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Notation in vector calculus

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 $\mathbf {A} =(\mathbf {A} _{x},\mathbf {A} _{y},\mathbf {A} _{z})$, and a scalar field $\varphi$ is $\varphi =f(x,y,z)\,$.

First, a differential operator, or a Hamilton operator ∇ which is called nabla is symbolically defined in a form of a vector,

$\nabla =\left({\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},{\frac {\partial }{\partial z}}\right)$,

where the terminology symbolically refrects that the operator ∇ will also be treated as an ordinary vector.

∇φ

• Gradient: The gradient $\mathrm {grad} \phi \,$ of the scalar field φ is a vector, which is symbolically expressed by the multiplication of ∇ and scalar field φ,

$\mathrm {grad} \,\varphi =\left({\frac {\partial \varphi }{\partial x}},{\frac {\partial \varphi }{\partial y}},{\frac {\partial \varphi }{\partial z}}\right)$ ,

$=\left({\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},{\frac {\partial }{\partial z}}\right)\varphi$ ,

$=\nabla \varphi$ .

∇∙A

• Divergence: The divergence $\mathrm {div} \,\mathbf {A} \,$ of the vector A is a scalar, which is symbolically expressed by the dot product of ∇ and the vector A,

$\mathrm {div\,} \mathbf {A} ={\partial A_{x} \over \partial x}+{\partial A_{y} \over \partial y}+{\partial A_{z} \over \partial z}$ ,

$=\left({\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},{\frac {\partial }{\partial z}}\right)\cdot \mathbf {A}$,

$=\nabla \cdot \mathbf {A}$ .

∇²φ

• Laplacian: The Laplacian $\mathrm {div} \,\mathrm {grad} \,\varphi \,$ of the scalar field $\varphi$ is a scalar, which is symbolically expressed by the scalar multiplication of ∇² and the scalar field φ,

$\mathrm {div} \,\mathrm {grad} \,\varphi \,=\nabla \cdot (\nabla \varphi )$

$=(\nabla \cdot \nabla )\varphi =\nabla ^{2}\varphi =\Delta \varphi$ ,

where, $\Delta =\nabla ^{2}$ is called a Laplacian operator.

∇×A

• Rotation: The rotation $\mathrm {curl} \,\mathbf {A} \,$, or $\mathrm {rot} \,\mathbf {A} \,$, of the vector is a vector, which is symbolically expressed by the cross product of ∇ and the vector A,

$\mathrm {curl} \,\mathbf {A} =\left({\partial A_{z} \over {\partial y}}-{\partial A_{y} \over {\partial z}},{\partial A_{x} \over {\partial z}}-{\partial A_{z} \over {\partial x}},{\partial A_{y} \over {\partial x}}-{\partial A_{x} \over {\partial y}}\right)$,

$=\left({\partial A_{z} \over {\partial y}}-{\partial A_{y} \over {\partial z}}\right)\mathbf {i} +\left({\partial A_{x} \over {\partial z}}-{\partial A_{z} \over {\partial x}}\right)\mathbf {j} +\left({\partial A_{y} \over {\partial x}}-{\partial A_{x} \over {\partial y}}\right)\mathbf {k}$,

$={\begin{vmatrix}\mathbf {i} &\mathbf {j} &\mathbf {k} \\[5pt]{\cfrac {\partial }{\partial x}}&{\cfrac {\partial }{\partial y}}&{\cfrac {\partial }{\partial z}}\\[12pt]A_{x}&A_{y}&A_{z}\end{vmatrix}}$ ,

$=\nabla \times \mathbf {A}$ .

These notations with the operator ∇ mentioned above are very powerful as in symbolic operations. For example product rule in ordinary differentiation, $(fg)'=f'g+fg'\,$ in the Lagrange's notation, can directly be applied to the gradient of the multiplication of scalar fields φ and ψ, and the rule is expressed $\nabla (\phi \psi )=(\nabla \phi )\psi +\phi (\nabla \psi )\,$ as exactly same as ordinary one.