MATH11
Find potential function for conservative vector field F(x,y,z)=<P,Q,R>
1. Integrate P with respect to x f(x,y,z) = ∫P + g(y,z) 2. Take partial derivative with respect to y; solve for gy. fy = ∫Py + gy(y,z) = Q 3. Integrate gy with respect to y. g(y,z) = ∫gy + h(z) 4. Take partial derivative with respect to z; solve for h'(z). gz = gz(y,z) + h'(z) = R 5. Integrate h'(z) with respect to z. ∫h'(z) = h(z) 6. Add up to get potential function. f(x,y,z) = ∫P + g(y,z) + h(z) + C
Lagrange steps
1. Set up system of equations ▽f = λ▽g g(x0,y0) = 0 2. Solve for x0 and y0 3. Largest value of f maximizes function; smallest value minimizes
Continuity conditions
1. f(a,b) exists 2. lim(x,y)→(a,b) f(x,y) = exists 3. lim(x,y)→(a,b) f(x,y) = f(a,b)
Average Value of a Function
1/Area ∬R f(x,y)dxdy
2nd partial derivative test
D = fxx(a,b)fyy(a,b) - (fxy(a,b))^2 D>0 fxx>0 : local min D>0 fxx<0 : local max D<0 : saddle D=0 : inconclusive
Directional Derivative
Duf(x,y) = fx(x,y)cosϴ+fy(x,y)sinϴ
Linear approximation
L(x,y) = f(a,b) + fx(a,b)(x-a) + fy(a,b)(y-b)
Cross partial property of conservative vector fields
Py = Qx, Qz = Ry, Pz = Rx if vector field is conservative
Flux integral
Surface integral over vector field ∬S F∙NdS = ∬S F(r(u,v))∙(tu x tv) dA
Equation of a plane
a(x-x0)+b(y-y0)+c(z-z0)=0 r=<x,y,z> r0=(x0,y0,z0) n=<a,b,c>
component of u onto v
comp_v^u = (u·v)/||v||
Dot product
cosθ = (u·v)/||u||/||v||
Angle between planes
cosθ = |n1·n2|/||n1||/||n2||
Curl equation
curlF = <(Ry-Qz),(Pz-Rx),(Qx-Py)> = ∇ x F
Distance from point to line
d = ||PM x v||/||v||
Distance from point to plane
d = ||QP·n||/||n||
How to find if a vector field is not conservative
dP/dy = dQ/dx , dQ/dz = dR/dy , dP/dz = dR/dx
Jacobian
det | dx/du dy/du | | dx/dv dy/dv |
Divergence equation
divF = Px + Qy + Rz = ∇∙F
Implicit differentiation
dy/dx = -(df/dx)/(df/dy)
Total differential
dz = fx(x,y)dx + fy(x,y)dy
Differentiability equation
f(x,y) = f(x0,y0) + fx(x0,y0) + fy(x0,y0) + E(x,y)
Critical point definition
fx = fy = 0 OR fx or fy = DNE
Clairaut's Theorem
fxy=fyx
Evaluating a vector line integral
integral from a to b of F(r(t))⋅r'(t) dt
Evaluating a scalar line integral
integral from a to b of f(r(t)) ||r'(t)||dt
E(x,y)
lim(x,y)→(x0,y0) of E(x,y)/√((x-x0)^2+(y-y0)^2) = 0
Max/Min directional derivatives
max Duf = ||▽f|| min Duf = -||▽f||
Gradient
normal to level curve
projection of u onto v
proj_v^u = (u·v)/||v||^2 * v
Line equation
r = r0 + tv = <x0 + at, y0 + bt, z0 + ct>
Cross product
sinθ = ||u x v||/||u||/||v||
Property of cross products
u x (v+w) = u x v + u x w
cartesian→cylindrical
x=rcosϴ y=rsinϴ z=z r = √x^2+y^2
cartesian→spherical
x=ρsinϕcosθ y=ρsinϕsinθ z=ρcosϕ ρ^2 = x^2+y^2+z^2 r = ρ^2sinϕ
Equation of a cone
x^2+y^2=z^2
Tangent plane equation
z = z0 + fx(x0,y0)(x-x0) + fy(x0,y0)(y-y0)
Equation of a hemisphere
z^2=c-x^2-y^2
Vector line integral
∫C F*dr
Stokes' Theorem
∫C F∙dr = ∬S curlF∙dS
Flux over a curve
∫C F⋅n ds = ∫C F(r(t))⋅n(t) dt
Scalar Line Integral
∫C f(x,y)ds
Fundamental theorem for line integrals
∫C ∇f∙dr = f(r(b)) - f(r(a)) r(t): a ≤ t ≤ b ∫C ∇f∙dr = f(r(t))∙r'(t)dt
Path independence
∫C1 F∙dr = ∫C2 F∙dr c1 and c2 share same endpoints
Green's theorem flux form
∬C Px + Qy dA = ∬C F∙NdS
Surface area of S
∬D ||tu x tv|| dA tu = <dx/du,dy/du,dz/du> and tv = <dx/dv,dy/dv,dz/dv>
Change of variables equation
∬R f(x,y)dA = ∬S f(g(u,v),h(u,v)) |d(x,y)/d(u,v)| dudv
Surface integral
∬S F∙dS = ∬S F∙NdS
Calculate scalar surface integrals
∬S f(x,y,z) ds = ∬D f(r(u,v)) ||tu x tv|| dA
Divergence theorem
∭E divF dV = ∬S F∙dS
Green's theorem circulation form
∮C F∙dr = ∬D (Qx-Py)dA
Lagrange multiplier
▽f = λ▽g