Finding the n in stokes theorem.

[Mdhiggenz]

Homework Statement

Hey guys,

I'm having trouble finding the n in stokes theorem.

For example,

F(x,y,z)= z²i+2xj-y³; C is the circle x² + y²=1 in the xy-plane with counterclockwise orientation looking down the positive z-axis.

∫∫CurlF*n

I know the curl is -3y²i+2zj+2k

The book found that n=k and just multiplied it out. I don't understand where they got that value.

Thanks

Homework Equations

The Attempt at a Solution

 

[LCKurtz]

Homework Statement

Hey guys,

I'm having trouble finding the n in stokes theorem.

For example,

F(x,y,z)= z²i+2xj-y³; C is the circle x² + y²=1 in the xy-plane with counterclockwise orientation looking down the positive z-axis.

∫∫CurlF*n

I know the curl is -3y²i+2zj+2k

The book found that n=k and just multiplied it out. I don't understand where they got that value.

Thanks

Imagine grabbing the z axis with your right hand with your thumb pointing up the axis. Are you fingers going counterclockwise when you look down at them. If so, your thumb is pointing the direction of the normal, which in this case is k. It's the "right-hand rule".

 

[Mdhiggenz]

Thanks for the response, Let's say it was moving in the negative z direction, thus going counterclockwise would it just be -k? Also the book doesn't seem to explain that whole topic very well.

 

[LCKurtz]

Thanks for the response, Let's say it was moving in the negative z direction, thus going counterclockwise would it just be -k? Also the book doesn't seem to explain that whole topic very well.

? A particle going in the negative z direction is not going counterclockwise or clockwise. It is going in a straight line.

The right hand rule states that if you have motion around a closed plane curve and you point your right hand fingers along the curve in the direction of motion, your thumb will point in the direction of the corresponding normal to orient the surface for Stokes' theorem.