SELECTED FORMULAS FOR VECTOR CALCULUS
DERIVATIVES
∇=i∂
∂x +j∂
∂y +k∂
∂z (“del” or “nabla” ooperator) F(x, y, z)=F1(x, y, z )i+F2(x, y, z)j+F3(x, y, z )k
∇φ(x, y, z)=grad φ(x, y, z )=∂φ
∂x i+∂φ
∂y j+∂φ
∂z k∇•F(x, y, z)=div F(x, y, z)= ∂F1
∂x +∂F2
∂y +∂F3
∂z
∇×F(x, y, z)=curl F(x, y, z )=
ijk
∂
∂x ∂
∂y ∂
∂z
F1F2F3
=∂F3
∂y −∂F2
∂z i+∂F1
∂z −∂F3
∂x j+∂F2
∂x −∂F1
∂y k
∇(φψ)=φ∇ψ+ψ∇φ∇•(F×G)=(∇×F)•G−F•(∇×G)
∇•(φF)=(∇φ)•F+φ(∇•F)∇×(F×G)=F(∇•G)−G(∇•F)−(F•∇ )G+(G•∇)F
∇×(φF)=(∇φ)×F+φ(∇×F)∇(F•G)=F×(∇×G)+G×(∇×F)+(F•∇)G+(G•∇)F
∇×(∇φ)=0(curlgrad =0)∇•(∇×F)=0 (div curl =0)
∇2φ(x, y, z)=∇•∇φ(x, y, z)=div grad φ=∂2φ
∂x2+∂2φ
∂y2+∂2φ
∂z2∇×(∇×F)=∇(∇•F)−∇
2F(curl curl =graddiv −laplacian)
DERIVATIVES AND INTEGRALS
Zb
a
f0(t)dt =f(b)−f(a) (the one-dimensional Fundamental Theorem)
ZC
grad φ•dr=φr(b)−φr(a),ifCis the curve r=r(t), a≤t≤b
ZZR∂F2
∂x −∂F1
∂y dA =IC
F•dr=IC
F1(x, y)dx +F2(x, y)dy,whereCis the positively oriented boundary of R(Green’s Theorem)
ZZS
curl F •b
NdS =IC
F•dr=IC
F1(x, y, z)dx +F2(x, y, z )dy +F3(x, y, z)dz,whereCis the oriented boundary of S(Stokes’s Theorem)
ZZZD
div F dV =Z
ZS
F•b
NdS,whereSis the closed boundary of D, with outward unit normal b
N (Divergence Theorem)
ZZZD
curl F dV =−Z
ZS
F×b
NdS ZZZD
grad φdV =Z
ZS
φb
NdS
SURFACE NORMALS AND AREA ELEMENTS
r=r(u, v) (parametrized surface): n=±∂r
∂u ×∂r
∂vdS=±∂r
∂u ×∂r
∂vdu dv
G(x, y, z) = 0 (smooth level surface): n=±∇G(x, y, z)dS=±∇G(x, y, z)
|∂G/∂z|dxdy
dS=n
|n•k|dx dy =n
|n•j|dx dz =n
|n•i|dy dz for other projections dS =|dS|
CYLINDRICAL COORDINATES
Transformation: x=rcosθ, y =rsin θ, z =zPosition vector: r=rcos θi+rsin θj+zk
Volume element: dV =rdrdθdz Surface area element (on r=a): dS =adθdz
SPHERICAL COORDINATES
Transformation: x=ρsinφcos θ, y =ρsin φsin θ, z =ρcosφPosition vector: r=ρsin φcos θi+ρsin φsin θj+ρcos φk
Volume element: dV =ρ2sinφdρdφdθ Surface area element (on ρ=a): dS =a2sin φdθdφ
CURVES IN 3-SPACE
Curve: r=r(t)=x(t)i+y(t)j+z(t)kVelocity: v=dr
dt =vb
TSpeed: v=|v|=ds
dt
Arc length: s=Zt
t0
vdt Acceleration: a=dv
dt =d2r
dt2Acceleration components: a=dv
dt b
T+v2κb
N
Unit tangent: b
T=v
vBinormal: b
B=v×a
|v×a|Normal: b
N=b
B×b
T=db
T/dt
|db
T/dt|
Curvature: κ=|v×a|
v3Radius of curvature: ρ=1
κTorsion: τ=(v×a)•(da/dt)
|v×a|2
Frenet-Serret formulas: db
T
ds =κb
N,db
N
ds =−κb
T+τb
B,db
B
ds =−τb
N
Modified from R. A. Adams, Calculus , A Complete Course, Addison-Wesley, 2003.
