calc 3 test 2

Formula for Curvature (basic)

‖T’(t)‖ / ‖r’(t)‖

Formula for Curvature (mathy)

‖r’(t) x r’’(t)‖ / ‖r’(t)‖³

Formula for Curvature (ℝ² curve f(x))

|f’’(x)| / (1+(f’(x))²)^3/2

Ellipsoid

x² + y² + z² = 1

Elliptic Paraboloid

z = x² + y²

Hyperbolic Paraboloid

z = x² - y²

Cone

x² + y² = z²

Hyperboloid of Two Sheets

-x² - y² + z² = 1

Hyperboloid of One Sheet

x² + y² - z² = 1

Unit Normal Vector

T’(t) / ‖T’(t)‖

Binormal Vector

T(t) x N(t)

Important polar formulas

r²=x²+y²

x in polar

rcosθ

y in polar

rsinθ