User:Guy vandegrift/T/LineIntegralFormula

Do not delete: Transcludes into Physics equations $$\vec F \cdot d\vec\ell = F_xdx+F_ydy = F_r dr + F_\theta rd\theta$$

$$\vec F \cdot d\vec\ell = \left(F_x\hat i + F_y\hat j\right)\cdot \left( \hat i dx + \hat j dy\right)$$ $$= F_x dx (\hat i\cdot \hat i) + F_y dy (\hat j\cdot \hat j) + (junk)(\hat j \cdot \hat i)$$

where $$\hat i\cdot \hat i =\hat j\cdot \hat j = 1$$ and $$\hat i\cdot\hat j = 0$$.

For polar coordinates, use $$d\vec\ell = \hat r dr + rd\theta \hat \theta$$