Discuss the below:
Q: 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