Using relations like this one, e j ˙ ˙ = ∑ ∑ k = 1 d ∂ ∂ ∂ ∂ q k e j q ˙ ˙ k = ∑ ∑ k = 1 d ∑ ∑ i = 1 d Γ Γ k i j q ˙ ˙ i e k , which allows all the time derivatives to be evaluated. For example, for the velocity: v = d d t r = ∑ ∑ k = 1 d q ˙ ˙ k e k + ∑ ∑ k = 1 d q k e k ˙ ˙ = ∑ ∑ k = 1 d q ˙ ˙ k e k + ∑ ∑ j = 1 d q j e j ˙ ˙ , = ∑ ∑ k = 1 d q ˙ ˙ k e k + ∑ ∑ k = 1 d ∑ ∑ j = 1 d ∑ ∑ i = 1 d q j Γ Γ k i j e k q ˙ ˙ i = ∑ ∑ k = 1 d ( q ˙ ˙ k + ∑ ∑ j = 1 d ∑ ∑ i = 1 d q j Γ Γ k i j q ˙ ˙ i ) e k , with the Γ-notation for the Christoffel symbols replacing the bracket notation. Using the same approach, the acceleration is then a = d d t v = ∑ ∑ k = 1 d v ˙ ˙ k e k + ∑ ∑ k = 1 d v k e k ˙ ˙ = ∑ ∑ k = 1 d ( v ˙ ˙ k + ∑ ∑ j = 1 d ∑ ∑ i = 1 d v j Γ Γ k i j q ˙ ˙ i ) e k . Looking at the relation for acceleration, the first summation contains the time derivatives of velocity, which would be associated with acceleration if these were Cartesian coordinates, and the second summation (the one with Christoffel symbols) contains terms related to the way the unit vectors change with time.