Math2270 FINAL

Plane Equation (3 var)

Normal Vector: <a,b,c>
Point: (x₀,y₀,z₀) a(x-x₀) + b(y-y₀) + c(z-z₀) = 0

Find the Tangent Plane (f(x,y,z))

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

Find the Tangent Plane (z = f(x,y))

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

Find the Linear Approximation L(x₀,y₀) =

L(x₀,y₀) = F(x₀,y₀) + δf/δx(a,b)(x-x₀) + δf/δy(a,b)(y-y₀)
Find tangent plane, plug in approxed values

Find the direction f increases most rapidly at (x,y,z)

⛛f(x,y,z) / ||⛛f(x,y,z)||

How fast is f increasing?

||⛛f(x,y,z)||

Find the Normal Vector

ax + by + cz: OR u × v OR ⛛f(x₀,y₀,z₀)

Find the plane at (x₀,y₀,z₀)

⛛f(x₀,y₀,z₀) → a(x-x₀) + b(y-y₀) +c(z-z₀) = 0

Area of a Parallelogram with corners ABCD

||AB × AC||

Area of a Triangle with corners ABC

1/2||AB × AC||

Find the Plane containing A, B, C

Point: (x₀,y₀,z₀) Normal Vector: AB × AC a(x-x₀) + b(y-y₀) +c(z-z₀) = 0

Distance from point to plane

|n ·  AD| / ||n||

Distance from point to line

(||v × AC||)/||v||

Vector Equations

r(t)=r₀​+tv
r(0) is the point
v is the direction vector

Symmetric Equations

t = (x-x₀)/a = (y-y₀)/b = (z-z₀)/c

Parametric Equations

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

Rate of Change (chain rule)

F(r(t)) = ⛛f(r(t)) · r’(t) = (δf/δx)(dx/dt) + (δf/δy)(dy/dt)

Rate of Change in terms of ⛛T & R(t)

⛛T(r(0))

Surface when r=1

Cylinder

Surface when z=r

Cone

Surface when z=r²

Paraboloid

Surface when P = 1

Sphere

Surface when φ = π/4

Cone

Surface when φ = π/2

The Plane z=0

Surface when θ = π/2

Half of the yz-plane

Find Absolute Extrema (no lagrange)

Use gradiant to find critical points Parameterize boundary, plug into f Solve d/dt = 0 Plug t into parameter Solve for z, find max and min

Find Absolute Extrema (lagrange)

Use gradiant to find critical points ⛛f(x,y,z) = λ⛛g(x,y,z) Solve for points Find max/min

Hessian Matrix

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

Local Min

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

Local Max

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

Saddle Point

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

Inconclusive

Unit Vector

v/||v||

Magnitude

√(x² + y² + z²)

v + w

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

v - w

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

v · w Dot Product

ax + by + cz

v · v

||v||²

v · w angle

||v||||w||cos(θ)

projw(v)

w((v · w)/(||w||²))

projv(w)

v((v · w)/(||v||²))

v × w cross product

i j k
v1 v2 v3
w1 w2 w3

||v × w||

||v||||w||sin(θ)

Area of a Parallelapiped

u · (v × w)

Arc Length

s(t) = ∫₀^t||r’(u)||du

Speed

||v(t)|| (parameterized)

Unit Tangent Vector

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

df/dt =

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

Gradient ⛛f(x,y) =

⛛f(x,y) = <δf/δx(x,y), δf/δy(x,y)>

Directional Derivative Df(a,b)v =

Df(a,b)v = v/||v|| · ⛛f(x,y)

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

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

Critical Points

⛛f(x,y) = 0, solve for x,y,z.

Lagrange

⛛f(x,y,z) = λ⛛g(x,y,z)

Center of mass (x0,y0)

x₀ = 1/M∫∫_DxdA, y₀ = 1/M∫∫_DydA,

Work: Kinetic Energy Gained

(m/2) [||r’(t)||]_a^b

Work: Potential Energy Lost

Work = f(r(b)) - f(r(a))

Average value of f(x,y)

(1/Area(D)) ∫∫_D f(x,y)dA

Area of D

∫∫_D 1

Volume of E

∫∫∫_E 1

T(u,v) =

(x(u,v), y(u,v))

Jacobian: J(u,v) =

| δx/δu δx/δv |
| δy/δu δy/δv | = (δx/δu)(δy/δv) - (δx/δv)(δy/δu)

∫∫_T(D) f(x,y)dA =

∫∫_D f(x(u,v), y(u,v)) * |J(u,v)|

Polar Jacobian

r

Cylindrical Jacobian

r

Spherical Jacobian

Ρ²sinφ

T(r,θ) =

(x = rcosθ, y = rsinθ)

T(r,θ,z) =

(x = rcosθ, y = rsinθ, z=z)

T(Ρ,φ,θ)

(x = Ρsinφcosθ, y = Ρsinφsinθ, z = Ρcosφ)

Polar/Cylindrical Coords: x =

x = rcosθ

Polar/Cylindrical Coords: y =

y = rsinθ

Polar/Cylindrical Coords: r =

r = √(x²+y²)

Polar/Cylindrical Coords: θ =

θ = arctan(y/x)

Spherical Coords: r =

r = Ρsinφ

Spherical Coords: Ρ =

Ρ = √(x²+y²+z²)

Spherical Coords: φ =

φ = arccos(z/P)

Spherical Coords: θ =

θ = arctan(y/x)

If E is symmetric about x=0 and F(x,y,z) is odd on x:

∫∫∫_E f(x,y,z)dV = 0

Normalizing a vector field

F / ||F||

Conservative Vector:

F = ⛛ f

If a vector field is conservative, simply connected, f(x,y):

δQ/δx = δP/δy

If a vector field is conservative, simply connected, f(x,y,z):

δR/δy = δQ/δz, δP/δz = δR/δx, δQ/δx = δP/δy

Line Integral

∫_a^b f(r(t)) * ||r’(t)||

Integral of a vector field

∫_a^b F · dr

Integral of a vector field (along an parameterized curve)

∫_a^b F(r(t)) · r’(t)

Fundamental Theorem of Line Integrals:

If F = ⛛f and C is (x₀,y₀) → (x₁,y₁), then ∫_cF · dr = f(x₁,y₁) - f(x₀,y₀)

FToLI: If C is a closed path:

∫c⛛f · dr = 0

cos(0)

1

sin(0)

0

cos(π/4)

(√2)/2

sin(π/4)

(√2)/2

cos(π/2)

0

sin(π/2)

1

cos(π)

-1

sin(π)

0

cos(3π/2)

0

sin(3π/2)

-1

Trig Identity: cos²(x) =

(1 + cos(2x)) / 2

Trig Identity: sin²(x) =

(1 - cos(2x)) / 2

Green’s Theorem

∫_δDF · dR = ∫∫_D δQ/δx - δP/δy dA

Area using Green’s Theorem: Flux is

F = <-y/2,x/2>, <-y,0>, <0,x>