User:Kurosuke88/Temp1

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, $$(f g)' = f' g+f g'\,$$ 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.