# Physical field

Corresponding Wikipedia article: Vector field

The physical field is the vector function of a physical space point (coordinate).

The scalar field is a scalar function of a physical space point. The scalar is like a vector with the equal projections.

The spatial derivative of a field $$\mathbf{\overrightarrow{F}}$$ is the physical field vector, which projections are the derivatives of the field $$\mathbf{\overrightarrow{F}}$$ projections over the spatial lines. The spatial derivative is denoted by an operator $$del$$ or Nabla ($$\nabla$$), which is defined for an integer dimensionality. This notation may be confused with a divergence. For the one-dimensional space, it becomes an ordinary derivative, as the divergence also.

The gradient is a spatial derivative of a scalar field. It is denoted as $$grad$$ or $$\nabla$$.

The following linear operators over a physical field $$\mathbf{\overrightarrow{F}}$$ have the geometric definitions, where a coordinate system is used only in the special cases:

Physical meaning Operator Notation Definition Dimensionality
Vorticity Circulation $\oint{\mathbf{\overrightarrow{F}}\mathrm{d}\mathbf{\overrightarrow{l}}}$ $\geq 1$
Curl $curl\mathbf{\overrightarrow{F}}$ (vector projection)
$\lim_{S\to 0}\frac{\mathbf{\overrightarrow{n}}}{S}\oint{\mathbf{\overrightarrow{F}}\mathrm{d}\mathbf{\overrightarrow{l}}}$
$\lim_{S\to 0}\frac{1}{S}\oint{\mathbf{\overrightarrow{r}}\times\mathbf{\overrightarrow{F}}\mathrm{d}\alpha}$
$\geq 3$
$curl\mathbf{\overrightarrow{F}}=\nabla\times\mathbf{\overrightarrow{F}}$ $\begin{pmatrix}\mathbf{\overrightarrow{e}}_x & \mathbf{\overrightarrow{e}}_y & \mathbf{\overrightarrow{e}}_z \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ F_x & F_y & F_z\end{pmatrix}$ $3$
Source and sink Flux $\int{\mathbf{\overrightarrow{F}}\mathrm{d}\mathbf{\overrightarrow{S}}}$ $>1$
Divergence $div\mathbf{\overrightarrow{F}}$ $\lim_{V\to 0}\frac{1}{V}\oint{\mathbf{\overrightarrow{F}}\mathrm{d}\mathbf{\overrightarrow{S}}}$ $>1$
$div\mathbf{\overrightarrow{F}}=\nabla\cdot\mathbf{\overrightarrow{F}}$ $\sum{\frac{\partial F_n}{\partial x_n}}$ integer

$$\mathrm{d}\mathbf{\overrightarrow{l}}$$ is a tangent vector of the circulation curve;
$$\mathbf{\overrightarrow{n}}$$ is a normal vector of the circulation plane, which is directed by the right-hand rule;
$$\mathbf{\overrightarrow{r}}$$ is a position vector on the circulation curve;
$$\mathrm{d}\alpha$$ is an angle of an arc on the circulation curve;
$$S$$ is an area of the circulation surface;
$$\mathrm{d}S$$ is an area of the flux surface multiplied by its normal, which is directed outwards from the closed surface;
$$V$$ is a volume of the closed surface.

The Kelvin–Stokes theorem links a curve circulation to the flux through the surface bounded by this curve: $\oint{\mathbf{\overrightarrow{F}}\mathrm{d}\mathbf{\overrightarrow{l}}}=\int{curl\mathbf{\overrightarrow{F}}\mathrm{d}\mathbf{\overrightarrow{S}}}$ The divergence (Gauss–Ostrogradsky) theorem links a flux through a closed surface to the volumetric integral bounded by this surface: $\oint{\mathbf{\overrightarrow{F}}\mathrm{d}\mathbf{\overrightarrow{S}}}=\oint{div\mathbf{\overrightarrow{F}}\mathrm{d}V}$

Solenoidal (vortical) $curl\mathbf{\overrightarrow{F}}\neq 0\;\;\;\;\;div\mathbf{\overrightarrow{F}}=0$ $curl\mathbf{\overrightarrow{F}}=0$ $curl\mathbf{\overrightarrow{F}}=0\;\;\;\;\;div\mathbf{\overrightarrow{F}}=0\;\;\;\;\;\mathbf{\overrightarrow{F}}=const$

The vector potential is an abstract vector field, which substitutes the physical vortical field for simplicity. The potential $$\mathbf{\overrightarrow{A}}$$ is defined so: $\mathbf{\overrightarrow{F}}=curl\mathbf{\overrightarrow{A}}=\nabla\times\mathbf{\overrightarrow{A}}$ The potential (scalar potential) is a scalar field, which substitutes the physical potential field for simplicity. The potential $$P$$ is defined so: $\mathbf{\overrightarrow{F}}=grad\;P=\nabla P$ As the circulation of a potential field on any closed curve is zero, the potential is equal to the integral over an arbitrary unclosed curve: $P=\int{\mathbf{\overrightarrow{F}}\mathrm{d}\mathbf{\overrightarrow{l}}}+const$ The field of a scalar potential is not physical, and its value is relative.

## Vector analysis formulas

The formulas of vector analysis, which use the nabla operator as a vector, are proven and applicable only for the 3-dimensional space. For example:

 $div\;curl\mathbf{\overrightarrow{F}}=0$ $\nabla\cdot(\nabla\times\mathbf{\overrightarrow{F}})=0$ $curl\;grad\;P=0$ $\nabla\times(\nabla P)=0$ $\Delta P=div\;grad\;P$ $\Delta P=\nabla\cdot(\nabla P)=\nabla^2P$ $curl\;curl\;P=grad\;div\;P-\Delta P$ $\nabla\times\nabla P=\nabla(\nabla\cdot P)-\nabla^2P$ $curl\;P\mathbf{\overrightarrow{F}}=(grad\;P)\times\mathbf{\overrightarrow{F}}+P\;curl\mathbf{\overrightarrow{F}}$ $\nabla\times(P\mathbf{\overrightarrow{F}})=(\nabla P)\times\mathbf{\overrightarrow{F}}+P(\nabla\times\mathbf{\overrightarrow{F}})$ $div\;P\mathbf{\overrightarrow{F}}=(grad\;P)\mathbf{\overrightarrow{F}}+P\;div\mathbf{\overrightarrow{F}}$ $\nabla\cdot(P\mathbf{\overrightarrow{F}})=(\nabla P)\mathbf{\overrightarrow{F}}+P(\nabla\cdot\mathbf{\overrightarrow{F}})$ $grad(\mathbf{\overrightarrow{F}}_1\mathbf{\overrightarrow{F}}_2)=(\mathbf{\overrightarrow{F}}_2\cdot\nabla)\mathbf{\overrightarrow{F}}_1+(\mathbf{\overrightarrow{F}}_1\cdot\nabla)\mathbf{\overrightarrow{F}}_2+\mathbf{\overrightarrow{F}}_2\times curl\mathbf{\overrightarrow{F}}_1+\mathbf{\overrightarrow{F}}_1\times curl\mathbf{\overrightarrow{F}}_2$ $\nabla(\mathbf{\overrightarrow{F}}_1\mathbf{\overrightarrow{F}}_2)=(\mathbf{\overrightarrow{F}}_2\cdot\nabla)\mathbf{\overrightarrow{F}}_1+(\mathbf{\overrightarrow{F}}_1\cdot\nabla)\mathbf{\overrightarrow{F}}_2+\mathbf{\overrightarrow{F}}_2\times(\nabla\times\mathbf{\overrightarrow{F}}_1)+\mathbf{\overrightarrow{F}}_1\times(\nabla\times\mathbf{\overrightarrow{F}}_2)$ $curl(\mathbf{\overrightarrow{F}}_1\times\mathbf{\overrightarrow{F}}_2)=(\mathbf{\overrightarrow{F}}_2\cdot\nabla)\mathbf{\overrightarrow{F}}_1-(\mathbf{\overrightarrow{F}}_1\cdot\nabla)\mathbf{\overrightarrow{F}}_2-\mathbf{\overrightarrow{F}}_2 div\mathbf{\overrightarrow{F}}_1+\mathbf{\overrightarrow{F}}_1 div\mathbf{\overrightarrow{F}}_2$ $\nabla\times(\mathbf{\overrightarrow{F}}_1\times\mathbf{\overrightarrow{F}}_2)=(\mathbf{\overrightarrow{F}}_2\cdot\nabla)\mathbf{\overrightarrow{F}}_1-(\mathbf{\overrightarrow{F}}_1\cdot\nabla)\mathbf{\overrightarrow{F}}_2-\mathbf{\overrightarrow{F}}_2(\nabla\cdot\mathbf{\overrightarrow{F}}_1)+\mathbf{\overrightarrow{F}}_1(\nabla\cdot\mathbf{\overrightarrow{F}}_2)$ $div(\mathbf{\overrightarrow{F}}_1\times\mathbf{\overrightarrow{F}}_2)=\mathbf{\overrightarrow{F}}_2 curl\mathbf{\overrightarrow{F}}_1-\mathbf{\overrightarrow{F}}_1 curl\mathbf{\overrightarrow{F}}_2$ $\nabla\cdot(\mathbf{\overrightarrow{F}}_1\times\mathbf{\overrightarrow{F}}_2)=\mathbf{\overrightarrow{F}}_2(\nabla\times\mathbf{\overrightarrow{F}}_1)-\mathbf{\overrightarrow{F}}_1(\nabla\times\mathbf{\overrightarrow{F}}_2)$

$$\Delta$$ is a Laplace operator (Laplacian).