Ch. 12:

i.)Vector Knowledge

ii) Dot Product (a.k.a. Scalar product)

iii) Cross Product (a.k.a. Vector product)

iv) Equations for lines & line segments

a.) l(t) = a + t ( b – a )

b.) to make it a line segment, 0 <= t <= 1.

v) Equations / normal vectors for planes

vi) Quadric surfaces

a.) Cross sections

b.) Contours

Ch. 14:

i.) Partial Differentiationii.) Implicit Differentiationii.) Tangent Planes and Linearization

a.) How to discern them conceptually and mathematically.b.) z - z0 = ƒx(x0,y0)(x-x0)+ƒy(x0,y0)(y-y0)c.) Fx(x0,y0,z0)(x-x0) + Fy(x0,y0,z0)(y-y0) + Fz(x0,y0,z0)(z-z0) = 0d.) Differential Theorem: If the partial derivatives fx and fy exist near (a,b) and are continuous at (a,b) then f is differentiable at (a,b)

iii.) The Chain Rule

a.) (dz/dt) = (dz/dx)(dx/dt) + (dz/dy)(dy/dt)iv.) Directional Derivatives & Gradient Vectors

a.) Directional Derivatives: ƒx(x0,y0) = lim(h->0) [ƒ(x0 + h,y0) - ƒ(x0,y0)]/hb.) Grad f = <ƒx(x,y),ƒy(x,y)> = (∂ƒ/∂x)i + (∂ƒ/∂y)j + (∂ƒ/∂z)kc.) Duƒ(x,y) = Gradƒ(x,y,z)*u

v.) *Maximum and Minimum Values

a.) Second Derivatives Test:

D = | ƒxx ƒxy | | ƒyx ƒyy | = ƒxxƒyy - (ƒxy)^2

- If D > 0 and ƒxx(a,b) > 0, then ƒ(a,b) is a local minimum- If D > 0 and ƒxx(a,b) < 0, then ƒ(a,b) is local maximum- If D < 0 then ƒ(a,b) is neither- If D = 0 then ƒ(a,b) is saddle point

vi.) *Lagrange Multipliers

a.) Grad ƒ(x0,y0,z0) = Lambda Grad g(x0,y0,z0)

b.) Grad ƒ(x0,y0,z0) = Lambda Grad g(x0,y0,z0) µ Grad h(x0,y0,z0)

Ch. 15:

i.) Double Integrals over Rectangles

a.) ∫∫ƒ(x,y)dAb.) ∫∫ƒ(x,y)dA = ∫a-b∫c-d ƒ(x,y)dxdy = ∫c-d∫a-b ƒ(x,y)dxdy

ii.) Iterated Integrals

a.) ∫∫ƒ(x,y)dA

b.) Remember to change variables, create sketches

c.) Broken Down: ∫∫Dƒ(x,y)dA = ∫∫D1ƒ(x,y)dA + ∫∫D2ƒ(x,y)dAiii.) Double Integrals + Polars

a.) Coversions: r^2 = x^2+y^2, x =rcosø, y = rsinøb.) Fudge Factor = r

c.) General Form: ∫∫ƒ(rcosø,rsinø)rdrdø

iv.) Applications of Double Integrals

a.) Center of Mass:

i.) ii.) x = My/m = 1/m∫∫xp(x,y)dAiii.) y = Mx/m = 1/m∫∫yp(x,y)dA

b.) Moment of Inertia:

iv.) Ix = ∫∫y^2p(x,y)dAv.) Iy = ∫∫x^2p(x,y)dAvi.) I0 = ∫∫(x^2+y^2)p(x,y)dA

v.) Surface Area

a.) A(s) = ∫∫√[1+(∂z/∂x)^2+(∂z/∂y)^2]dA

vi.) Triple Integrals

a.) ∫∫∫ƒ(x,y,z)dV

b.) Remember to change variables accordingly, use drawings.

c.) Mass and Moments:

i.) m = ∫∫∫p(x,y,z)dV

ii.) x = Myz/m = 1/m∫∫∫xp(x,y,z)dAiii.) y = Mxz/m = 1/m∫∫∫yp(x,y,z)dAiv.) z = Mxy/m = 1/m∫∫∫zp(x,y,z)dA

d.) Moment of Inertia:

i.) Ix = ∫∫∫(y^2 + z^2)p(x,y,z)dAii.) Iy = ∫∫∫(x^2 + z^2)p(x,y,z)dAiii.) Iz = ∫∫∫(x^2+y^2)p(x,y,z)dA

e.) Cylindrical Coordinates:

i.) Conversions: x = rcosø, y = rcosø, z=z; r^2 = x^2+y^2, tanø = y/xii.) ∫∫∫ƒ(rcosø,rsinø,z)rdrdødz

f.) Spherical Coordinates:

i.) Conversions: x = psin¢cosø, y = psin¢cosø, z = pcos¢; p^2 = x^2 + y^2 +z^2ii.) Definitions: p = like radius, outward from origin; ¢ down from z axis, down on both sides of axis; ø on xy plane around z in circle.iii.) General Form: ∫∫∫ƒ(psin¢cosø,psin¢cosø,pcos¢)p^2sin¢^2dpdød

Ch. 16:

i.) Vector Fields

a.) A function F that assigns point (x,y,z) to three dimensional vector F(x,y,z)

ii.) Line Integral

a.) Scalar Form: ∫cƒ(r)|r'(t)|ds

b.) Vector Form ∫cF(r)•(r'(t))ds

c.) Parameterize - find curve for line by setting variables = t, use F to find "field" acting on it.

iii.) Fundamental Theory of Line Integrals

a.) ∫cGradƒ(r)•dr = ƒ(r(b)) - ƒ(r(a))b.) ∫cF•dr is independent of path in D if and only if ∫cF•dr = 0 for every closed path C in D. c.) If Gradƒ = F, then a field is conservative; if ∂P/∂y = ∂Q/∂x F is conservative.d.) To find a function F from ƒ, take the integrals of each term, put repeats into equation only once.

iv.) Green's Theorem

a.) For positively oriented, simple closed curve bounded by C, D = :

∫c Pdx + Qdy = ∫∫(dQ/dx - dP/dy)dAb.) Area Cases:

A = 1/2∫c xdy-ydx

v.) Curl and Divergence Theorem

a.) curlF = Grad x F - used for vectorsb.) If curlF = 0, F is conservativec.) divF = Grad • F - used for scalars

vi.) Parametric Surfaces

a.) Find parameterization: r(u,v) = x(u,v)i + y(u,v)j + z(u,v)k (One variable = constant)b.) Cross partials: A(s) = ∫∫D|ru x rv| dA

vii.) Surface Integrals

a.) Used to find the surface area.b.) Scalar: ∫cƒ(r)||ru x rv||dA

c.) Vector Form ∫cF(r)•|ru x rv|dA

-Synonymous to: ∫cF•dS = ∫∫D(-P(∂g/∂x) - Q(∂g/∂y) + R)dAviii.) Stokes Theorem

a.) Any surface "delimited" by an open space has the same flux as opening.

b.) ∫cF•dr = ∫∫s curl F•dS

ix.) Divergence Theorem

a.) If E is the simple solid region and S is the boundary surface of E, given positive orientation.

b.) ∫∫s F•dS = ∫∫∫E div F dV