Test 2 Calculus 3 Formulas

Curvature

k(kappa) = 1/|v| * |(dT/dt)|

aT

d|v|/dt

aN

√(|a|² - (aT)²)

Unit Tangent Vector

T = v / |v|

Unit Normal Vector

N = (dT/dt) / (|dT/dt|)

Unit Binormal Vector

B = T x N

chain rule for dw/dt

(δw/δx)(dx/dt) + (δw/δy)(dy/dt)

dy/dx is the same as

-Fx/Fy

dF/dx

Fx+Fy(dy/dx)

gradient vector of f(x,y)

<fx, fy>

Duf

gradient vector f * unit vector

Tangent Plane at point P

fx(P₀)(x-x₀) + fy(P₀)(y-y₀) + fz(P₀)(z-z₀)

Normal Line at point P

x=x₀+fx(P₀)t

L(x,y)

f(x₀,y₀) + fx(x₀,y₀)(x-x₀) + fy(x₀,y₀)(y-y₀)

f has a local maximum at f(a,b) if

fxx < 0 and fxxfyy - (fxy)² > 0

determinant/Hessian

fxxfyy - (fxy)²

f has a local minimum at (a,b) if

fxx > 0 and fxxfyy - (fxy)² > 0

f has a saddle point at (a,b) if

fxxfyy - (fxy)² < 0

with the osculating plane, what vector will you need to find the plane equation

Binormal

with the normal plane, which vector will you need to find the plane equation

Unit Tangent Vector

with the rectifying plane, what vector will you need to find the plane equation

Unit Normal Vector