# Homework Help: Finding the unit normal of a vector field

1. May 31, 2012

### robertjford80

I'm on the last chapter of a 1200 page calc book, I'm really psyched.

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

3. The attempt at a solution

The method I learned for finding the unit normal of a vector field, n, is take the derivative of the equation and divide that by the magnitude of the derivative. This technique works for questions 3 and 4 above, but for 1,2 and 5, the solution manual just says n = k. I can't figure out why n = k. What are they talking about?

Last edited: May 31, 2012
2. May 31, 2012

### robertjford80

I also need to know what d sigma = dxdy means.

3. May 31, 2012

### dimension10

dσ=dx dy simply means that the differential of area is equal to the differential of x multiplied by the differential of y.

4. May 31, 2012

### dimension10

Because the curve is parallel to the xy plane.

5. May 31, 2012

### sharks

$\hat n = \vec k$ means that the unit normal vector is parallel to the z-axis, in this case, it is in the vertically upward direction.

6. May 31, 2012

### robertjford80

how do you find n if the curve is parallel to the xy plane, does it just equal k?

7. May 31, 2012

### dimension10

Yes.

8. May 31, 2012

### dimension10

d sigma is dx dy.

9. May 31, 2012

### dimension10

Or in other words, the curve is parallel to the xy plane.

10. May 31, 2012

### sharks

n could be k or -k, depending on the orientation of the surface. n is the outward unit normal vector, and if the problem states "counterclockwise from above" then n is pointing upward, otherwise if it states ""counterclockwise from below"" then n = -k. If the surface lies in any plane (x=0, y=0 or z=0), then there are only 2 possible orientations for n, without any need for calculations, as you can deduce n directly.

If the surface lies in:
plane z=0, n is either k or -k
plane y=0, n is either j or -j
plane x=0, n is either i or -i

In all 3 cases, the positive value denotes moving along the positive direction parallel the respective axis, and similarly the negative value denotes moving in the negative direction parallel to the axis.

Last edited: May 31, 2012
11. May 31, 2012

### dimension10

The OP said that the problem says n=k, so n is just k.

12. May 31, 2012

### robertjford80

but what does d sigma refer to? how do you calculate it? what does it mean?

13. May 31, 2012

### sharks

$d\sigma$ refers to the differential area, which when expressed in terms of Cartesian coordinates, converts to dxdy, as you are projecting the surface onto the x-y plane.

14. May 31, 2012

### dimension10

As sharks has said, it is the differential of area.

$${\rm{d}}\sigma = \left\| {\frac{{\partial \vec r}}{{\partial s}} \times \frac{{\partial \vec r}}{{\partial t}}} \right\|{\rm{d}}s{\rm{ d}}t$$

In your case, $\left\| {\frac{{\partial \vec r}}{{\partial s}} \times \frac{{\partial \vec r}}{{\partial t}}} \right\| = 1$ and $s=x,t=y$