MATH2111 Higher Several Variable Calculus (2 Viewers)

InteGrand

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Re: Several Variable Calculus



I might be misinterpreting the question but I thought that they're basically asking for 1/4 * area of curve of intersection, which happens to be an ellipse through (0,0,8) passing through (0,8,4) and (8,0,4).

If I'm right, how do I use surface integrals to get to the answer of ? And if I'm wrong, how do I get back on the right path?




 
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leehuan

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Re: Several Variable Calculus

Yep makes perfect sense, thanks as always :)

Out of curiosity, in general does setting x and y to be the parameters make things easier? Or are there scenarios where you'd pick a different choice of parameters.
 

InteGrand

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Re: Several Variable Calculus

Yep makes perfect sense, thanks as always :)

Out of curiosity, in general does setting x and y to be the parameters make things easier? Or are there scenarios where you'd pick a different choice of parameters.


 

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Re: Several Variable Calculus



I might be misinterpreting the question but I thought that they're basically asking for 1/4 * area of curve of intersection, which happens to be an ellipse through (0,0,8) passing through (0,8,4) and (8,0,4).

If I'm right, how do I use surface integrals to get to the answer of ? And if I'm wrong, how do I get back on the right path?
Obtaining the value of the major radius is slightly tricky, but it's not too bad.

So, the major radius is the hyotenuse of a right-angled triangle running parallel to one of the co-ordinate planes with one of the sides trivially being of length 8.

Now, by parametrising the equation of the circle using trigonometry, it is easy to find the maximum and minimum values of the z-ordinate along boundary of the ellipse, via the method of Auxiliary Transformations (See: Harmonic Addition Theorem). These values turn out to be 8 ± 4√5, which trivially implies the other side length of the triangle is 4√5. By simple computation, the major radius turns out to be 12 units.

Thus, the area of the ellipse is 12 × 8 × π and so the quarter-area that is desired turns out to be 24 π square units.
 

leehuan

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Re: Several Variable Calculus

I'm going from cylindrical to spherical in this made-up question.



 

leehuan

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Re: Several Variable Calculus

Ah true, that's unfortunate. Forgot that pho follows a different rule for different values of phi. Would I be forced to splitting the integral up into three separate integrals then?
 

InteGrand

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Re: Several Variable Calculus

Ah true, that's unfortunate. Forgot that pho follows a different rule for different values of phi. Would I be forced to splitting the integral up into three separate integrals then?
Yeah, pretty much. Two of those integrals would be equal by symmetry, of course.
 

InteGrand

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Re: Several Variable Calculus

Imagine it as polar coordinates where the radius is e^x and angle is y ==> radius > 0 (and can attain any positive value) and angle can be anything ==> any point except (0, 0) is in the image (since any nonzero point can be expressed in polar coordinates with a strictly positive radius and some angle).
 
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leehuan

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Re: Several Variable Calculus



I was able to do this question correctly using the typical approach and the answer is sinh(1) - 1. Here's an approach that seems to be wrong.





Which line is the mistake in and why is it incorrect?
 

InteGrand

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Re: Several Variable Calculus



I was able to do this question correctly using the typical approach and the answer is sinh(1) - 1. Here's an approach that seems to be wrong.





Which line is the mistake in and why is it incorrect?
You ended up with the negative of your previous answer because you used the oppositely oriented normal.
 

leehuan

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Re: Several Variable Calculus

You ended up with the negative of your previous answer because you used the oppositely oriented normal.
Oh. In general, how do we determine which normal vector has the correct orientation?
 

leehuan

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Re: Several Variable Calculus





(Also ignore the previous question, even the lecturer agreed it was lacking information)
 
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InteGrand

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Re: Several Variable Calculus





(Also ignore the previous question, even the lecturer agreed it was lacking information)
The basic ideas are:

- When x = 0 or 1 (the endpoints of the domain), f_n (x) = 0 for every n, so the convergence at these x is clear.

- For any given x in (0, 1) (interior of the domain), the fact that exponential decay dominates power function growth yields the result.
 

leehuan

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Re: Several Variable Calculus

Got that part out.

 

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