MATH2700 - Exam 2

Calc 3

(1/Area(D)) ∫∫_Df(x,y)dA

Average value of f(x,y)

x₀ = 1/M∫∫_DxdA, y₀ = 1/M∫∫_DydA,

Center of mass (x₀,y₀)

∫∫_D 1

Area of D

∫∫∫_E 1

Volume of E

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

T(u,v) =

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

Jacobian: J(u,v) =

∫∫_D f(x(u,v), y(u,v)) * |J(u,v)|

∫∫_T(D) f(x,y)dA =

r

Polar Jacobian

r

Cylindrical Jacobian

Ρ²sinφ

Spherical Jacobian

(x = rcosθ, y = rsinθ)

T(r,θ) =

(x = rcosθ, y = rsinθ, z=z)

T(r,θ,z) =

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

T(Ρ,φ,θ)

x = rcosθ

Polar/Cylindrical Coords: x =

y = rsinθ

Polar/Cylindrical Coords: y =

r = √(x²+y²)

Polar/Cylindrical Coords: r =

θ = arctan(y/x)

Polar/Cylindrical Coords: θ =

r = Ρsinφ

Spherical Coords: r =

Ρ = √(x²+y²+z²)

Spherical Coords: Ρ =

φ = arccos(z/P)

Spherical Coords: φ =

θ = arctan(y/x)

Spherical Coords: θ =

∫∫∫_E f(x,y,z)dV = 0

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

F / ||F||

Normalizing a vector field

F = ⛛f

Conservative Vector:

δP/δy = δQ/δx

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

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

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

∫_a^b f(r(t)) · ||r’(t)||

Line Integral

∫_a^b F · dr

Integral of a vector field

∫_a^b F(r(t)) · r’(t)

Integral of a vector field (along an oriented curve)

∫_a^b F(r(t)) · <y’(t), -x(t)>

Flux of F through oriented curve C:

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

Fundamental Theorem of Line Integrals:

∫_c⛛f · dr = 0

FToLI: If C is a closed path:

(m/2) [||r’(t)||]_a^b

Work: Kinetic Energy Gained

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

Work: Potential Energy Lost