Key Equations

Key Equations

Vector-valued function

$\text{r}(t)=f\left(t\right)\text{i}+g\left(t\right)\text{j}\text{or}\text{r}\left(t\right)=f\left(t\right)\text{i}+g\left(t\right)\text{j}+h\left(t\right)\text{k},\text{or}\text{r}\left(t\right)=\langle f\left(t\right),g\left(t\right)\rangle \text{or}\text{r}\left(t\right)=\langle f\left(t\right),g\left(t\right),h\left(t\right)\rangle$

Limit of a vector-valued function

$\underset{t\to a}{\text{lim}}\text{r}\left(t\right)=\left[\underset{t\to a}{\text{lim}}f\left(t\right)\right]\text{i}+\left[\underset{t\to a}{\text{lim}}g\left(t\right)\right]\text{j}\text{or}\underset{t\to a}{\text{lim}}\text{r}\left(t\right)=\left[\underset{t\to a}{\text{lim}}f\left(t\right)\right]\text{i}+\left[\underset{t\to a}{\text{lim}}g\left(t\right)\right]\text{j}+\left[\underset{t\to a}{\text{lim}}h\left(t\right)\right]\text{k}$

Derivative of a vector-valued function	$r^{\prime }\left(t\right)=\underset{\text{Δ}t\to 0}{\text{lim}}\frac{\text{r}\left(t+\text{Δ}t\right)-\text{r}\left(t\right)}{\text{Δ}t}$
Principal unit tangent vector	$\text{T}\left(t\right)=\frac{r^{\prime }\left(t\right)}{\text{‖}{r}^{\prime }\left(t\right)\text{‖}}$
Indefinite integral of a vector-valued function	${\displaystyle \int \left[f\left(t\right)\text{i}+g\left(t\right)\text{j}+h\left(t\right)\text{k}\right]dt}=\left[{\displaystyle \int f\left(t\right)dt}\right]\text{i}+\left[{\displaystyle \int g\left(t\right)dt}\right]\text{j}+\left[{\displaystyle \int h\left(t\right)dt}\right]\text{k}$
Definite integral of a vector-valued function	${\displaystyle {\int }_a^b\left[f\left(t\right)\text{i}+g\left(t\right)\text{j}+h\left(t\right)\text{k}\right]dt}=\left[{\displaystyle {\int }_a^{b}f\left(t\right)dt}\right]\text{i}+\left[{\displaystyle {\int }_a^{b}g\left(t\right)dt}\right]\text{j}+\left[{\displaystyle {\int }_a^{b}h\left(t\right)dt}\right]\text{k}$

Arc length of space curve	$s={\displaystyle {\int }_a^b\sqrt{{\left[f^{\prime }\left(t\right)\right]}^2+{\left[g^{\prime }\left(t\right)\right]}^2+{\left[h^{\prime }\left(t\right)\right]}^2}}dt={\displaystyle {\int }_a^b\Vert r^{\prime }\left(t\right)\Vert }dt$
Arc-length function	$s\left(t\right)={\displaystyle {\int }_a^t\sqrt{{\left(f^{\prime }\left(u\right)\right)}^2+{\left(g^{\prime }\left(u\right)\right)}^2+{\left(h^{\prime }\left(u\right)\right)}^2}}du\text{or}s\left(t\right)={\displaystyle {\int }_a^t\Vert r^{\prime }\left(u\right)\Vert du}$
Curvature	$\kappa =\frac{\Vert T^{\prime }\left(t\right)\Vert }{\Vert r^{\prime }\left(t\right)\Vert }\text{or}\kappa =\frac{\Vert r^{\prime }\left(t\right)\times r^{″}\left(t\right)\Vert }{{\Vert r^{\prime }\left(t\right)\Vert }^3}\text{or}\kappa =\frac{\left|y^{″}\right|}{{\left[1+{\left(y^{\prime }\right)}^2\right]}^{3\text{/}2}}$
Principal unit normal vector	$\text{N}\left(t\right)=\frac{T^{\prime }\left(t\right)}{\Vert T^{\prime }\left(t\right)\Vert }$
Binormal vector	$\text{B}\left(t\right)=\text{T}\left(t\right)\times \text{N}\left(t\right)$

Velocity	$\text{v}\left(t\right)=r^{\prime }\left(t\right)$
Acceleration	$\text{a}\left(t\right)=v^{\prime }\left(t\right)=\text{r″}\left(t\right)$
Speed	$v\left(t\right)=\Vert \text{v}\left(t\right)\Vert =\Vert r^{\prime }\left(t\right)\Vert =\frac{ds}{dt}$
Tangential component of acceleration	$a_{\text{T}}=\text{a}·\text{T}=\frac{\text{v}·\text{a}}{\Vert \text{v}\Vert }$
Normal component of acceleration	$a_{\text{N}}=\text{a}·\text{N}=\frac{\Vert \text{v}\times \text{a}\Vert }{\Vert \text{v}\Vert }=\sqrt{{\Vert \text{a}\Vert }^2-a_{\text{T}}^2}$