Smooth Curve - Three Dimensional Space

A smooth curve is a curve for which ^→r' (t) is continuous and ^→r' (t) ≠ 0 for any t except probably at the endpoints. A helix is a smooth curve, for instance.

At last, there is requirement of discuss integrals of vector functions. By using both limits and derivatives like a guide it shouldn't be too surprising that we as well comprise the following for integration for indefinite integrals

∫ r^→ (t) = {∫ f (t)dt, ∫g (t)dt, ∫ h(t) dt} + c^→

∫ r^→ (t) = ∫ f (t)dt i^→ + ∫g (t)dt j^→ + ∫ h(t) dt} k^→  + c^→

and the following for definite integrals.

∫^b_a r^→ (t) dt = {∫^b_a f (t) dt, ∫^b_a g (t) dt, ∫^b_a h(t) dt}

∫^b_a r^→ (t) dt = ∫^b_a f (t) dt i^→ + ∫^b_a g (t) dt j^→ + ∫^b_a h(t) dt k^→

Along with the indefinite integrals we put in a constant of integration to ensure that it was clear that the constant in this case requires being a vector in place of a regular constant.