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.

Homework Statement

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]

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

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

 

[dimension10]

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

Homework Statement

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?

Because the curve is parallel to the xy plane.

 

[DryRun]

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?

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

 

[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]

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

Yes.

 

[dimension10]

what about d sigma?

d sigma is dx dy.

 

[dimension10]

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

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

 

[DryRun]

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

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]

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.

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

 

[robertjford80]

d sigma is dx dy.

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

 

[DryRun]

$d\sigma$ 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]

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

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 $s=x,t=y$