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Mathematics-Online problems:

Interactive Problem 535: Potential Exercise with Parameter

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Given the curves $ C_1,\,C_2$ with the parametrization

\begin{displaymath}\begin{array}{rlll} & C_1(t)=(\cos t, t^2 - \pi t, \sin t)^t... ...nd} & C_2(t) = (t,2,1)^t & \text{with} & t\in[0,1] \end{array}\end{displaymath}

and the vector field $ g$ with $ g(x,y,z)=(y^2z^3,\alpha xyz^3,3xy^2z^2 + \alpha )^t$ with the parameter $ \alpha \in \mathbb{R}$.

Calculate $ \operatorname{rot} g$:

$ \operatorname{rot} g=$ $ \Big($$ xyz^2+$ $ \alpha xyz^2$ , , $ yz^3+$ $ \alpha yz^3\Big)^t$

Calculate $ C_1'(t)$ and $ C_2'(t)$:


$ C_1'(t)=$ $ \Big($$ \sin t$ , $ t-\pi$ , $ \cos t\Big)^t$          $ C_2'(t)=$ $ \Big($ , , $ \Big)^t$

Calculate the integral $ I$ = $ \int\limits_{C_2} g\, \mathrm{d}x$.

$ I= $.

Calculate the value of the parameter $ \alpha$, for which the vector field has a potential:

$ \alpha =$ .

Find the potential function $ u$ corresponding to that $ \alpha$:

$ u=$ $ xy^2z^3+$ $ y+$ $ z$

and calculate for this $ \alpha$ the value of the integral $ I$ = $ \int\limits_{C_1} g\, \mathrm{d}x$:

$ I= $.


   

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  automatically generated: 8/11/2017