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.
Derivative of a vector valued function
edit
Let
a
(
x
)
{\displaystyle \mathbf {a} (x)\,}
be a vector function that can be represented as
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\,}
is a scalar.
Then the derivative of
a
(
x
)
{\displaystyle \mathbf {a} (x)\,}
with respect to
x
{\displaystyle x\,}
is
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}~.}
Note: In the above equation, the unit vectors
e
i
{\displaystyle \mathbf {e} _{i}}
(i=1,2,3) are assumed constant.
If
a
(
x
)
{\displaystyle \mathbf {a} (x)\,}
and
b
(
x
)
{\displaystyle \mathbf {b} (x)\,}
are two vector functions, then from the chain rule we get
d
(
a
⋅
b
)
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
t
=
d
a
d
t
⋅
(
b
×
c
)
+
a
⋅
(
d
b
d
t
×
c
)
+
a
⋅
(
b
×
d
c
d
t
)
{\displaystyle {\begin{aligned}{\cfrac {d({\mathbf {a} }\cdot {\mathbf {b} })}{x}}&={\mathbf {a} }\cdot {\cfrac {d\mathbf {b} }{dx}}+{\cfrac {d\mathbf {a} }{dx}}\cdot {\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} }\cdot {({\mathbf {b} }\times {\mathbf {c} })}]}{dt}}&={\cfrac {d\mathbf {a} }{dt}}\cdot {({\mathbf {b} }\times {\mathbf {c} })}+{\mathbf {a} }\cdot {\left({\cfrac {d\mathbf {b} }{dt}}\times {\mathbf {c} }\right)}+{\mathbf {a} }\cdot {\left({\mathbf {b} }\times {\cfrac {d\mathbf {c} }{dt}}\right)}\end{aligned}}}
Scalar and vector fields
edit
Let
x
{\displaystyle \mathbf {x} \,}
be the position vector of any point in space. Suppose that
there is a scalar function (
g
{\displaystyle g\,}
) that assigns a value to each point in space. Then
g
=
g
(
x
)
{\displaystyle g=g(\mathbf {x} )\,}
represents a scalar field . An example of a scalar field is the temperature . See Figure4(a).
If there is a vector function (
a
{\displaystyle \mathbf {a} \,}
) that assigns a vector to each point in space, then
a
=
a
(
x
)
{\displaystyle \mathbf {a} =\mathbf {a} (\mathbf {x} )\,}
represents a vector field . An example is the displacement field. See Figure 4(b).
Gradient of a scalar field
edit
Let
φ
(
x
)
{\displaystyle \varphi (\mathbf {x} )\,}
be a scalar function. Assume that the partial derivatives of the function are continuous in some region of space. If the point
x
{\displaystyle \mathbf {x} \,}
has coordinates (
x
1
,
x
2
,
x
3
{\displaystyle x_{1},x_{2},x_{3}\,}
) with respect to the basis (
e
1
,
e
2
,
e
3
{\displaystyle \mathbf {e} _{1},\mathbf {e} _{2},\mathbf {e} _{3}\,}
), the gradient of
φ
{\displaystyle \varphi \,}
is defined as
∇
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 }}{}}
as an operator of the form
∇
=
∂
∂
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}~.}
Divergence of a vector field
edit
If we form a scalar product of a vector field
u
(
x
)
{\displaystyle \mathbf {u} (\mathbf {x} )\,}
with the
∇
{\displaystyle {\boldsymbol {\nabla }}{}}
operator, we get a 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 }}\cdot {\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 }}\cdot {\mathbf {u} }\equiv u_{i,i}~.}
If
∇
⋅
u
=
0
{\displaystyle {\boldsymbol {\nabla }}\cdot {\mathbf {u} }=0}
, then
u
{\displaystyle \mathbf {u} \,}
is called a 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.
Curl of a vector field
edit
The curl of a vector field
u
(
x
)
{\displaystyle \mathbf {u} (\mathbf {x} )\,}
is a vector defined as
∇
×
u
=
det
|
e
1
e
2
e
3
∂
∂
x
1
∂
∂
x
2
∂
∂
x
3
u
1
u
2
u
3
|
{\displaystyle {\boldsymbol {\nabla }}\times {\mathbf {u} }=\det {\begin{vmatrix}\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{vmatrix}}}
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.
Laplacian of a scalar or vector field
edit
The Laplacian of a scalar field
φ
(
x
)
{\displaystyle \varphi (\mathbf {x} )\,}
is a scalar defined as
∇
2
φ
:=
∇
⋅
∇
φ
=
∂
2
φ
∂
x
1
+
∂
2
φ
∂
x
2
+
∂
2
φ
∂
x
3
.
{\displaystyle \nabla ^{2}{\varphi }:={\boldsymbol {\nabla }}\cdot {{\boldsymbol {\nabla }}{\varphi }}={\frac {\partial ^{2}\varphi }{\partial x_{1}}}+{\frac {\partial ^{2}\varphi }{\partial x_{2}}}+{\frac {\partial ^{2}\varphi }{\partial x_{3}}}~.}
The Laplacian of a vector field
u
(
x
)
{\displaystyle \mathbf {u} (\mathbf {x} )\,}
is a vector defined as
∇
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}~.}