Physical field
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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) | \[curl\mathbf{\overrightarrow{F}}\neq 0\;\;\;\;\;div\mathbf{\overrightarrow{F}}=0\] |
|---|---|
| Conservative (potential) | \[curl\mathbf{\overrightarrow{F}}=0\] |
| Uniform | \[curl\mathbf{\overrightarrow{F}}=0\;\;\;\;\;div\mathbf{\overrightarrow{F}}=0\;\;\;\;\;\mathbf{\overrightarrow{F}}=const\] |
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).
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