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Calculus and Beyond Homework Help
Vector line integral notation.
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[QUOTE="LCKurtz, post: 4302703, member: 198114"] Suppose you have a force field ##\vec F(x,y,z)## and a curve parameterized by ##\vec r(t) =\langle x(t),y(t),z(t)\rangle,\ a\le t \le b##. Since ##\frac {d\vec r}{dt}## is parallel to the curve, if you want to calculate the work done by the force moving along the curve you would calculate the integral$$ W=\int_a^b \vec F(x(t),y(t),z(t))\cdot \frac{d\vec r}{dt}\, dt =\int_a^b \vec F(x(t),y(t),z(t))\cdot \langle \frac{dx}{dt},\frac {dy}{dt}\frac{dz}{dt}\rangle\, dt$$ This is sometimes written in the differential form, using ##\frac{d\vec r}{dt}dt=d\vec r## and ##\frac{dx}{dt}dt = dx## etc, as ##\int_C\vec F \cdot d\vec r##. You can think of ##d\vec r = \langle dx, dy,dz\rangle##. Whatever notation you use, remember that it means ##\int_a^b\vec F \cdot \frac {d\vec r}{dt}\, dt##. For surface integrals, it is a good idea to use capital "##S##" as in ##d\vec S = \hat n\cdot dS## so as not to confuse arc length notation with surface area notation. [/QUOTE]
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Vector line integral notation.
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