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...
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...
(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...
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...
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...