Vector notation is ubiquitous in the modern literature on solid mechanics, fluid mechanics, biomechanics, nonlinear finite elements and a host of other subjects in mechanics. A student has to be familiar with the notation in order to be able to read the literature. In this section we introduce the notation that is used, common operations in vector algebra, and some ideas from vector calculus.
A vector is an object that has certain properties. What are these properties? We usually say that these properties are:
a vector has a magnitude (or length)
a vector has a direction. To make the definition of the vector object more precise we may also say that vectors are objects that satisfy the properties of a vector space .
The standard notation for a vector is lower case bold type (for example
a
{\displaystyle \mathbf {a} \,}
In Figure 1(a) you can see a vector
a
{\displaystyle \mathbf {a} }
basis (
e
1
,
e
2
{\displaystyle \mathbf {e} _{1},\mathbf {e} _{2}\,}
a
=
a
1
e
1
+
a
2
e
2
{\displaystyle \mathbf {a} =a_{1}\mathbf {e} _{1}+a_{2}\mathbf {e} _{2}\,}
where
e
1
{\displaystyle \mathbf {e} _{1}\,}
e
2
{\displaystyle \mathbf {e} _{2}\,}
orthonormal unit vectors . Recall that unit vectors are vectors of length 1. These vectors are also called basis vectors.
You could also represent the same vector
a
{\displaystyle \mathbf {a} \,}
g
1
,
g
2
{\displaystyle \mathbf {g} _{1},\mathbf {g} _{2}\,}
(
b
1
,
b
2
)
{\displaystyle (b_{1},b_{2})\,}
a
=
b
1
g
1
+
b
2
g
2
.
{\displaystyle \mathbf {a} =b_{1}\mathbf {g} _{1}+b_{2}\mathbf {g} _{2}\,~.}
Note that the basis vectors
g
1
{\displaystyle \mathbf {g} _{1}\,}
g
2
{\displaystyle \mathbf {g} _{2}\,}
linearly independent , that is, it should not be possible for us to represent one solely in terms of the others.
In three dimensions, using an orthonormal basis , we can write the vector
a
{\displaystyle \mathbf {a} \,}
a
=
a
1
e
1
+
a
2
e
2
+
a
3
e
3
{\displaystyle \mathbf {a} =a_{1}\mathbf {e} _{1}+a_{2}\mathbf {e} _{2}+a_{3}\mathbf {e} _{3}\,}
where
e
3
{\displaystyle \mathbf {e} _{3}\,}
e
1
{\displaystyle \mathbf {e} _{1}\,}
e
2
{\displaystyle \mathbf {e} _{2}\,}
Figure 1: A vector and its basis.
Some vector operations are shown in Figure 2.
Figure 2: Vector operations.
If
a
{\displaystyle \mathbf {a} \,}
b
{\displaystyle \mathbf {b} \,}
c
=
a
+
b
{\displaystyle \mathbf {c} =\mathbf {a} +\mathbf {b} \,}
The two vectors can also be subtracted from one another to give another vector
d
=
a
−
b
{\displaystyle \mathbf {d} =\mathbf {a} -\mathbf {b} \,}
Multiplication of a vector
b
{\displaystyle \mathbf {b} \,}
λ
{\displaystyle \lambda \,}
You can form a unit vector
b
^
{\displaystyle {\hat {\mathbf {b} }}\,}
b
{\displaystyle \mathbf {b} \,}
|
b
|
{\displaystyle |\mathbf {b} |\,}
b
^
=
b
|
b
|
.
{\displaystyle {\hat {\mathbf {b} }}={\cfrac {\mathbf {b} }{|\mathbf {b} |}}~.}
The scalar product or inner product or dot product of two vectors is defined as
a
∙
b
=
|
a
|
|
b
|
cos
(
θ
)
{\displaystyle \mathbf {a} \bullet \mathbf {b} =|\mathbf {a} ||\mathbf {b} |\cos(\theta )}
where
θ
{\displaystyle \theta \,}
If
a
{\displaystyle \mathbf {a} \,}
b
{\displaystyle \mathbf {b} \,}
θ
=
π
/
2
{\displaystyle \theta =\pi /2\,}
cos
(
θ
)
=
0
{\displaystyle \cos(\theta )=0\,}
a
∙
b
=
0
{\displaystyle {\mathbf {a} }\bullet {\mathbf {b} }=0}
The dot product therefore has the geometric interpretation as the length of the projection of
a
{\displaystyle \mathbf {a} \,}
b
^
{\displaystyle {\hat {\mathbf {b} }}\,}
The scalar product leads to a scalar quantity and can also be written in component form (with respect to a given basis) as
a
∙
b
=
a
1
b
1
+
a
2
b
2
+
a
3
b
3
=
∑
i
=
1..3
a
i
b
i
.
{\displaystyle {\mathbf {a} }\bullet {\mathbf {b} }=a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}=\sum _{i=1..3}a_{i}b_{i}~.}
If the vector is
n
{\displaystyle n}
a
∙
b
=
∑
i
=
1..
n
a
i
b
i
.
{\displaystyle {\mathbf {a} }\bullet {\mathbf {b} }=\sum _{i=1..n}a_{i}b_{i}~.}
Using the Einstein summation convention, we can also write the scalar product as
a
∙
b
=
a
i
b
i
.
{\displaystyle {\mathbf {a} }\bullet {\mathbf {b} }=a_{i}b_{i}~.}
Also notice that the following also hold for the scalar product
a
∙
b
=
b
∙
a
{\displaystyle {\mathbf {a} }\bullet {\mathbf {b} }={\mathbf {b} }\bullet {\mathbf {a} }}
a
∙
(
b
+
c
)
=
a
∙
b
+
a
∙
c
{\displaystyle {\mathbf {a} }\bullet {(\mathbf {b} +\mathbf {c} )}={\mathbf {a} }\bullet {\mathbf {b} }+{\mathbf {a} }\bullet {\mathbf {c} }}
The vector product (or cross product) of two vectors
a
{\displaystyle \mathbf {a} \,}
b
{\displaystyle \mathbf {b} \,}
c
{\displaystyle \mathbf {c} \,}
c
=
a
×
b
=
|
a
|
|
b
|
sin
(
θ
)
c
^
{\displaystyle \mathbf {c} ={\mathbf {a} }\times {\mathbf {b} }=|\mathbf {a} ||\mathbf {b} |\sin(\theta ){\hat {\mathbf {c} }}}
where
θ
{\displaystyle \theta \,}
a
{\displaystyle \mathbf {a} \,}
b
{\displaystyle \mathbf {b} \,}
c
^
{\displaystyle {\hat {\mathbf {c} }}\,}
a
{\displaystyle \mathbf {a} \,}
b
{\displaystyle \mathbf {b} \,}
Figure 3: Vector product of two vectors.
In terms of the orthonormal basis
(
e
1
,
e
2
,
e
3
)
{\displaystyle (\mathbf {e} _{1},\mathbf {e} _{2},\mathbf {e} _{3})\,}
a
×
b
=
|
e
1
e
2
e
3
a
1
a
2
a
3
b
1
b
2
b
3
|
.
{\displaystyle {\mathbf {a} }\times {\mathbf {b} }={\begin{vmatrix}\mathbf {e} _{1}&\mathbf {e} _{2}&\mathbf {e} _{3}\\a_{1}&a_{2}&a_{3}\\b_{1}&b_{2}&b_{3}\end{vmatrix}}~.}
In index notation, the cross product can be written as
a
×
b
≡
ε
i
j
k
e
i
a
j
b
k
.
{\displaystyle {\mathbf {a} }\times {\mathbf {b} }\equiv \varepsilon _{ijk}\mathbf {e} _{i}a_{j}b_{k}~.}
where
ε
i
j
k
{\displaystyle \varepsilon _{ijk}}
Some useful vector identities are given below.
a
×
b
=
−
b
×
a
{\displaystyle {\mathbf {a} }\times {\mathbf {b} }=-{\mathbf {b} }\times {\mathbf {a} }}
a
×
(
b
+
c
)
=
a
×
b
+
a
×
c
{\displaystyle {\mathbf {a} }\times {(\mathbf {b} +\mathbf {c} })={\mathbf {a} }\times {\mathbf {b} }+{\mathbf {a} }\times {\mathbf {c} }}
a
×
(
b
×
c
)
=
b
(
a
∙
c
)
−
c
(
a
∙
b
)
{\displaystyle {\mathbf {a} }\times {({\mathbf {b} }\times {\mathbf {c} })}=\mathbf {b} ({\mathbf {a} }\bullet {\mathbf {c} })-\mathbf {c} ({\mathbf {a} }\bullet {\mathbf {b} })}
(
a
×
b
)
×
c
=
b
(
a
∙
c
)
−
a
(
b
∙
c
)
{\displaystyle {({\mathbf {a} }\times {\mathbf {b} })}\times {\mathbf {c} }=\mathbf {b} ({\mathbf {a} }\bullet {\mathbf {c} })-\mathbf {a} ({\mathbf {b} }\bullet {\mathbf {c} })}
a
×
a
=
0
{\displaystyle {\mathbf {a} }\times {\mathbf {a} }=\mathbf {0} }
a
∙
(
a
×
b
)
=
b
∙
(
a
×
b
)
=
0
{\displaystyle {\mathbf {a} }\bullet {({\mathbf {a} }\times {\mathbf {b} })}={\mathbf {b} }\bullet {({\mathbf {a} }\times {\mathbf {b} })}=\mathbf {0} }
(
a
×
b
)
∙
c
=
a
∙
(
b
×
c
)
{\displaystyle {({\mathbf {a} }\times {\mathbf {b} })}\bullet {\mathbf {c} }={\mathbf {a} }\bullet {({\mathbf {b} }\times {\mathbf {c} })}}
So far we have dealt with constant vectors. It also helps if the vectors are allowed to vary in space. Then we can define derivatives and integrals and deal with vector fields. Some basic ideas of vector calculus are discussed below.
Let
a
(
x
)
{\displaystyle \mathbf {a} (x)\,}
a
(
x
)
=
a
1
(
x
)
e
1
+
a
2
(
x
)
e
2
+
a
3
(
x
)
e
3
{\displaystyle \mathbf {a} (x)=a_{1}(x)\mathbf {e} _{1}+a_{2}(x)\mathbf {e} _{2}+a_{3}(x)\mathbf {e} _{3}\,}
where
x
{\displaystyle x\,}
Then the derivative of
a
(
x
)
{\displaystyle \mathbf {a} (x)\,}
x
{\displaystyle x\,}
d
a
(
x
)
d
x
=
lim
Δ
x
→
0
a
(
x
+
Δ
x
)
−
a
(
x
)
Δ
x
=
d
a
1
(
x
)
d
x
e
1
+
d
a
2
(
x
)
d
x
e
2
+
d
a
3
(
x
)
d
x
e
3
.
{\displaystyle {\cfrac {d\mathbf {a} (x)}{dx}}=\lim _{\Delta x\rightarrow 0}{\cfrac {\mathbf {a} (x+\Delta x)-\mathbf {a} (x)}{\Delta x}}={\cfrac {da_{1}(x)}{dx}}\mathbf {e} _{1}+{\cfrac {da_{2}(x)}{dx}}\mathbf {e} _{2}+{\cfrac {da_{3}(x)}{dx}}\mathbf {e} _{3}~.}
If
a
(
x
)
{\displaystyle \mathbf {a} (x)\,}
b
(
x
)
{\displaystyle \mathbf {b} (x)\,}
d
(
a
∙
b
)
d
x
=
a
∙
d
b
d
x
+
d
a
d
x
∙
b
d
(
a
×
b
)
d
x
=
a
×
d
b
d
x
+
d
a
d
x
×
b
d
[
a
∙
(
b
×
c
)
]
d
x
=
d
a
d
x
∙
(
b
×
c
)
+
a
∙
(
d
b
d
x
×
c
)
+
a
∙
(
b
×
d
c
d
x
)
{\displaystyle {\begin{aligned}{\cfrac {d({\mathbf {a} }\bullet {\mathbf {b} })}{dx}}&={\mathbf {a} }\bullet {\cfrac {d\mathbf {b} }{dx}}+{\cfrac {d\mathbf {a} }{dx}}\bullet {\mathbf {b} }\\{\cfrac {d({\mathbf {a} }\times {\mathbf {b} })}{dx}}&={\mathbf {a} }\times {\cfrac {d\mathbf {b} }{dx}}+{\cfrac {d\mathbf {a} }{dx}}\times {\mathbf {b} }\\{\cfrac {d[{\mathbf {a} }\bullet {({\mathbf {b} }\times {\mathbf {c} })}]}{dx}}&={\cfrac {d\mathbf {a} }{dx}}\bullet {({\mathbf {b} }\times {\mathbf {c} })}+{\mathbf {a} }\bullet {\left({\cfrac {d\mathbf {b} }{dx}}\times {\mathbf {c} }\right)}+{\mathbf {a} }\bullet {\left({\mathbf {b} }\times {\cfrac {d\mathbf {c} }{dx}}\right)}\end{aligned}}}
Let
x
{\displaystyle \mathbf {x} \,}
scalar function (
g
{\displaystyle g\,}
g
=
g
(
x
)
{\displaystyle g=g(\mathbf {x} )\,}
represents a scalar field . An example of a scalar field is the temperature . See Figure4(a).
Figure 4: Scalar and vector fields.
If there is a vector function (
a
{\displaystyle \mathbf {a} \,}
a
=
a
(
x
)
{\displaystyle \mathbf {a} =\mathbf {a} (\mathbf {x} )\,}
represents a vector field . An example is the displacement field. See Figure 4(b).
Let
φ
(
x
)
{\displaystyle \varphi (\mathbf {x} )\,}
x
{\displaystyle \mathbf {x} \,}
x
1
,
x
2
,
x
3
{\displaystyle x_{1},x_{2},x_{3}\,}
e
1
,
e
2
,
e
3
{\displaystyle \mathbf {e} _{1},\mathbf {e} _{2},\mathbf {e} _{3}\,}
gradient of
φ
{\displaystyle \varphi \,}
∇
φ
=
∂
φ
∂
x
1
e
1
+
∂
φ
∂
x
2
e
2
+
∂
φ
∂
x
3
e
3
.
{\displaystyle {\boldsymbol {\nabla }}{\varphi }={\frac {\partial \varphi }{\partial x_{1}}}~\mathbf {e} _{1}+{\frac {\partial \varphi }{\partial x_{2}}}~\mathbf {e} _{2}+{\frac {\partial \varphi }{\partial x_{3}}}~\mathbf {e} _{3}~.}
In index notation,
∇
φ
≡
φ
,
i
e
i
.
{\displaystyle {\boldsymbol {\nabla }}{\varphi }\equiv \varphi _{,i}~\mathbf {e} _{i}~.}
The gradient is obviously a vector and has a direction. We can think of the gradient at a point being the vector perpendicular to the level contour at that point.
It is often useful to think of the symbol
∇
{\displaystyle {\boldsymbol {\nabla }}{}}
∇
=
∂
∂
x
1
e
1
+
∂
∂
x
2
e
2
+
∂
∂
x
3
e
3
.
{\displaystyle {\boldsymbol {\nabla }}{}={\frac {\partial }{\partial x_{1}}}~\mathbf {e} _{1}+{\frac {\partial }{\partial x_{2}}}~\mathbf {e} _{2}+{\frac {\partial }{\partial x_{3}}}~\mathbf {e} _{3}~.}
If we form a scalar product of a vector field
u
(
x
)
{\displaystyle \mathbf {u} (\mathbf {x} )\,}
∇
{\displaystyle {\boldsymbol {\nabla }}{}}
scalar quantity called the
divergence of the vector field. Thus,
∇
∙
u
=
∂
u
1
∂
x
1
+
∂
u
2
∂
x
2
+
∂
u
3
∂
x
3
.
{\displaystyle {\boldsymbol {\nabla }}\bullet {\mathbf {u} }={\frac {\partial u_{1}}{\partial x_{1}}}+{\frac {\partial u_{2}}{\partial x_{2}}}+{\frac {\partial u_{3}}{\partial x_{3}}}~.}
In index notation,
∇
∙
u
≡
u
i
,
i
.
{\displaystyle {\boldsymbol {\nabla }}\bullet {\mathbf {u} }\equiv u_{i,i}~.}
If
∇
∙
u
=
0
{\displaystyle {\boldsymbol {\nabla }}\bullet {\mathbf {u} }=0}
u
{\displaystyle \mathbf {u} \,}
divergence-free field.
The physical significance of the divergence of a vector field is the rate at which some density exits a given region of space. In the absence of the creation or destruction of matter, the density within a region of space can change only by having it flow into or out of the region.
The curl of a vector field
u
(
x
)
{\displaystyle \mathbf {u} (\mathbf {x} )\,}
vector defined as
∇
×
u
=
[
e
1
e
2
e
3
∂
∂
x
1
∂
∂
x
2
∂
∂
x
3
u
1
u
2
u
3
]
{\displaystyle {\boldsymbol {\nabla }}\times {\mathbf {u} }={\begin{bmatrix}\mathbf {e} _{1}&\mathbf {e} _{2}&\mathbf {e} _{3}\\{\frac {\partial }{\partial x_{1}}}&{\frac {\partial }{\partial x_{2}}}&{\frac {\partial }{\partial x_{3}}}\\u_{1}&u_{2}&u_{3}\\\end{bmatrix}}}
The physical significance of the curl of a vector field is the amount of rotation or angular momentum of the contents of a region of space.
The Laplacian of a scalar field
φ
(
x
)
{\displaystyle \varphi (\mathbf {x} )\,}
scalar defined as
∇
2
φ
:=
∇
∙
∇
φ
=
∂
2
φ
∂
x
1
2
+
∂
2
φ
∂
x
2
2
+
∂
2
φ
∂
x
3
2
.
{\displaystyle \nabla ^{2}{\varphi }:={\boldsymbol {\nabla }}\bullet {{\boldsymbol {\nabla }}{\varphi }}={\frac {\partial ^{2}\varphi }{\partial x_{1}^{2}}}+{\frac {\partial ^{2}\varphi }{\partial x_{2}^{2}}}+{\frac {\partial ^{2}\varphi }{\partial x_{3}^{2}}}~.}
The Laplacian of a vector field
u
(
x
)
{\displaystyle \mathbf {u} (\mathbf {x} )\,}
∇
2
u
:=
(
∇
2
u
1
)
e
1
+
(
∇
2
u
2
)
e
2
+
(
∇
2
u
3
)
e
3
.
{\displaystyle \nabla ^{2}{\mathbf {u} }:=(\nabla ^{2}{u_{1}})\mathbf {e} _{1}+(\nabla ^{2}{u_{2}})\mathbf {e} _{2}+(\nabla ^{2}{u_{3}})\mathbf {e} _{3}~.}
Let
u
(
x
)
{\displaystyle \mathbf {u} (\mathbf {x} )\,}
Ω
{\displaystyle \Omega \,}
Γ
{\displaystyle \Gamma \,}
divergence theorem states that
∫
Ω
∇
∙
u
d
V
=
∫
Γ
n
∙
u
d
A
{\displaystyle {\int _{\Omega }{\boldsymbol {\nabla }}\bullet {\mathbf {u} }~dV=\int _{\Gamma }{\mathbf {n} }\bullet {\mathbf {u} }~dA}}
where
n
{\displaystyle \mathbf {n} \,}
In index notation,
∫
Ω
u
i
,
i
d
V
=
∫
Γ
n
i
u
i
d
A
{\displaystyle \int _{\Omega }u_{i,i}~dV=\int _{\Gamma }n_{i}u_{i}~dA}
Figure 5: Volume for application of the divergence theorem.
Some frequently used identities from vector calculus are listed below.
∇
∙
(
a
+
b
)
=
∇
∙
a
+
∇
∙
b
{\displaystyle {\boldsymbol {\nabla }}\bullet {(\mathbf {a} +\mathbf {b} )}={\boldsymbol {\nabla }}\bullet {\mathbf {a} }+{\boldsymbol {\nabla }}\bullet {\mathbf {b} }}
∇
×
(
a
+
b
)
=
∇
×
a
+
∇
×
b
{\displaystyle {\boldsymbol {\nabla }}\times {(\mathbf {a} +\mathbf {b} )}={\boldsymbol {\nabla }}\times {\mathbf {a} }+{\boldsymbol {\nabla }}\times {\mathbf {b} }}
∇
∙
(
φ
a
)
=
(
∇
φ
)
∙
a
+
φ
(
∇
∙
a
)
{\displaystyle {\boldsymbol {\nabla }}\bullet {(\varphi \mathbf {a} )}=({\boldsymbol {\nabla }}{\varphi })\bullet \mathbf {a} +\varphi ({\boldsymbol {\nabla }}\bullet {\mathbf {a} })}
∇
×
(
φ
a
)
=
(
∇
φ
)
×
a
+
φ
(
∇
×
a
)
{\displaystyle {\boldsymbol {\nabla }}\times {(\varphi \mathbf {a} )}={({\boldsymbol {\nabla }}{\varphi })}\times {\mathbf {a} }+\varphi ({\boldsymbol {\nabla }}\times {\mathbf {a} })}
∇
∙
(
a
×
b
)
=
b
∙
(
∇
×
a
)
−
a
∙
(
∇
×
b
)
{\displaystyle {\boldsymbol {\nabla }}\bullet {({\mathbf {a} }\times {\mathbf {b} })}={\mathbf {b} }\bullet {({\boldsymbol {\nabla }}\times {\mathbf {a} })}-{\mathbf {a} }\bullet {({\boldsymbol {\nabla }}\times {\mathbf {b} })}}
∇
(
φ
a
)
=
a
⊗
(
∇
φ
)
+
φ
∇
a
{\displaystyle {\boldsymbol {\nabla }}(\varphi ~\mathbf {a} )=\mathbf {a} \otimes ({\boldsymbol {\nabla }}\varphi )+\varphi ~{\boldsymbol {\nabla }}\mathbf {a} }
![{\displaystyle {\begin{aligned}{\cfrac {d({\mathbf {a} }\bullet {\mathbf {b} })}{dx}}&={\mathbf {a} }\bullet {\cfrac {d\mathbf {b} }{dx}}+{\cfrac {d\mathbf {a} }{dx}}\bullet {\mathbf {b} }\\{\cfrac {d({\mathbf {a} }\times {\mathbf {b} })}{dx}}&={\mathbf {a} }\times {\cfrac {d\mathbf {b} }{dx}}+{\cfrac {d\mathbf {a} }{dx}}\times {\mathbf {b} }\\{\cfrac {d[{\mathbf {a} }\bullet {({\mathbf {b} }\times {\mathbf {c} })}]}{dx}}&={\cfrac {d\mathbf {a} }{dx}}\bullet {({\mathbf {b} }\times {\mathbf {c} })}+{\mathbf {a} }\bullet {\left({\cfrac {d\mathbf {b} }{dx}}\times {\mathbf {c} }\right)}+{\mathbf {a} }\bullet {\left({\mathbf {b} }\times {\cfrac {d\mathbf {c} }{dx}}\right)}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2786e7528bee7e1d53f05023c774fc00ed1c0858)
