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§1.6 Vectors and Vector-Valued Functions

Contents

§1.6(i) Vectors

1.6.1 a =(a1,a2,a3),
b =(b1,b2,b3).

Dot Product (or Scalar Product)

1.6.2 ab=a1b1+a2b2+a3b3.

Magnitude and Angle of Vector a

1.6.3 a=aa,
1.6.4 cosθ=abab;

θ is the angle between a and b.

Unit Vectors

1.6.5 i =(1,0,0),
j =(0,1,0),
k =(0,0,1),
1.6.6 a=a1i+a2j+a3k.

Cross Product (or Vector Product)

1.6.7 i×j =k,
j×k =i,
k×i =j,
1.6.8 j×i =-k,
k×j =-i,
i×k =-j.
1.6.9 a×b=|ijka1a2a3b1b2b3|=(a2b3-a3b2)i+(a3b1-a1b3)j+(a1b2-a2b1)k=ab(sinθ)n,

where n is the unit vector normal to a and b whose direction is determined by the right-hand rule; see Figure 1.6.1.

See accompanying text
Figure 1.6.1: Vector notation. Right-hand rule for cross products. Magnify

Area of parallelogram with vectors a and b as sides =a×b.

Volume of a parallelepiped with vectors a, b, and c as edges =|a(b×c)|.

1.6.10 a×(b×c) =b(ac)-c(ab),
1.6.11 (a×b)×c =b(ac)-a(bc).

§1.6(ii) Vectors: Alternative Notations

The following notations are often used in the physics literature; see for example Lorentz et al. (1923, pp. 122–123).

Einstein Summation Convention

Much vector algebra involves summation over suffices of products of vector components. In almost all cases of repeated suffices, we can suppress the summation notation entirely, if it is understood that an implicit sum is to be taken over any repeated suffix. Thus pairs of indefinite suffices in an expression are resolved by being summed over (or “traced” over).

Example

1.6.12 ajbj=j=13ajbj=ab.

Next,

1.6.13 e1 =(1,0,0),
e2 =(0,1,0),
e3 =(0,0,1);

compare (1.6.5). Thus ajej=a.

Levi-Civita Symbol

1.6.14 ϵj,k,={+1,if j,k, is even permutation of 1,2,3,-1,if j,k, is odd permutation of 1,2,3,0,otherwise.

Examples

1.6.15 ϵ1,2,3 =ϵ3,1,2
=1,
ϵ2,1,3 =ϵ3,2,1
=-1,
ϵ2,2,1 =0.
1.6.16 ϵj,k,ϵ,m,n=δj,mδk,n-δj,nδk,m,

where δj,k is the Kronecker delta.

1.6.17 ej×ek=ϵj,k,e;

compare (1.6.8).

1.6.18 ajej×bkek=ϵj,k,ajbke;

compare (1.6.7)–(1.6.8).

Lastly, the volume of a parallelepiped with vectors a, b, and c as edges is |ϵj,k,ajbkc|.

§1.6(iii) Vector-Valued Functions

Del Operator

1.6.19 =ix+jy+kz.

The gradient of a differentiable scalar function f(x,y,z) is

1.6.20 gradf=f=fxi+fyj+fzk.

The divergence of a differentiable vector-valued function F=F1i+F2j+F3k is

1.6.21 divF=F=F1x+F2y+F3z.

The curl of F is

1.6.22 curlF=×F=|ijkxyzF1F2F3|=(F3y-F2z)i+(F1z-F3x)j+(F2x-F1y)k.
1.6.23 (fg)=fg+gf,
1.6.24 (f/g)=(gf-fg)/g2,
1.6.25 (fF)=f(F)+Ff,
1.6.26 (F×G)=G(×F)-F(×G),
1.6.27 (×F)=divcurlF=0,
1.6.28 ×(fF)=f(×F)+(f)×F,
1.6.29 ×(f)=curlgradf=0,
1.6.30 2f=(f),
1.6.31 2(fg)=f2g+g2f+2(fg),
1.6.32 (f×g)=0,
1.6.33 (fg-gf)=f2g-g2f,
1.6.34 ×(×F)=curlcurlF=(F)-2F.

§1.6(iv) Path and Line Integrals

Note: The terminology open and closed sets and boundary points in the (x,y) plane that is used in this subsection and §1.6(v) is analogous to that introduced for the complex plane in §1.9(ii).

c(t)=(x(t),y(t),z(t)), with t ranging over an interval and x(t),y(t),z(t) differentiable, defines a path.

1.6.35 c(t)=(x(t),y(t),z(t)).

The length of a path for atb is

1.6.36 abc(t)t.

The path integral of a continuous function f(x,y,z) is

1.6.37 cfs=abf(x(t),y(t),z(t))c(t)t.

The line integral of a vector-valued function F=F1i+F2j+F3k along c is given by

1.6.38 cFs=abF(c(t))c(t)t=ab(F1xt+F2yt+F3zt)t=cF1x+F2y+F3z.

A path c1(t), t[a,b], is a reparametrization of c(t), t[a,b], if c1(t)=c(t) and t=h(t) with h(t) differentiable and monotonic. If h(a)=a and h(b)=b, then the reparametrization is called orientation-preserving, and

1.6.39 cFs=c1Fs.

If h(a)=b and h(b)=a, then the reparametrization is orientation-reversing and

1.6.40 cFs=-c1Fs.

In either case

1.6.41 cfs=c1fs,

when f is continuous, and

1.6.42 cfs=f(c(b))-f(c(a)),

when f is continuously differentiable.

The geometrical image C of a path c is called a simple closed curve if c is one-to-one, with the exception c(a)=c(b). The curve C is piecewise differentiable if c is piecewise differentiable. Note that C can be given an orientation by means of c.

Green’s Theorem

Let

1.6.43 F(x,y)=F1(x,y)i+F2(x,y)j

and S be the closed and bounded point set in the (x,y) plane having a simple closed curve C as boundary. If C is oriented in the positive (anticlockwise) sense, then

1.6.44 S(F2x-F1y)A=CFs=CF1x+F2y.

Sufficient conditions for this result to hold are that F1(x,y) and F2(x,y) are continuously differentiable on S, and C is piecewise differentiable.

The area of S can be found from (1.6.44) by taking F(x,y)=-yi, xj, or -12yi+12xj.

§1.6(v) Surfaces and Integrals over Surfaces

A parametrized surface S is defined by

1.6.45 Φ(u,v)=(x(u,v),y(u,v),z(u,v))

with (u,v)D, an open set in the plane.

For x, y, and z continuously differentiable, the vectors

1.6.46 Tu=xu(u0,v0)i+yu(u0,v0)j+zu(u0,v0)k

and

1.6.47 Tv=xv(u0,v0)i+yv(u0,v0)j+zv(u0,v0)k

are tangent to the surface at Φ(u0,v0). The surface is smooth at this point if Tu×Tv0. A surface is smooth if it is smooth at every point. The vector Tu×Tv at (u0,v0) is normal to the surface at Φ(u0,v0).

The area A(S) of a parametrized smooth surface is given by

1.6.48 A(S)=DTu×Tvuv,

and

1.6.49 Tu×Tv=((x,y)(u,v))2+((y,z)(u,v))2+((x,z)(u,v))2.

The area is independent of the parametrizations.

For a sphere x=ρsinθcosϕ, y=ρsinθsinϕ, z=ρcosθ,

1.6.50 Tθ×Tϕ=ρ2|sinθ|.

For a surface of revolution, y=f(x), x[a,b], about the x-axis,

1.6.52 A(S)=2πab|f(x)|1+(f(x))2x,

and about the y-axis,

1.6.53 A(S)=2πab|x|1+(f(x))2x.

The integral of a continuous function f(x,y,z) over a surface S is

1.6.54 Sf(x,y,z)S=Df(Φ(u,v))Tu×Tvuv.

For a vector-valued function F,

1.6.55 SFS=DF(Tu×Tv)uv,

where S is the surface element with an attached normal direction Tu×Tv.

A surface is orientable if a continuously varying normal can be defined at all points of the surface. An orientable surface is oriented if suitable normals have been chosen. A parametrization Φ(u,v) of an oriented surface S is orientation preserving if Tu×Tv has the same direction as the chosen normal at each point of S, otherwise it is orientation reversing.

If Φ1 and Φ2 are both orientation preserving or both orientation reversing parametrizations of S defined on open sets D1 and D2 respectively, then

1.6.56 Φ1(D1)FS=Φ2(D2)FS;

otherwise, one is the negative of the other.

Stokes’s Theorem

Suppose S is an oriented surface with boundary S which is oriented so that its direction is clockwise relative to the normals of S. Then

1.6.57 S(×F)S=SFs,

when F is a continuously differentiable vector-valued function.

Gauss’s (or Divergence) Theorem

Suppose S is a piecewise smooth surface which forms the complete boundary of a bounded closed point set V, and S is oriented by its normal being outwards from V. Then

1.6.58 V(F)V=SFS,

when F is a continuously differentiable vector-valued function.

Green’s Theorem (for Volume)

For f and g twice-continuously differentiable functions

1.6.59 V(f2g+fg)V=SfgnA,

and

1.6.60 V(f2g-g2f)V=S(fgn-gfn)A,

where g/n=gn is the derivative of g normal to the surface outwards from V and n is the unit outer normal vector.