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1 Algebraic and Analytic MethodsTopics of Discussion

§1.6 Vectors and Vector-Valued Functions

  1. §1.6(i) Vectors
  2. §1.6(ii) Vectors: Alternative Notations
  3. §1.6(iii) Vector-Valued Functions
  4. §1.6(iv) Path and Line Integrals
  5. §1.6(v) Surfaces and Integrals over Surfaces

§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|=(a2b3a3b2)𝐢+(a3b1a1b3)𝐣+(a1b2a2b1)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).


1.6.12 ajbj=j=13ajbj=𝐚𝐛.


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


1.6.15 ϵ123 =ϵ312=1,
ϵ213 =ϵ321=1,
ϵ221 =0.
1.6.16 ϵjkϵmn=δj,mδk,nδj,nδk,m,

where δj,k is the Kronecker delta.

1.6.17 𝐞j×𝐞k=ϵjk𝐞;

compare (1.6.8).

1.6.18 aj𝐞j×bk𝐞k=ϵjkajbk𝐞;

compare (1.6.7)–(1.6.8).

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

§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|=(F3yF2z)𝐢+(F1zF3x)𝐣+(F2xF1y)𝐤.
1.6.23 (fg)=fg+gf,
1.6.24 (f/g)=(gffg)/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 (fggf)=f2gg2f,
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).

A path is defined by 𝐜(t)=(x(t),y(t),z(t)), with t ranging over an interval and x(t),y(t),z(t) differentiable. Letting

1.6.35 𝐜(t)=(x(t),y(t),z(t)),

then the length of a path for atb is

1.6.36 ab𝐜(t)dt.

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

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

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

1.6.38 𝐜𝐅d𝐬=ab𝐅(𝐜(t))𝐜(t)dt=ab(F1dxdt+F2dydt+F3dzdt)dt=𝐜F1dx+F2dy+F3dz.

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 𝐜𝐅d𝐬=𝐜1𝐅d𝐬.

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

1.6.40 𝐜𝐅d𝐬=𝐜1𝐅d𝐬.

In either case

1.6.41 𝐜fds=𝐜1fds,

when f is continuous, and

1.6.42 𝐜fd𝐬=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


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(F2xF1y)dA=C𝐅d𝐬=CF1dx+F2dy.

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


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×𝐓vdudv,


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))2dx,

and about the y-axis,

1.6.53 A(S)=2πab|x|1+(f(x))2dx.

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

1.6.54 Sf(x,y,z)dS=Df(𝚽(u,v))𝐓u×𝐓vdudv.

For a vector-valued function 𝐅,

1.6.55 S𝐅d𝐒=D𝐅(𝐓u×𝐓v)dudv,

where d𝐒 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)𝐅d𝐒=𝚽2(D2)𝐅d𝐒;

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 anticlockwise relative to the normals of S. Then

1.6.57 S(×𝐅)d𝐒=S𝐅d𝐬,

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)dV=S𝐅d𝐒,

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)dV=SfgndA,


1.6.60 V(f2gg2f)dV=S(fgngfn)dA,

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