Curves in euclidean 3-space
In Euclidean three-space, let p be the point with coordinates (x,y,z) = (1,0,-1). Consider the following curves that pass through p:
Curve 1: xi (λ) = (λ, (λ-1)2, - λ)
Curve 2: xi (μ) = (cos μ , sin μ, μ-1)
Curve 3: xi (σ) = (σ2, σ3 + σ2, σ)
The curves are parametrized by the parameters that vary, at least in principle,
from -∞ to +∞
(a) Calculate the components of the tangent vectors to these curves at p in the coordinate basis {∂x , ∂y , ∂z}
(b) Let a particular function f be defined on this 3-space, f = x2 + y2 - yz.
Calculate the function's rate of change as it varies along each of these curves,
i.e., find df/dλ, df/dμ, df/dσ
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Solution Preview
To find the tangent vector you just differentiate the components of the vector w.r.t. the parameter and then normalize the result. Let's denote the position vector by V. Then if:
V = [lambda, (lambda-1)^2, - lambda]
dV/dlambda = [1, 2(lambda-1), - 1]
at point P the parameter lambda = 1, so the derivative there is [1, 0, - 1]. The tangent vector is obtained by normalizing this to 1: 1/sqrt(2) [1, 0, - 1]
Next case:
V = [cos(mu), sin(mu), mu-1]
dV/d mu = [-sin(mu), cos(mu), 1]
At P the parameter mu = 0, so there the derivative is [0,1,1] and the ...
Solution Summary
The function's rate of change as it varies along each of the curves is calculated. Particular functions in three-space is determined. Tangent vectors are analyzed.