Question

How can the integration formula for circulation be multiplied by the unit tangent vector "T" if for example circulation were to represent fluid flow? Isnt that supposed to be a velocity vector instead?

Answer 1

??

Answer 2

what multiplication ... dot product or cross product??
v must be given .... eg \( v = <-y, x>\) or some path \( \vec r = <x(t),y(t)>\) as function of time be given.
\( \oint \vec v.\hat{n} ds\) gives the flux though the path
\( \oint \vec v.\hat{t} ds\) gives the ....(not sure) along the path, if v were F ... it would be work done along moving the path!!

Answer 3

Maybe that wasnt the right question. My tutor/math professor told me in a way that line integrals describe mass and when i saw how line integrals are used to calcluate work along a curve from point A to B that made sense to me. \[\int\limits_{a}^{b} F *T dr\] where F is intrerpeted as force and T as the distance or arc length of a small interval. This all makes sense when describing work but how can it also desribe a flow integrals when they are bacially "motion?"

Answer 4

to add on from my last question, i am ust wondering as to why T is used instead of magnitude to measure circulation's movement?
I am thinking it is because F can either be used as a force or as a velocity vector?

Answer 5

@experimentX way off topic here...how did you make the vector arrow and ^ symbol over your letters? not being able to do this is SO annoying when addressing vector based queries. thanks!

Answer 6

@eseidl \hat {} ..

Answer 7

testing...
\hat{T}

Answer 8

\[\hat{T}\]

Answer 9

ah...sweet

Answer 10

and the arrow?

Answer 11

\vec {}

Answer 12

\[\vec{r}\]awesome :) thanks alot man

Answer 13

From the definition of mechanics \( \vec F * \vec r \) gives the work done ... i.e. work done is defined as the product of Force to the component of displacement along it.

Since we are moving in a certain path ... in a field, \( \vec F .\hat t\) gives the component of Force along the the path ... and \( \vec F .\hat t ds \) gives the work done ... and the closed loop integral gives the work done in moving all along the path!!