Answer to Question #144511 in Differential Geometry | Topology for Hussaina

Question

Given that \\(R=sin ¡t i+cos ¡t j+tk\\), find \\( (d^2 R)/(dt^2 )\\).

Answer 1

First of all let us notice the fact that the basis vectors of $\mathbb{R}^3 (i, j, k)$ are constant (as opposed to local vectors $e_r,e_\theta, e_z$ of the cylindrical coordinates base, for example). Therefore the derivation of $R(t)$ comes down to deriving every coordinate. Secondly, let us write $R(t)$ as a vector for more obviousness:

$R(t)=\begin{pmatrix}\sin(\hat{a}t) \\ \cos(\hat{a}t) \\ t\end{pmatrix}$

Now let's calculate the first and then the second derivative:

$\frac{d}{dt} R(t) = \begin{pmatrix} \hat{a}\cos(\hat{a}t) \\ -\hat{a}\sin(\hat{a}t) \\ 1 \end{pmatrix}$

$\frac{d^2}{dt^2} R(t) = \begin{pmatrix} -\hat{a}^2 \sin(\hat{a}t) \\ -\hat{a}^2 \cos(\hat{a}t) \\ 0 \end{pmatrix}$