I Physical significance of integral of F cross dr

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1. May 1, 2016

arpon

In the vector calculus course, I calculated integrals like,
$\int \vec F \times \vec{dr}$
Does this kind of integrals have physical significance or practical application other than Biot-Savart's Law?

2. May 1, 2016

Andrew Mason

In Newtonian mechanics the integral represents torque or rate of change of angular momentum.

The cross-product as a mathematical tool was invented probably because of its practical application. Although its development as a mathematical tool was post-Newton, it is very useful in mechancs.

AM

3. May 1, 2016

arpon

Isn't torque defined as $\vec r \times \vec F$ ?

4. May 1, 2016

PeroK

$\vec r \times \vec F = - \vec F \times \vec r$

5. May 1, 2016

Andrew Mason

Yes. That is just a convention. The difference is the sign or direction of the torque vector.

AM

6. May 1, 2016

arpon

That was not my point. $d \vec r$ represents infinitesimal change in position vector, while $\vec r$ represents position vector. Could you please give me a practical example where the net torque is calculated by $\int \vec F \times d \vec r$ ?

7. May 1, 2016

PeroK

The line integral doesn't represent the torque on a body. The total torque on a body would be a volume integral:

$\vec{\tau} = -\int \vec F \times \vec r dV$

I'm not sure when you would use the line integral.

8. May 2, 2016

Andrew Mason

I see your point. PeroK is quite right that $\int \vec F \times d \vec r$ does not represent torque. I am not sure what it would represent. I also don't see how it applies even to the Biot-Savart Law.

AM