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    Curves in euclidean 3-space

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    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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    https://brainmass.com/physics/scalar-and-vector-operations/curves-euclidean-space-145092

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    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.

    $2.49

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