Calc III Quadratic Shapes

(x^2/a^2)+(y^2/b^2)+(z^2/c^2)=1

Ellipsoid (All traces are ellipses)

z=(x^2/a^2)+(y^2/b^2)

Elliptic Paraboloid (Made of ellipses and parabolas)

(x^2/a^2)+(y^2/b^2)-(z^2/c^2)=1

Hyperboloid of one sheet (connected)

(-x^2/a^2)-(y^2/b^2)+(z^2/c^2)=1

Hyperboloid of two sheets (disconnected)

(x^2/a^2)+(y^2/b^2)=(z^2/c^2)

Elliptic Cone (Two inverted cones touching tips)

(x^2/a^2)-(y^2/b^2)=z

Hyperbolic paraboloid (Two hyperbolas connected)

Directional Derivative

F_x A + F_y B + F_z C

∂z/∂x

\-Fx/Fz

∂z/∂y

\-Fy/Fz

Gradient

Tangent Plane

Fx(x-x_0) + Fy(y-y_0) -(z-z_0)

T (Unit Tangent Vector)

T = r’(t) / |r’(t)|

K (Curvature)

k = |T’(t)| / |r’(t)|

K (Curvature Alternate Form)

k = |r’’(t) X r’(t)| / |r’(t)|^3

Length of Curve

L = ∫|r’(t)|dt from (A to B)