finding the unit normal of a vector field

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?

robertjford80

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

dimension10

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

dimension10

Because the curve is parallel to the xy plane.

sharks

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

robertjford80

what about d sigma?

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

dimension10

Yes.

dimension10

d sigma is dx dy.

dimension10

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

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.

dimension10

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

robertjford80

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

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.

dimension10

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

[tex]{\rm{d}}\sigma = \left\| {\frac{{\partial \vec r}}{{\partial s}} \times \frac{{\partial \vec r}}{{\partial t}}} \right\|{\rm{d}}s{\rm{ d}}t[/tex]

In your case, [itex]\left\| {\frac{{\partial \vec r}}{{\partial s}} \times \frac{{\partial \vec r}}{{\partial t}}} \right\| = 1[/itex] and ##s=x,t=y##