled by the length of the vector gives the actual vector .

 yes, the x-component was wrong. now, i get the gradient as -4i -12j +4k whose magnitude is sqrt(176). if the unit normal vector is supposed to be (gradient/magnitude of gradient), i still am not arrivng at the answer !!
 hang on hang on !! i got it .. simple math.. sqrt(176) = 4*sqrt(11). sorry guys.. my mistake.. thanks for the help :)

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 Quote by msslowlearner i dont think it should have length 1. its the vector/mod vector, which is not 1. its the smallest multipliable unit of the vector, when multipled by the length of the vector gives the actual vector .
Huh?
I don't understand you. Could you elaborate?

Note that your textbook says: (-1/rt 11, -3/rt 11, 1/rt 11)
$$(\frac {-1} {\sqrt {11}}, \frac {-3} {\sqrt {11}}, \frac {1} {\sqrt {11}})$$