I am struggling with this question. So if the j and k components were reversed, it would be a Cylinder, but in this form I'm unsure.
I considered the different values of t (as it goes from 0 to pi) and how j is irrespective to t but aren't really sure what else to do from here or how to go about sketching the surface.
I am struggling with this question. So if the j and k components were reversed, it would be a Cylinder, but in this form I'm unsure.
I considered the different values of t (as it goes from 0 to pi) and how j is irrespective to t but aren't really sure what else to do from here or how to go about sketching the surface.
Whether or not you swap the j and k components, this isn't a cylinder. (It is a surface of revolution).
Swapping the components really shouldn't throw you. Suppose we have a cylinder of radius R and height H, with the axis of the cylinder in the direction of k.
r(θ,t)=Rcosθi+y=Rsinθj+tk, where 0≤θ<2π,0≤t≤H
Swapping j and k will just change this to a cylinder of radius R and height H, with the axis of the cylinder in the direction of j (which is hopefully no real surprise).
I am struggling with this question. So if the j and k components were reversed, it would be a Cylinder, but in this form I'm unsure.
I considered the different values of t (as it goes from 0 to pi) and how j is irrespective to t but aren't really sure what else to do from here or how to go about sketching the surface.
Thank you for both of your replies, I really appreciate it first and foremost. This really isn't a module I'm particularly strong at - I'm more of a Stats kind of guy, but somehow ended up here so please bear with me
So the a[1+(s/L)^2] would become a at s=0, so the curve would contract as s increases until s=0 and then would expand as s increases from 0 to h, back to when s=-h.
Thank you for both of your replies, I really appreciate it first and foremost. This really isn't a module I'm particularly strong at - I'm more of a Stats kind of guy, but somehow ended up here so please bear with me
So the a[1+(s/L)^2] would become a at s=0, so the curve would contract as s increases until s=0 and then would expand as s increases from 0 to h, back to when s=-h.
Yes. If you define b = 1/L^2, then a[1+(s/L)^2] is just a(1+bs^2), which is just a parabola.
I know it can be confusing when 2 people give different advice about a question, but to my mind the key thing is to recognize that what you have here is a surface of revolution. In general, when you can define the surface in the form r(s,t)=f(s)sinti+f(s)costj+sk (or some variation obtained by permuting i, j, k), you have a surface of revolution, and you can visualise the surface by imagining plotting f(s) against s to get a profile, and then rotating that profile 360 degrees around the k axis to make a "vase" shape. (It's the k-axis in the example I'm giving because that is the coordinate corresponding directly to s).
Yes. If you define b = 1/L^2, then a[1+(s/L)^2] is just a(1+bs^2), which is just a parabola.
I know it can be confusing when 2 people give different advice about a question, but to my mind the key thing is to recognize that what you have here is a surface of revolution. In general, when you can define the surface in the form r(s,t)=f(s)sinti+f(s)costj+sk (or some variation obtained by permuting i, j, k), you have a surface of revolution, and you can visualise the surface by imagining plotting f(s) against s to get a profile, and then rotating that profile 360 degrees around the k axis to make a "vase" shape. (It's the k-axis in the example I'm giving because that is the coordinate corresponding directly to s).
I am still slightly unsure as to how to do this.
Plotting f(s) against s gives a parabola crossing at f(0)=a with f(-h) = f(h). How do I rotate around, in my case, the j, axis?
Sorry, I just took them on my phone and that's how they came out.
So, you're supposed to be taking the modulus of ds x dt. I don't know if you've just done that (very) wrong, or if you've done something else entirely, but that's where everything falls apart. (I.e. everything before "A = an integral" looks vaguely plausible, but that line is totally wrong).
Hint/observation: it's at the point of taking the modulus that you'll get the square root you need to match the given integral.
So, you're supposed to be taking the modulus of ds x dt. I don't know if you've just done that (very) wrong, or if you've done something else entirely, but that's where everything falls apart. (I.e. everything before "A = an integral" looks vaguely plausible, but that line is totally wrong).
Hint/observation: it's at the point of taking the modulus that you'll get the square root you need to match the given integral.
So, out of curiosity rather than ragging on you (we all have brain farts!); am I right in thinking you effectively decided that |ai +bj + ck| = a + b + c?
(I know TSR can be a pain about attachment orientation, but I feel I lose at least 20 IQ points trying to follow something at 90 degrees, so I might be misinterpreting).