What's the integral of a unit vector?

[MaestroBach]

Homework Statement

Determine the integral of phi hat with respect to phi.

Relevant Equations

N/A

So I'm trying to figure out the integral of phi hat with respect to phi in cylindrical coordinates. My assumption was that the unit vector would just pass through my integral... is that correct? (I reached this point in life without ever thinking about how vectors go through integrals, and google wasn't the best help).

 

[PeroK]

You'll need to define the integral you want to calculate. The polar unit vectors are functions of position, so you can integrate a unit vector along a curve.

 

[MaestroBach]

You'll need to define the integral you want to calculate. The polar unit vectors are functions of position, so you can integrate a unit vector along a curve.

If I perform a line integral in the direction of my unit vector, then what happens to the unit vector?

 

[PeroK]

If I perform a line integral in the direction of my unit vector, then what happens to the unit vector?

Please state your example mathematically. What you've written is too vague.

 

[haruspex]

Try expressing the unit vector in Cartesian coordinates but expressing its components in terms of φ.

 

[HallsofIvy]

If I perform a line integral in the direction of my unit vector, then what happens to the unit vector?

NOTHING happens to the unit vector. The unit vector in a given direction is a constant vector and doesn't change.

 

[WWGD]

Do you mean you are integrating a vector field with constant length 1? If so, what is the direction given? Is it constant?

 

[haruspex]

NOTHING happens to the unit vector. The unit vector in a given direction is a constant vector and doesn't change.

No, the direction of ##\hat\theta## changes along the curve.

Since the thread appears to have fizzled out, here's my answer, using my hint in post #5.
##\hat\phi=-\sin(\phi)\hat x+\cos(\phi)\hat y##
Since ##\hat x## and ##\hat y## are constant we can integrate immediately:
##\int\hat\phi.d\phi=[\cos(\phi)\hat x+\sin(\phi)\hat y]=\Delta\cos(\phi)\hat x+\Delta\sin(\phi)\hat y=\Delta \hat r##

 

[Delta2]

I believe something that relates to this thread: If I say that the integral of a constant in magnitude and tangential in direction vector along a closed curve is zero, am I correct?

 

[haruspex]

I believe something that relates to this thread: If I say that the integral of a constant in magnitude and tangential in direction vector along a closed curve is zero, am I correct?

And more generally, the integral of the unit tangential vector along a smooth path is the displacement from start to finish.