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Thanks for your patience. It's still a challenge for me to make sense of proofs in concrete terms-- work in progress. I went back earlier into the lecture and see now how (after computing the cross product) the binormal vector must be (0,0,1) in this particular case. The trouble all...
So would the α(s) (a circle) I've chosen be wrong or would it be the vector (1/sqrt(2)(1,1,0). It seems like the vector is constant and the circle has zero torsion which is what I'm hung up about. The proof makes sense, but there's a disconnect with a specific example.
And further, I can see...
Couldn't the vector also be v= (1/sqrt(2), 1/sqrt(2), 0), which then wouldn't equal a constant? So the dot product would be 1/sqrt(2)(cos(t) + sin(t)). Or perhaps that choice of v violates a rule?
The lecturer has used α(s) in the past to be any curve in R^3. I'm not sure why dotting that curve with a constant value represents a plane. I can see how the dot product will equal a scalar constant, but think of a plane as something as ax+by+cz=d, but am having trouble envisioning that from...
I was watching a lecture that made the conclusion about the torsion being equal to zero necessitated that the path was planar. The argument went as follows:
-Torsion = 0 => B=v, which is a constant
-(α⋅v)'=(T⋅v)'= 0 => α⋅v= a, which is a constant (where α is a function describing the path and...
Thanks for your reply. In past searches I have found a lot of basic problems (calculate the line integral of this function, use the divergence theorem to solve some integral, take the cross product of these vectors). Rarer finds seem to be the sets that have less obvious paths forward. I did...
I've taken multivariable/vector calc and can do most of the basic operations and have an OK understanding of the fundamental concepts, but certainly can't "see it" like I can calc I and II. In those subjects, I often feel competent to take on any problem I come across because the concepts are...
OK cool! So I see it this way then: If z=r, it simplifies to r^k(e^(r^2)-1)/(4r^(2k). Using the expansion e^(r^2) ≈ 1 +r^2, we get the expression to become r^2/(4r^(2k)= 1/4(r^(2-k), where as long as 2-k>0, the limit is zero.
Wait, now I'm doubting myself and thinking I made a mistake with the z=0, r -> 0 part. Won't the numerator actually become 0(e^(r^2)-1)/r^(2k), which is again 0 and independent of k?
Thanks once again! The numerator then becomes (1+r^2)-1= r^2. So the expression reduces to r^2/r^(2k). In order for the limit to go to zero, then then numerator should have the higher degree. So, the expression equals r^(2-2k), and when k<1, the limit would equal zero (very small number to...
Ooh, yeah, thanks. Substituting those parameters gives me:
z^k(e^r^2 -1)/(r^2+z^2)^k
So for the limit (r=0, z->0) would be z^k(0)/z^(2k)= 0/z^k, since (e^(r^2)-1)=0 at r=0. As z->0, isn't this of the form 0/0? I would think L'Hopital, but am a little unsure because there isn't really a...
The divergence would be zero, which probably indicates something meaningful! I know the divergence of the curl must be zero, and that a curl of zero means that F is conservative. If it were the opposite (curl zero, divergence non-zero), I would use the concept that f=∫c F⋅ds = f(b)-f(a) for...
(a) I thought perhaps a parameterization would be the place to begin given all the squared terms.
x=rcos(u)sin(v)
y=rsin(u)sin(v)
z=rcos(v)
That would yield: r^k(cos(v))^k*(e^(r^2*(sin(v))^2))/(r^(2k))
Canceling a r^k at each level: (cos(v))^k*(e^(r^2*(sin(v))^2))/(r^(k))
I'm not sure how...
Thanks everyone who contributed here. It's all very helpful. I think I'm seeing what I missed the first time around (parameterization was backward, logically) and am playing with a few of the suggestions. But as some have said, I think I may have picked a tricky problem (though interesting!).
Thanks for pointing that out. Although with line integrals, do we still have to use the Jacobian? In the final three options I was trying to use Stokes' Theorem and solve something of the form ∫c F⋅ds. It feels like there should be a good method to use a line integral, but I don't know if any...
I've tried a few ways of solving this, both directly and by using Stokes' Theorem. I may be messing up what the surface is in the first place
F= r x (i + j+ k) = (y-z, z-x, x-y)
Idea 1: Solve directly. So ∇ x F = (-2,-2,-2). I was unsure on which surface I could use for the normal vector...
Also, for curl, we sometimes were told to plug in points, but I seem to be getting an inconsistency. I tried:
20,10: negative (because tends toward clockwise)
10, 20: positive
-10,20: negative
-20,10: positive
-20,-10: positive
-10,-20: negative
10, -20: positive
20, -10: negative
The only...
For divergence: We learned to draw a circle at different locations and to see if gas is expanding/contracting. Whenever the y-coordinate is positive, the gas seems to be expanding, and it's contracting when negative. I find it hard to tell if the gas is expanding or contracting as I go to the...