Multi Exam 2

Arc Length

L = integral of ||r’(t)||dt (arc length)

s(t) = integral of ||r’(u)||du (arc length function)

dr/ds

unit tangent vector

LENGTH = 1 because UNIT VECTOR

curvature

weird k = |r’(t) x r’’(t)|

. |r’(t)|³

Curvature of a Line

IS 0 Dumb Dumb

Principal Unit Normal Vector

N(t) = T’(t)

. |T’(t)|

Radius of Curvature

weird p = 1/weird k

Oscillating Center

r(t) + weird p*N(t)

dw/du

dw/du = dw/dx * dx/du + dw/dy * dy/du + dw/dz * dz/du

u

HAS TO BE UNIT VECTOR

Most Rapid Increase

unit vector: gradient/|gradient| where is positive

max rate = |gradient|

Tangent Plane

r-r₀ = f_x (x₀,y₀,z₀)(x-x₀) + f_y (x₀,y₀,z₀)(y-y₀) + f_z (x₀,y₀,z₀)(z-z₀)

Second Derivative Tesst

f_xx < 0 and f_xxf_yy - (f_xy)² > 0 : Local Max

f_xx > 0 and f_xxf_yy - (f_xy)² > 0 : Local Min

f_xxf_yy - (f_xy)² < 0 : Saddle Point

Anything Else: Inconclusive

Hessian Matrix

|f_xx f_xy|

|f_xy f_yy|

= f_xxf_yy - f_xy²

Lagrange Multipliers

gradient f = lambda * gradient g