Multivariable Calc Unit 1 Formulas

Spherical coordinates, x =

ρ sin φ cos θ,

Spherical coordinates, y =

ρ sin φ sin θ

Spherical coordinates, z =

ρ cos φ

Spherical coordinates p² =

x² + y² + z²

Cylindrical Coordinates x =

r cos θ

Cylindrical Coordinates y =

r sin θ

Cylindrical Coordinates r² =

x² + y²

Vector projection (u onto v)

(u·v / ∥v∥²)v

Distance from a point to a plane

|ax0+by0+cz0+d| / √(a² + b² + c²)

u dot v

(∥u∥)(∥v∥)cosθ

Equation of a circle

(x-h)² + (y-k)² = r²

Cauchy-Schwarz Inequality

|udotv|<= (∥u∥)(∥v∥)

Triangle Inequality

∥u + v∥ ≤ ∥u∥ + ∥v∥

Equation of a cone

x² + y² = z²

Equation of a sphere

x² + y² + z² = r²

∥u + v∥ =

∥u∥² + ∥v∥²

Area of a parallelogram

|a x b| (magnitude of cross product)

Area of a parallelepiped

|(a x b)•c| or

|a1 a2 a3|

|b1 b2 b3|

|b1 b2 b3|

Area of triangle from verticies

½ |a x b| ( ½ magnitude of cross product)