Notation in vector calculus
edit
Vector calculus concerns differentiation and integration of vector or scalar fields particularly in a three-dimensional Euclidean space , and uses specific notations of differentiation. In a Cartesian coordinate o-xyz , assuming a vector field A is
A
=
(
A
x
,
A
y
,
A
z
)
{\displaystyle \mathbf {A} =(\mathbf {A} _{x},\mathbf {A} _{y},\mathbf {A} _{z})}
scalar field
φ
{\displaystyle \varphi }
φ
=
f
(
x
,
y
,
z
)
{\displaystyle \varphi =f(x,y,z)\,}
First, a differential operator, or a Hamilton operator ∇ which is called nabla is symbolically defined in a form of a vector,
∇
=
(
∂
∂
x
,
∂
∂
y
,
∂
∂
z
)
{\displaystyle \nabla =\left({\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},{\frac {\partial }{\partial z}}\right)}
where the terminology symbolically refrects that the operator ∇ will also be treated as an ordinary vector.
∇φ
Gradient
g
r
a
d
ϕ
{\displaystyle \mathrm {grad} \phi \,}
multiplication of ∇ and scalar field φ ,
g
r
a
d
φ
=
(
∂
φ
∂
x
,
∂
φ
∂
y
,
∂
φ
∂
z
)
{\displaystyle \mathrm {grad} \,\varphi =\left({\frac {\partial \varphi }{\partial x}},{\frac {\partial \varphi }{\partial y}},{\frac {\partial \varphi }{\partial z}}\right)}
=
(
∂
∂
x
,
∂
∂
y
,
∂
∂
z
)
φ
{\displaystyle =\left({\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},{\frac {\partial }{\partial z}}\right)\varphi }
=
∇
φ
{\displaystyle =\nabla \varphi }
∇∙A
Divergence
d
i
v
A
{\displaystyle \mathrm {div} \,\mathbf {A} \,}
A is a scalar, which is symbolically expressed by the dot product of ∇ and the vector A ,
d
i
v
A
=
∂
A
x
∂
x
+
∂
A
y
∂
y
+
∂
A
z
∂
z
{\displaystyle \mathrm {div\,} \mathbf {A} ={\partial A_{x} \over \partial x}+{\partial A_{y} \over \partial y}+{\partial A_{z} \over \partial z}}
=
(
∂
∂
x
,
∂
∂
y
,
∂
∂
z
)
⋅
A
{\displaystyle =\left({\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},{\frac {\partial }{\partial z}}\right)\cdot \mathbf {A} }
=
∇
⋅
A
{\displaystyle =\nabla \cdot \mathbf {A} }
∇2 φ
Laplacian
d
i
v
g
r
a
d
φ
{\displaystyle \mathrm {div} \,\mathrm {grad} \,\varphi \,}
φ
{\displaystyle \varphi }
2 and the scalar field φ ,
d
i
v
g
r
a
d
φ
=
∇
⋅
(
∇
φ
)
{\displaystyle \mathrm {div} \,\mathrm {grad} \,\varphi \,=\nabla \cdot (\nabla \varphi )}
=
(
∇
⋅
∇
)
φ
=
∇
2
φ
=
Δ
φ
{\displaystyle =(\nabla \cdot \nabla )\varphi =\nabla ^{2}\varphi =\Delta \varphi }
where,
Δ
=
∇
2
{\displaystyle \Delta =\nabla ^{2}}
Laplacian operator . ∇×A
Rotation
c
u
r
l
A
{\displaystyle \mathrm {curl} \,\mathbf {A} \,}
r
o
t
A
{\displaystyle \mathrm {rot} \,\mathbf {A} \,}
cross product of ∇ and the vector A ,
c
u
r
l
A
=
(
∂
A
z
∂
y
−
∂
A
y
∂
z
,
∂
A
x
∂
z
−
∂
A
z
∂
x
,
∂
A
y
∂
x
−
∂
A
x
∂
y
)
{\displaystyle \mathrm {curl} \,\mathbf {A} =\left({\partial A_{z} \over {\partial y}}-{\partial A_{y} \over {\partial z}},{\partial A_{x} \over {\partial z}}-{\partial A_{z} \over {\partial x}},{\partial A_{y} \over {\partial x}}-{\partial A_{x} \over {\partial y}}\right)}
=
(
∂
A
z
∂
y
−
∂
A
y
∂
z
)
i
+
(
∂
A
x
∂
z
−
∂
A
z
∂
x
)
j
+
(
∂
A
y
∂
x
−
∂
A
x
∂
y
)
k
{\displaystyle =\left({\partial A_{z} \over {\partial y}}-{\partial A_{y} \over {\partial z}}\right)\mathbf {i} +\left({\partial A_{x} \over {\partial z}}-{\partial A_{z} \over {\partial x}}\right)\mathbf {j} +\left({\partial A_{y} \over {\partial x}}-{\partial A_{x} \over {\partial y}}\right)\mathbf {k} }
=
|
i
j
k
∂
∂
x
∂
∂
y
∂
∂
z
A
x
A
y
A
z
|
{\displaystyle ={\begin{vmatrix}\mathbf {i} &\mathbf {j} &\mathbf {k} \\[5pt]{\cfrac {\partial }{\partial x}}&{\cfrac {\partial }{\partial y}}&{\cfrac {\partial }{\partial z}}\\[12pt]A_{x}&A_{y}&A_{z}\end{vmatrix}}}
=
∇
×
A
{\displaystyle =\nabla \times \mathbf {A} }
These notations with the operator ∇ mentioned above are very powerful as in symbolic operations. For example product rule in ordinary differentiation,
(
f
g
)
′
=
f
′
g
+
f
g
′
{\displaystyle (fg)'=f'g+fg'\,}
φ and ψ , and the rule is expressed
∇
(
ϕ
ψ
)
=
(
∇
ϕ
)
ψ
+
ϕ
(
∇
ψ
)
{\displaystyle \nabla (\phi \psi )=(\nabla \phi )\psi +\phi (\nabla \psi )\,}
![{\displaystyle ={\begin{vmatrix}\mathbf {i} &\mathbf {j} &\mathbf {k} \\[5pt]{\cfrac {\partial }{\partial x}}&{\cfrac {\partial }{\partial y}}&{\cfrac {\partial }{\partial z}}\\[12pt]A_{x}&A_{y}&A_{z}\end{vmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e72ffe52834409e7c6d201ea06ed04800bb7e85)
