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Evaluate the integralJ J ( - 1)win(,v) dydz, wherc min(r,y) is the minimum value of and y: variable such that f" and 9 are continuous. Show Let f,9:R + R be f...

Question

Evaluate the integralJ J ( - 1)win(,v) dydz, wherc min(r,y) is the minimum value of and y: variable such that f" and 9 are continuous. Show Let f,9:R + R be functions of one that f(0) + 9(0) - f(2) _ 9(2) +2f'(2) + 29(0) K Koe) ~9"(y)) dydr = surface is defined by 2a cos 0 for 0 <0<{ Let a > 0. In spherical coordinates surface function of Find the volume of the solid enclosedl by the spacx PA(R) , with inner product Consider the iuner product Vp,q e P(R). (p,4) J Ar)ot)

Evaluate the integral J J ( - 1)win(,v) dydz, wherc min(r,y) is the minimum value of and y: variable such that f" and 9 are continuous. Show Let f,9:R + R be functions of one that f(0) + 9(0) - f(2) _ 9(2) +2f'(2) + 29(0) K Koe) ~9"(y)) dydr = surface is defined by 2a cos 0 for 0 <0<{ Let a > 0. In spherical coordinates surface function of Find the volume of the solid enclosedl by the spacx PA(R) , with inner product Consider the iuner product Vp,q e P(R). (p,4) J Ar)ot) basis from the busis proces construct An orthonormal Use the Gram-Schmidt {1,2,1} approximation R(R) to the pnrt (a), give the least squares Using your AnSWer function f(r) on the iuterval [0, 1]. the following result without proof: Mea



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Evaluate the line integral in Stokes" Theorem to evaluate the surface integral $\iint_{S}(\nabla \times \mathbf{F}) \cdot \mathbf{n} d S .$ Assume that n points in the positive z-direction. $\mathbf{F}=\mathbf{r} /|\mathbf{r}| ; S$ is the paraboloid $x=9-y^{2}-z^{2},$ for $0 \leq x \leq 9$ (excluding its base), where $\mathbf{r}=\langle x, y, z\rangle$

Problem Number 18. Let us be a surface CB. It's closed boundary and death. A vector field Stokes theorem says that the line and the surface integral are equal. In other words, we can say that the surface integral off the current off the vector field F N ds equals the integral off the line Integral left the d x to evaluate the surface integral. It is enough to evaluate the line integral and to evaluate the non integral. First, we compute f dot all their 15. The Cursi is the circle. Why square Plus that squared equals nine with a counterclockwise orientation with which to the polar coordinates y equals Why, honey, why equals three cool sinti and that equals three sign Okay and parameter realization is are off key equals zero three to sign t and three. So I'm g the 4th, 4th t bigger than would equal zero and less than or equal to pi. Our dash off key equals zero negative three signed t 70 and city cool sinti. The field victims in polar coordinates is f equals are over the norm of our have their 50 effect off History equals one third, but apply zero three co sovereignty and three sovereignty for F R. There's 50 equals. We now need to substitute by our like this equation multiple. Oy this equation she by this question okay, And we can get final valuables. Gee, usual coarsen G three Cosentino deployed zero negative city. You saw it. Jean. Three co societies is equal to negative three scientific co sovereignty plus three course I'm t both three pull sign P which is equal toe zero. And now the circulation integral the Asian from 0 to 2 pi off off. Tr national t equals the integration off Zero for the final value is off this integral zero.

Problem Number 17. Let s be a surface CB is closed boundary and F A vector Field Stokes theorem says that the line and the surface integral are equal. In other words, we can see that the surface integral off the girl with the vector field F and D s equals the line integral of F d. X to evaluate the surface integral. It is enough to evaluate the line integral and to evaluate the line integral. First, we compute as those orders of tea. The curve C is the ellipse export to over four plus why squared over nine, which is equal to one with a counterclockwise orientation we switched to the polar coordinates X equals to co sign T and why equals three shine T and the parameter ization is our t equals to co sign t three sign T and zero Ford G bigger than or equal zero and less than or equal to pi or the death of T equals negative toe Sigh Inti save Cool Sign T and zero The field factory important coordinate is f equals X y, and that, as for key equals to cosign t three sign T and zero f the orders of T equals this equation off ploy. This equation we get negative four close I in T science T plus nine cool sovereignty sign teeth, which is equal to five society. So, Yankee, the circulation integral is the integration from zero till two pi off have 50 that our national t d t. Which is this equation. If we interviewed this equation, we get 5/2 month supply scientists squared, and then we substitute by our end conditions to point and zero to get final value of this integral is zero.

In order to find the value for the service interval. We can also evaluate a line integral along the coop see of F dotted with de Gaulle. So it's given in the problem that oh surface is X squared over four plus y squared off with nine plus z squared equals wanting. It's the upper half of this will absorb. So in order to find a boundary coop you If we just take on X Y trace a curve is going to be equal to X squared over four plus y squared over nine equals one. Now when we evaluate this inter cool, we have to put everything in terms of one valuable because this is a single interval. So if we look at a coup, few we concedes pretty similar to the Pythagorean identity which is co sign squared of theta plus sine squared of theta is equal to one. So this means if we have ah x be equal to two co sine theta and know why be equal to three science data. Z, of course, is equal to zero as well. In the X y plane, we have successfully report motorized both x and Y and Z in terms of Fada as it matches equation too. So now we have a question for all which is equal to to co sign theta to sign data need zero three signed data and that means r D r di Fada. Do you think that is going to be equal to negative two sine theta three Coast data zero all Times D data. But now for F, it's given in the problem that f is equal to X comma. Why commas e? But because we have reporting which rise both x, y and Z here we can see that it's equal to two times coastline Fada three times sine theta and zero. This means ah, half guarded with D all is going to be equal to negative four Khost. They'd assign data plus nine Khost Theda sine theta Paul zero. Because your times your zero all times Day data. This, of course, is equal to five coast data sign data. So all that's left now is for us to evaluate the integral. So if we shrink this in order to get more room to maneuver, we have ah interval of five coast Fada signed beta de photo. And because we're taking it with respect to theta and we have to traverse this cove once counterclockwise will be going from zero to to pie. Now, if we do a simple you substitution here where u is equal to sine theta, it means d u is equal to co sign theta de Seita. We Can we wait? Oh Inter Goal as five. You do you? Of course, this means we have to change our limits. So you of two pi deal with a sign of two pi which is zero and you of zero sign is your which is also Dio So you get the integral from 0 to 0 a five year do you and this is equal to zero so there is no answer.

Mhm. Yeah, if it is imposed a lesson, Jacob Jacob had stigmatized the surface of the soul. G get taken out off. But please political the area Our deal towards this solution a person. Thanks. Like white Jake. Thank you. Taken the Indians and the flux. It's not that effort. That's a girl from Okay. Lets start every day. Oh, look. Quite computer agents. Yeah. Okay. Deep after that, we have used Yeah. Yeah. See? So if that will be the off duty with cheese off G equals Enjoy. Okay. Okay. Over the region. See? Eight. And yes, we have. That's this is a description question. Thank you.


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