course Mth 275
I have not received responses to my submissions for the last few sections. Did you receive them?
These should have been posted on 6/26, probably around the time you submitted this. Let me know if not. I'm back in town, hopefully for the duration, and expect to be caught up by tomorrow. Then I'll respond to the question about u x (u x (u x v)). There is a lot to that explanation, and a lot of insight to be gained into the nature of the cross and dot products and the interaction of vectors in 3-dimensional space.
Section 11.1:This section is fairly straightforward. Drawing the graphs is sometimes difficult (I’m no artist, and perspective doesn’t come naturally to me). Problem 18, in particular, was difficult to draw (although I could visualize the graph)—x and y graphed a circle with radius 1/sqrt(2), and z = cos^2 t.
Fortunately I have the text at this location and was able to look up that problem. The i and j components are equal (both being sqrt(2)/2 sin(t)) so in planes parallel to the x-y plane you have a back-and-forth harmonic oscillation along a line segment of length 2 which runs at a 45-degree angle from the third quadrant to the first. The harmonic oscillation begins at t = 0 at the origin, oscillates through the first-quadrant segment, back to the origin and into the 3d quadrant, then back to the origin, repeating with a cycle of t = 2 pi.
The k component in the meantime is cos^2(t). The magnitude of the displacement from origin in the x-y plane is sin(t). If we denote this displacement as c and the z component as z, we have c^2 + z = sin^2(t) + cos^2(t) = 1. c^2 + z = 1 is the equation of a parabola in the c-z plane having vertex at (0, 1) and crossing the c axis at (1, 0) and (-1, 0).
The point moves along this parabola, each cycle having period 2 pi and starting at (0, 1), moving to (1,0), back to (0, 1), to (-1, 0) and finally back again to (0, 1). The c component of the oscillation is, as specified before, harmonic.
(visuatlize the c-z plane as the plane defined by the '45-degree line' x = y in the x-y plane, and the z axis).
I have a question about problem 30, which asks for the vector function of the line of intersection of the planes 2x + y + 3z = 6 and x – y – z =1. I had no problem computing a vector, but it seems to me that there is an infinite number of vector functions that are solutions to the givens, depending upon which point on the line of intersection is chosen to provide the constant values in the parametric equations.
The line of intersection will have the form r(t) = r0 + t * v, where r0 is any vector from the origin to the line (i.e., you can use any point of the line of intersection for r0), t is a parameter that varies over all real numbers (e.g., t might be time, extending without bound into the past and into the future), and v is a vector parallel to the line (most easily calculated as the cross product of the two normal vectors; v can if desired be normalized to make it a unit vector).
Problem 63 asks whether the scalar multiples, sums of scalar multiples, dot products, and cross products of vector functions are continuous if the vector functions are continuous. Given that continuity and limits of vector functions appear to follow the same rules as scalar functions, I concluded that they are all continuous based on the theorem on continuity of sums, products, and quotients of functions. Is this reasoning correct?
It works out that way in most cases, but to prove continuity you have to use the definition of continuity. A function f(t) is continuous over a domain if for every a in the domain limit[t -> a] f(t) = f(a).
Continuity proofs then typically come down to epsilon-delta proofs involving limits. It is generally proved in the first semester of a first-year calculus course that the sum and product of two continuous functions are continuous functions, and those results can be used here to prove continuity.
For example if g(t) is a scalar function and v(t) a vector function in 2 dimensions, g(t) * v(t) = g(t) * (vx(t) i + vy(t) j) = g(t) vx(t) i + g(t) vy(t) j.
If v is continuous then it can be easily shown that its components vx(t) and vy(t) are continuous (if one component isn't continuous at a point it's very easy to show that the vector function isn't continuous at that point) . So the proof in this case comes down to showing that g(t) vx(t) and g(t) vy(t) are continuous; being the products of continuous functions, they are.
Similar strategies, but with more details, can be used for the four functions in these problems.