Calculus III

Vector Form of a Line

$x=x_0+ta$

Parametric Equations of a Line

$x=x_0+ta$ , $y=y_0+ta$ , $z=z_0+ta$

Symmetric Equations of a Line

$\frac{x-x_0}{a}=\frac{y-y_0}{b}=\frac{z-z_0}{c}$

Vector Equation of a Plane

$\vec{n}\cdot (\vec{r}-\vec{r}_0)=0$

Scalar Equation of a Plane

$a(x-x_0)+b(y-y_0)+c(z-z_0)=0$

or

$ax+by+cz=d$ where $d=ax_0+by_0+cz_0$

Common Surfaces and their Equations

Ellipsoid:

$\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$

Cone:

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=\frac{z^2}{c^2}$

Cylinder:

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

If the cylinder’s cross section is a circle, then the above equation becomes

$x^2+y^2=r^2$

Hyperboloid of One Sheet

$\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1$

Hyperboloid of Two Sheets

$-\frac{x^2}{a^2}-\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$

Hyperbolic Paraboloid

$\frac{x^2}{a^2}-\frac{y^2}{b^2}=z/c$

The Unit Tangent Vector

$\vec{T}(t)=\frac{\vec{r'}(t)}{\left \| \vec{r'(t)} \right \|}$

The Unit Normal Vector

$\vec{N}(t)=\frac{\vec{T'}(t)}{\left \| \vec{T'(t)} \right \|}$

The Binormal Vector

$\vec{B}(t)=\vec{T}(t)\times \vec{N}(t)$

Arc Length and Arc Length Function

The arc length is given by $L=\int_{a}^{b}\left \| \vec{r'}(t) \right \|dt$ and the arc length function is given by $s(t)=\int_{0}^{t}\left \| \vec{r'}(t) \right \|dt$

Curvature

$\kappa =\left | \frac{d\vec{T}}{ds} \right |=\frac{\left \| \vec{T'}(t) \right \|}{\left \| \vec{r'}(t) \right \|}=\frac{\left \| \vec{r'}(t)\times \vec{r''}(t) \right \|}{{\left \| \vec{r'}(t) \right \|}^3}=\frac{\left | f''(x) \right |}{(1+[f'(x)]^2)^{\frac{3}{2}}}$

Distance from a Point to a Line

The distance from the point $P_1$ and the line $l$ is given by

$d(P_1,l)=\frac{\left \| \overrightarrow{P_0P_1} \times \vec{d}\right \|}{\left \| \vec{d} \right \|}$

Distance from a Point to a Plane

The distance between a point $(x_1,y_1,z_1)$ and a plane $ax+by+cz=d$ is given by

$D=\frac{\left | ax_1+by_1+cz_1-d \right |}{\sqrt{a^2+b^2+c^2}}$

Angle Between two Planes

$latex \cos (\theta)=\left | \mathbf{u}_\textup{N}_1\cdot \mathbf{u}_\textup{N}_2 \right | &s=1$