Question

My physics professor just told me he could show me how to compute a curl without a coordinate system. Anyone know about this?

Answer 1

I thought curls \[\nabla \times \vec{F}(x_1,x_2,x_3)\] of some arbitrary vector field F had to have a coordinate system on those variables? How would we even interpret the curl with no coordinates?

Answer 2

Because then if you calculate things (for example, surface integrals) on them you need a means to parameterize a surface and all that.

Answer 3

I don't know what it means to calculate the curl of something that cannot be plotted via a coordinate system. Is there an example?

Answer 4

My problem is:
Is
\[\vec{F}=\frac{a}{r}\vec{e}_r; \vec{e}_r=\cos(\theta) \hat{i}+ \sin(\theta)\hat{j}\]
conservative?
And I asked if I should use r as the norm r=sqrt(x^2+y^2) and convert polar to cartesian or use the cylindrical coordinates with z=0? He said: "One does not need to explicitly use coordinates. But I'd need my chalkboard to show you that."

Answer 5

This is just a shot in the dark here, but Euler's method maybe? Or am I being dumb?

Answer 6

Sorry, I'm rapidly going from OS and actually studying bio at the same time >.>

Answer 7

If any vector field in polar coordinates has only an \( e_r \) component, it is conservative. That's not obvious, but for what it's worth, curl in polar coordinates has the following form and you can derive the result I just mentioned algebraically:

Answer 8

I'm pretty sure that constitutes as "explicitly using coordinates"?

Answer 9

I concede it.

Answer 10

My point is that you needn't go through the process of converting back to Cartesian coordinates.

Perhaps what this professor means is this: what curl intuitively means, and why in fact it's called curl. Curl measures the extent to which a vector field is or tending towards having loops, or "curls" around.

A vector field, such as F = (x,y,z) has no loops, no curling and indeed the curl of it is zero.

On the other hand, this vector field, F = (-y,x,0) loops around the z axis and in fact the curl of F is 2k.