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kylahwells
23.08.2020 •
Mathematics
Use Stokes' Theorem to evaluate
∫
C
F � dr
where F(x, y, z) = x2yi + 1/3x3j + xyk and C is the curve of intersection of the hyperbolic paraboloid z = y2 ? x2
and the cylinder x2 + y2 = 1 oriented counterclockwise as viewed from above.
Find parametric equations for C,Let x and y be in terms of t where
0 ? t ? 2?
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Ответ:
Step-by-step explanation:
Given that:
Here C is the curve of intersection of the hyperbolic parabolic
and the cylinder ![x^2 +y^2 =1](/tpl/images/0727/5819/83db4.png)
Using Stokes' Theorem
From above ;
S = the region under the surface
and above the circle ![x^2+y^2 =1](/tpl/images/0727/5819/5a7e0.png)
Suppose, we consider![f(x,y,z) =z-y^2+x^2](/tpl/images/0727/5819/bdf08.png)
therefore, S will be the level curve of f(x,y,z) = 0
Recall that:
∴
This implies that the unit vector![n = \dfrac{\bigtriangledown f}{|| \bigtriangledown ||}](/tpl/images/0727/5819/c7000.png)
So![\bigtriangledown f =](/tpl/images/0727/5819/9456d.png)
Also,![|| \bigtriangledown f ||= \sqrt{4x^2+4y^2+1}](/tpl/images/0727/5819/c4387.png)
Similarly ;
Then:
converting the integral to polar coordinates
This implies that:
⇒![\int_C F . dr = 2(\theta) ^{2 \pi} _{0} \begin {pmatrix} \dfrac{r^4}{4}^ \end {pmatrix}^1_0](/tpl/images/0727/5819/20e8d.png)
Therefore, the value of![\int_C F . dr = \pi](/tpl/images/0727/5819/2d07f.png)
The parametric equations for the curve of intersection of the hyperbolic paraboloid can be expressed as the equations of the plane and cylinder in parametric form . i.e
Set them equal now,
the Parametric equation of![C : x = cost , y = sin t, z = sin^2 t - cos^2 t , 0 \leq t \leq 2 \pi](/tpl/images/0727/5819/c0911.png)
Ответ: