Math 2270 Exam 1 Formulas

Math 2270 Calc 3

v/||v||

Unit Vector

√(x² + y² + z²)

Magnitude

<a+x, b+y, c+z>

v + w

<a-x, b-y, c-z>

v - w

ax + by + cz

v · w Dot Product

||v||²

v · v

||v||||w||cos(θ)

v · w angle

 w((v · w)/(||w||²))

projw(v)

 v((v · w)/(||v||²))

projv(w)

i j k

v1 v2 v3

w1 w2 w3

v × w cross product

||v||||w||sin(θ)

||v × w||

u · (v × w)

Area of a Parallelapiped

x(t) = x0 + ta,
y(t) = y0 + tb,
z(t) = z0 + tc

Parametric Equations

t = (x-x0)/a = (y-y0)/b = (z-z0)/c

Symmetric Equations

(||v × AC||)/||v||

Distance of a point to a line

|n ·  AC| / ||n||

Distance of a point to a plane

A vector orthogonal to the plane

ax + by + cz: <a,b,c>
OR
u × v

Normal Vector

Normal Vector: <a,b,c>
Point: (x0,y0,z0)

a(x-x0) + b(y-y0) + c(z-z0) = 0

Plane Equation

x = rcos(θ)

Polar Coordinates: x =

y = rsin(θ)

Polar Coordinates: y =

r² = x² + y²
r = √(x²+y²)

Polar Coordinates: r =

tan(θ) = y/x
θ = arctan(y/x)

Polar Coordinates: θ =

r = Ρsinφ

Spherical Coordinates: r =

x = rcosθ = Ρsinφcosθ

Spherical Coordinates: x =

y = rsinθ = Ρsinφsinθ

Spherical Coordinates: y =

z = Ρcosφ

Spherical Coordinates: z =

Ρ = √(x²+y²+z²)

Spherical Coordinates: Ρ =

φ = arccos(z/P)

Spherical Coordinates: φ =

θ = arctan(y/x)

Spherical Coordinates: θ =

s(t) = ∫(0,t)||v(u)||du

Arc Length

||v(t)|| (parameterized)

Speed

v(t)/||v(t)|| (parameterized)

Unit Tangent Vector

rx = <1,0,δf/δx(a,b)>

rx =

ry = <0,1,δf/δy(a,b)>

ry =

r = rx × ry = <-δf/δx(a,b),-δf/δy(a,b),1>

r =

<x-a, y-b, z-f(a,b)> · <-δf/δx(a,b),-δf/δy(a,b),1>

Tangent Plane

f(a,b) + δf/δx(a,b)(x-a) + δf/δy(a,b)(y-b)

Tangent Plane: z =

df/dt = (δf/δx * dx/dt) + (δf/δy * dy/dt) + (δf/δz * dz/dt) +

df/dt =

⛛f(x,y) = <δf/δx(x,y), δf/δy(x,y)>

Gradient ⛛f(x,y) =

Df(a,b)v = v/||v|| · ⛛f(x,y)

Directional Derivative Df(a,b)v Gradient

(1/||v||)Df(a,b)v

Directional Derivative Df(a,b)(v/||v||)

⛛f(x,y) = 0, solve for x,y,z.

Critical Points

⛛f(a,b)

Normal Vector at a Point (a,b)

| δ²f/δx²(a,b) δ²f/δxδy(a,b) |
| δ²f/δyδx(a,b) δ²f/δy²(a,b) |

det(H(a,b)(f)) = δ²f/δx²(a,b) * δ²f/δy²(a,b) - (δ²f/δxδy(a,b))²

Hessian Matrix

Local Min

Hessian: Det(H(a,b)(f)) > 0 & δ²f/δx²(a,b) > 0

Local Max

Hessian: Det(H(a,b)(f)) > 0 & δ²f/δx²(a,b) < 0

Saddle Point

Hessian: Det(H(a,b)(f)) < 0

Inconclusive

Hessian: Det(H(a,b)(f)) = 0

⛛f(x,y,z) = λ⛛g(x,y,z)

Legrange

⛛f(x,y,z) = <a,b,c>
a(x-x0) + b(y-y0) + c(z-0)

Tangent Plane at a point