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Mathematics-Online problems:

Interactive Problem 180: Line Integral, Potential

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Given the curve $ C$ with the parametrization $ C(t)=(\cos t, \sin t, \frac{t}{2\pi})$ with $ t\in[0,2\pi]$, and the vector field $ V$ with $ V(x,y,z)=(3x^2-y^2,-\alpha xy,-1)$ with the parameter $ \alpha \in \mathbb{R}$.


Calculate $ \mathrm{rot} V$:

$ \mathrm{rot} V\ =\ \displaystyle\Big($ , , $ \big($ - $ \alpha\big)y\displaystyle\Big)^{\operatorname t}$

Calculate $ C'(t)$:

$ C'(t)\ =\ \Big($ $ \sin t$ , $ \cos t$ , $ \big($ $ \pi\big)^{-1}\Big)^{\operatorname t}$

Calculate the integral $ I = \int\limits_C V \mathrm{d}r$.

$ \displaystyle\int\limits_0^{2\pi}$ $ \cos^2t\sin t$ + $ \sin^3t$ + $ \alpha\cos^2t\sin t$ + $ \Big($ $ \pi\Big)^{-1} ~\mathrm{d}t$

$ =$

There is a value of the parameter $ \alpha$, so that the vector field $ V$ has a potential. Find this $ \alpha$:

$ \alpha\ =\ $

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

$ u\ =\ $$ x^3$ + $ zx$ + $ y^2x$ + $ y^2$ + $ z$

and express in this case the value of the integral $ I$ using the potential:

$ I\ =\ $$ u\Big($ , , $ \Big)$ + $ u\Big($ , , $ \Big)$


   

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  automatically generated: 8/11/2017