Vector Algebra and Calculus
Components:
A =Ax î + Ay ĵ + Az k̂
∂φ
∂φ
∂φ
ĵ +
k̂
∇φ = î +
∂x
∂y
∂z
Scalar (“Dot”) Product:
A · B =Ax Bx + Ay By + Az Bz
∂Ax ∂Ay ∂Az
∇·A=
+
+
∂x
∂y
∂z
Vector (“Cross”) Product:
A × B =(Ay Bz − Az By )î + (Az Bx − Ax Bz )ĵ + (Ax By − Ay Bx )k̂
∂Ay ∂Ax
∂Az ∂Ay
∂Ax ∂Az
∇×A=
−
−
−
î +
ĵ +
k̂
∂y
∂z
∂z
∂x
∂x
∂y
Some Identities:
A · (B × C) = (A × B) · C)
A × (B × C) = (A · C)B − (A · B)C
It is sometimes convenient to represent the component labels of a vector by a
subscript which takes the values 1,2,3 for components x, y, z. Then we can write
the scalar product, for example, as
A·B=
3
X
Ai Bi .
i=1
Summation Convention: Repeated indices are automatically summed. Thus
A · B = Ai Bi .
1
Kronecker Delta:
δij =
1 for i = j
0 for i 6= j.
The Epsilon Symbol: Let
ǫijk
Then


 +1 (ijk) an even permutation of (123)
= −1 (ijk) an odd permutation of (123)


0
two or more of i, j, k are equal.
(A × B)i = ǫijk Aj Bk .
Identity:
ǫijk ǫmnk = δim δjn − δin δjm .
Caution: When applying vector identities to expressions involving the gradient
differential operator ∇ care must be taken that the derivatives act on all functions
to their right. For example:
∇ × (B × C) =
∂
∂
(BCi ) −
(Bi C)
∂xi
∂xi
Gauss Theorem:
Z
dV ∇ · E =
I
dA · E
Stokes Theorem:
Z
dA · ∇ × E =
I
dl · E
Potential Difference:
ZP2
dl · ∇φ = φ(P2 ) − φ(P1 )
P1
2