0
$\begingroup$

Let C be a circle of radius r centered at the origin in the XY-plane that is traced once counterclockwise. Provide an example of a vector field that would yield a positive value when computing a line integral over this curve.

I can't imagine how it would work if possible come with an example and the explanation.

$\endgroup$
0
$\begingroup$

An obvious example is $$ \vec F = - y \hat i + x \hat j$$ enter image description here If you would plot the $\vec F$, it would be vortex field with positive curl at the origin, so this implies if body is displaced along the circle the dot product between the tangent vector and the force would be positive as the angle between them would be zero, hence the work done would be positive.

If parametric form of the circle is $(\cos \theta, \sin \theta)$ ,then required integral is $$ W = \oint_C \vec F \cdot dr = \oint_C \vec F \cdot r'( \theta) d \theta = \int^{2 \pi}_0 (\sin^2 \theta + \cos^2 \theta ) d\theta = 2\pi $$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.