Key Equations - Calculus Volume 3

Vector-valued function	$r (t) = f (t) i + g (t) j or r (t) = f (t) i + g (t) j + h (t) k, or r (t) = 〈 f (t), g (t) 〉 or r (t) = 〈 f (t), g (t), h (t) 〉$
Limit of a vector-valued function	$lim_{t \to a} r (t) = [lim_{t \to a} f (t)] i + [lim_{t \to a} g (t)] j or lim_{t \to a} r (t) = [lim_{t \to a} f (t)] i + [lim_{t \to a} g (t)] j + [lim_{t \to a} h (t)] k$

Derivative of a vector-valued function	$r^{'} (t) = lim_{Δ t \to 0} \frac{r (t + Δ t) - r (t)}{Δ t}$
Principal unit tangent vector	$T (t) = \frac{r^{'} (t)}{‖ r^{'} (t) ‖}$
Indefinite integral of a vector-valued function	$\int [f (t) i + g (t) j + h (t) k] d t = [\int f (t) d t] i + [\int g (t) d t] j + [\int h (t) d t] k$
Definite integral of a vector-valued function	$\int_{a}^{b} [f (t) i + g (t) j + h (t) k] d t = [\int_{a}^{b} f (t) d t] i + [\int_{a}^{b} g (t) d t] j + [\int_{a}^{b} h (t) d t] k$

Arc length of space curve	$s = \int_{a}^{b} \sqrt{{[f^{'} (t)]}^{2} + {[g^{'} (t)]}^{2} + {[h^{'} (t)]}^{2}} d t = \int_{a}^{b} ‖ r^{'} (t) ‖ d t$
Arc-length function	$s (t) = \int_{a}^{t} \sqrt{{(f^{'} (u))}^{2} + {(g^{'} (u))}^{2} + {(h^{'} (u))}^{2}} d u or s (t) = \int_{a}^{t} ‖ r^{'} (u) ‖ d u$
Curvature	$κ = \frac{‖ T^{'} (t) ‖}{‖ r^{'} (t) ‖} or κ = \frac{‖ r^{'} (t) \times r^{″} (t) ‖}{{‖ r^{'} (t) ‖}^{3}} or κ = \frac{\| y^{″} \|}{{[1 + {(y^{'})}^{2}]}^{3 / 2}}$
Principal unit normal vector	$N (t) = \frac{T^{'} (t)}{‖ T^{'} (t) ‖}$
Binormal vector	$B (t) = T (t) \times N (t)$

Velocity	$v (t) = r^{'} (t)$
Acceleration	$a (t) = v^{'} (t) = r″ (t)$
Speed	$v (t) = ‖ v (t) ‖ = ‖ r^{'} (t) ‖ = \frac{d s}{d t}$
Tangential component of acceleration	$a_{T} = a \cdot T = \frac{v \cdot a}{‖ v ‖}$
Normal component of acceleration	$a_{N} = a \cdot N = \frac{‖ v \times a ‖}{‖ v ‖} = \sqrt{{‖ a ‖}^{2} - a_{T}^{2}}$