Vector Calculus

Quadric Surfaces

Note: Review Conic Sections

Surface	Standard Form	Tips to Recognize
Elliptic Paraboloid	$z-\ell =\frac{(x-h{)}^2}{a}+\frac{(y-k{)}^2}{b}$	2 Quadratics, 1 Linear, Same Signs
Hyperbolic Paraboloid	$z-\ell =\frac{(x-h{)}^2}{a}-\frac{(y-k{)}^2}{b}$	2 Quadratics, 1 Linear, Different Signs
Ellipsoid	$\frac{(x-h{)}^2}{a^2}+\frac{(y-k{)}^2}{b^2}+\frac{(z-\ell {)}^2}{c^2}=1$	3 Quadratics, Same Sign
Hyperboloid (1 Sheet)	$\frac{(x-h{)}^2}{a^2}+\frac{(y-k{)}^2}{b^2}-\frac{(z-\ell {)}^2}{c^2}=1$	3 Quadratics, Different Signs, 2 Positives, 1 Negative
Hyperboloid (2 Sheets)	$-\frac{(x-h{)}^2}{a^2}-\frac{(y-k{)}^2}{b^2}+\frac{(z-\ell {)}^2}{c^2}=1$	3 Quadratics, Different Signs, 2 Negatives, 1 Positive
Two Cones	$\frac{(x-h{)}^2}{a^2}+\frac{(y-k{)}^2}{b^2}-\frac{(z-\ell {)}^2}{c^2}=0$	3 Quadratics, Different Signs, No Constant