Encyclopedia Magnetica™

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====== Scalar, Vector, Tensor ======

|< 100% >|
| //[[user/Stan Zurek]], Scalar, Vector, Tensor, Encyclopedia Magnetica//, \\ @PAGEL@   |

**Scalar, vector, tensor** - a mathematical representation of a physical entity that may be
characterized by a [[magnitude]] and/or [[direction|directions]] associated with it. Scalars, vectors and tensors are quantities, which do not change if the [[system of coordinates]] is changed (e.g. between Cartesian, cylindrical, spherical).[(Fleisch_Tensors>[[https://isbnsearch.org/isbn/9780521171908|Daniel Fleisch, A Student's Guide to Vectors and Tensors, Cambridge University Press, Cambridge, 2012, ISBN 9780521171908]])][(Purcell>[[https://isbnsearch.org/isbn/9781107014022|E.M. Purcell, D.J. Morin, Electricity and magnetism, 3rd edition, Cambridge University Press, 2013, ISBN 9781107014022]])]

Scalars, vectors and tensors are widely used in mathematical treatment of [[electromagnetism|electromagnetic]] problems, as well as in the way the [[Maxwell's equations]] are typically formulated.[(Fleisch_Maxwell>[[https://isbnsearch.org/isbn/9780521877619|Daniel Fleisch, A Student’s Guide to Maxwell’s Equations, Cambridge University Press, Cambridge, 2008, ISBN 9780521877619]])]

[[Vector calculus]] (as well as tensor calculus) can be used with analytical equations, as well as numerical calculations such as [[finite element modelling]].[(COMSOL>[[https://www.comsol.com/multiphysics/introduction-to-physics-pdes-and-numerical-modeling|Physics, PDEs, Mathematical and Numerical Modeling, COMSOL, March 2019 https://www.comsol.com]], {accessed 2022-01-02})][(FEMM>[[https://www.femm.info/wiki/Documentation/|David Meeker, Finite Element Method Magnetics: Documentation]], {accessed 2021-06-25})]

Vectors can be analysed from the viewpoint of covariant and contravariant components, and can be transformed between various coordinate systems, including non-orthogonal ones. There are many detailed implications regarding vectors and calculations based of them, and it is best to study the relevant textbooks and literature, as well as practice with simpler cases before performing more complex calculations.[(Fleisch_Tensors)] This article contains only the most basic information.

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===== Overview =====

|< 100% 15% 20% 30% 35%>|
^  ^  Scalar \\ (tensor rank 0)  ^  Vector \\ (tensor rank 1)  ^  Tensor \\ (tensor rank 2)  ^
|  typical examples \\ in a Cartesian \\ system  |  <box 90% left #f0f0f0>[[file/scalar_encyclopedia_magnetica_png|{{scalar_encyclopedia_magnetica.png?100}}]]{{page>insert/by_SZ}}</box>  |  <box 80% left #f0f0f0>[[file/vector_encyclopedia_magnetica_png|{{vector_encyclopedia_magnetica.png?200}}]]{{page>insert/by_SZ}}</box>  |  <box 90% left #f0f0f0>[[file/tensor_encyclopedia_magnetica_png|{{tensor_encyclopedia_magnetica.png?250}}]]{{page>insert/by_SZ}}</box>  | 
|  mathematical \\ representation \\ with //x, y, z//   |  \\ $(x)$  |  $(x, y, z)$ or $\left( \begin{array}{c} x \\ y \\ z \end{array} \right)$  |  $\left( \begin{array}{c} x_{xx} & x_{xy} & x_{xz} \\ y_{yx} & y_{yy} & y_{yz} \\ z_{zx} & z_{zy} & z_{zz} \end{array} \right)$  |
|  mathematical \\ representation \\ with //x<sub>i</sub>//  |  \\ $(x_1)$  |  $(x_1, x_2, x_3)$ or $\left( \begin{array}{c} x_1 \\ x_2 \\ x_3 \end{array} \right)$  |  $\left( \begin{array}{c} x_{11} & x_{12} & x_{13} \\ x_{21} & x_{22} & x_{23} \\ x_{31} & x_{32} & x_{33} \end{array} \right)$  |
|  number of \\ values in 3D  |  **1**  |  **3**  |  **9**  |
|  meaning \\ of values  |  magnitude (no direction)  |  magnitude in a single direction  |  magnitudes of interactions in multiple directions  |


===== Scalar =====

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Example of a **scalar**: a length of some element
[[file/scalar_encyclopedia_magnetica_png|{{scalar_encyclopedia_magnetica.png}}]]
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| A **scalar** value has only an amplitude, without direction, but it can be positive, zero, or negative.  | 

Scalar can represent a single physical quantity, such as: [[mass]], [[temperature]], [[length]] (including length of a vector, or its [[vector modulus|modulus]]), or [[area]].

Under certain conditions, calculations based on vectors and tensors can be simplified to a scalar form, e.g. if all the vector quantities are parallel, or the angle between them is known explicitly.

Examples:
  * in a [[DC circuit]], there is no phase lag between voltage and current, and thus the [[Ohm's law]] can be used in the simple form, in which all the quantities are scalars: $I = \frac{V}{R}$
  * in an [[AC circuit]], there can be some phase lag between voltage and current, and this can be represented by using the [[vector notation]] or [[complex number|complex numbers]]. However, it is also possible to perform the calculations by taking into account the phase shift, with the aid of trigonometry, with scalar values, with the Ohm's law taking the form of $Z = \sqrt{R^2 + X^2}$ and $I = \frac{V}{Z}$

Mathematically, a scalar can be shown to be a tensor of rank zero, or to be a vector in a 1D space, so that its amplitude can be positive or negative. Scalars can be used also in spaces with more dimensions (2D, 3D, 4D). For example, [[rest mass]] is a scalar in a 4D [[time-space]].



===== Vector =====

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Example of a **vector** in 3D space, with three [[orthogonal]] coordinates (//x//, //y//, //z//)
[[file/vector_encyclopedia_magnetica_png|{{vector_encyclopedia_magnetica.png}}]]
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| A **vector** is a mathematical representation of a physical quantity, such that includes its magnitude and direction.[(Fleisch_Tensors)]  |

Vectors can represent such "directed" quantities as: [[force]] (acting along a given direction), [[speed]] or velocity (movement in a given direction), orientation of a [[surface]] (by defining a [[normal vector|vector perpendicular]] to it), and so on.

In a [[Cartesian system of coordinates]], a vector can be decomposed into a set of [[orthogonal components]], with as many components as there are dimensions of space. A value associated with each direction is a numerical multiplier of length for each **unit vector** $\hat{\imath}, \hat{\jmath}, \hat{k}$ (explained in more detail below). 

Vectors and vector components can be translated between different coordinate systems without changing their meaning or value of the represented physical quantity. Such transformations are mathematically strict if performed in an analytical way.[(Fleisch_Tensors)]

In 2D space there are 2 components, so it is possible to perform calculations similar to vector problems, but by using the tool of [[complex number|complex numbers]].[(Feynman)] Vectors can also be used in 4D spaces[(Purcell)], and in any arbitrary number of dimensions, as required.

The "chirality" of the coordinate system is important for vectors, because of implications such as [[right-hand rule]].[(Fleisch_Tensors)] In most calculations in the literature, if not specified then the right-hand system is implicitly assumed.

Vectors can be represented in a number of ways, with some examples shown in the table.

|< 100% >|
^  Dimensions \\ of space  ^  Examples of vector notation  ^
|  1D  |  $$ \vec{F} = \vec{F}(x)  = \vec{x} = x · \hat{\imath} $$  |
|  2D  |    $$ \vec{F} = \vec{F}(x, y) = \vec{F} \left( \begin{array}{c} x \\ y \end{array} \right)  = \vec{x} + \vec{y} = x · \hat{\imath} + y · \hat{\jmath}  $$  |
|  3D  |  $$ \vec{F} = \vec{F}(x, y, z) = \vec{F} \left( \begin{array}{c} x \\ y \\ z \end{array} \right) = \vec{x} + \vec{y} + \vec{z}  = x · \hat{\imath} + y · \hat{\jmath} + z · \hat{k}  $$  |
|  4D  |  $$ \vec{F} = \vec{F}(x_0, x_1, x_2, x_3) = \vec{F} \left( \begin{array}{c} x_0 \\ x_1 \\ x_2 \\ x_3 \end{array} \right) = \vec{x_0} + \vec{x_1} + \vec{x_2} + \vec{x_3} = x_0 · \hat{\imath} + x_1 · \hat{\jmath} + x_2 · \hat{k} + x_3 · \hat{l}  $$  |



==== Normal vector ====

Vectors can be also used to represent surface components. 

A type of vector can be defined, perpendicular to the flat plane which is tangent to the given surface at a given point. 

If this vector is then expressed as a **unit vector**, then it is known as a **normal vector** $\hat{n}$ ("normal" meaning "perpendicular"). Normal vectors are useful in **vector calculus**, especially in calculating and measurement of the [[flux of vector field]] ([[electric flux]] and [[magnetic flux]]).[(Fleisch_Tensors)] In a right-handed system of coordinates the polarity of a vector normal in the system is defined by the [[right-hand rule]] (as illustrated). In a left-handed system the polarity would have been reversed.

Similarly, a **tangential vector** can be also defined. Tangential components are useful in certain cases of measuring [[magnetic field strength]].

<box 30% left #f0f0f0>
Positive direction for [[circulation of a vector]] around the path //C// is also defined with the right hand in the right-hand system; **normal vector** $\mathbf{\hat{n}}$ denotes vector normal (perpendicular) to the surface $a$ 
[[file/vector_circulation_magnetica_png|{{vector_circulation_magnetica.png}}]]
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**Normal vectors** (blue) for each infinitesimal fragment of surface can be used to extract the perpendicular component of the vector field (red) - this allows calculation of [[flux of vector field]]
[[file/surface_integral_definition_chetvorno_png|{{surface_integral_definition_chetvorno.png}}]] \\ 
//<sup>by Chetvorno, [[https://en.wikipedia.org/wiki/File:Surface_integral_-_definition.svg|Wikimedia Commons]], [[https://creativecommons.org/licenses/by/4.0/|CC-BY-4.0]]</sup>//
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===== Tensor =====

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Example of a **tensor** - with a set of 3 orthogonal vectors interacting with the three orthogonal surfaces ([[normal vector|normal vectors]] are not shown)
[[file/tensor_encyclopedia_magnetica_png|{{tensor_encyclopedia_magnetica.png}}]]
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| A **tensor** is a mathematical representation of a physical quantity, such that includes magnitudes of quantities or interactions in multiple directions.[(Fleisch_Tensors)]  |

A tensor represents a more general way in which the various components can be handled in mathematics.
A scalar is a tensor rank 0, so requires only 1 independent value associated with it, regardless of the number of dimensions of a given space, so for 3D it is 3<sup>0</sup>=1.

A tensor rank 1 is a vector, and requires as many components as there are dimensions in a given space, thus for 3D it is 3<sup>1</sup>=3.

A tensor rank 2, requires more components, for 3D space it is 3<sup>2</sup>=9, and a tensor rank 3 would require 27 values (3<sup>3</sup>).[(Fleisch_Tensors)] 

Vectors (tensor rank 1) can represent actions and interactions along a single direction. However, tensors rank 2 can be used to represent actions and interactions in several directions in space. For example, forces can act along the analysed direction (compression or tension) as well as perpendicularly to it (shearing or twisting). These various directions of interaction cannot be represented by a single vector. But if there are forces acting on a unit volume of material, these interactions can be represented by taking into account each side of an [[infinitesimal]] cube and expressing the interactions between all 3D directions. 

Therefore, tensors are required for capturing anisotropic behaviour in all directions, and thus [[tensor permeability]] might be required in highly anisotropic 3D cases.

Mathematically, processing of tensors is similar to that of vectors, because multiple orthogonal components are used in matrix notation.[(Fleisch_Tensors)] Complexity is greatly increased though, because of the large number of independent components required for such calculations. 


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**Tensor field** has a tensor value at each point of space
[[file/tensor_field_magnetica_png|{{tensor_field_magnetica.png}}]]
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Magnitude ellipsoids of a tensor of [[susceptibility]] anisotropy ($k_i$ - radius in a given principal direction):[(Gupta)] \\ 
a) sphere ($k_1=k_2=k_3$) \\ 
b) oblate ellipsoid ($k_1 < k_2 = k_3$) \\ 
c) prolate ellipsoid ($k_1 > k_2 = k_3$) \\ 
d) triaxial ellipsoid ($k_1 \neq k_2 \neq k_3$) \\ 
[[file/ellipsoids_magnetica_png|{{ellipsoids_magnetica.png}}]]
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===== Unit vector =====

Unit vectors are a tool which is useful in vector calculus. A **unit vector** is parallel to some main vector, but it has a length of unity when expressed in the same units as the main vector. 

For example, for a main vector which can be described by the coordinate values of $\vec{F}$ = (//x//, //y//, //z//) = (2, 3, 4) the numbers denote the multiplication for each of the **unit vectors** (length of 2 in the //x// direction, 3 in //y// and 4 in //z//). 

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Example of a **vector** //F// and its relation to **unit vectors** //f//
[[file/unit_vector_magnetica_png|{{unit_vector_magnetica.png}}]]
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This is equivalent to a geometric addition of three component vectors so that: 
$\vec{F} = 2 ·\hat{i} +  3 ·\hat{j} +  4 ·\hat{k}$, where the unit vectors $\hat{i}, \hat{j}, \hat{k}$ are in the //x//, //y//, //z// directions, respectively. The "hat" notation $\hat{k}$ is used instead of the "arrow" notation $\vec{k}$ just to emphasize that the unit vectors constitute the directional bases - even though they are also "ordinary" vectors, just with length or modulus equal to 1 in a given system of coordinates. 

The unit vectors $\hat{i}, \hat{j}$ have the hat above the dot, and thus are often written without the dot, just with the hat: $\hat{\imath}, \hat{\jmath}$.

With using the unit vector notion, it is then implicitly understood that writing $\vec{F} = (2, 3, 4)$ means $\vec{F} = 2 ·\hat{\imath} +  3 ·\hat{\jmath} +  4 ·\hat{k}$, and similarly for other number of dimensions.[(Fleisch_Tensors)]

Calculations carried out with unit vectors are mathematically equivalent to those which use "full" vectors. The form is dictated mainly by the preference of the authors of the given publication. Therefore, equations which are equivalent mathematically can differ in various publications in the way the components are expressed, because of the approach taken during derivation of them.

===== Scalar, vector and tensor notation =====

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^  Typical notation[(Fleisch_Tensors)] of:  ^^
^  vectors  ^  scalars  ^
|  <html><div style="font-size: 140%;"> $$\vec{F}$$ </div></html> (arrow)  |  \\ \\ <html><div style="font-size: 140%;"> $$F$$ </div></html> (normal \\ font)  |
|  <html><div style="font-size: 140%;"> $$\underline{F}$$ </div></html> (underline)  |  :::  |
|  <html><div style="font-size: 140%;"> $$\mathbf{F}$$ </div></html> (bold)  |  :::  |
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There are several ways in which vectors are represented in mathematical equations. It is important to distinguish especially between the scalars and vectors, because some operations cannot be carried out in the same way on scalars and vectors, or the order of components can matter in vector operations.[(Fleisch_Tensors)]

Typically, scalars are denoted with a normal font used for the names of the variables. 

Vectors can be denoted with an "arrow" (for full vectors) or "hat" (for unit vectors) to include explicit information that a given quantity is indeed a vector. However, for convenience, vectors can be denoted simply 
by using a bold font, and this is widely used in scientific literature.[(Feynman>[[https://www.feynmanlectures.caltech.edu/II_36.html|Richard Feynman, Robert Leighton, Matthew Sands, Ferromagnetism, The Feynman Lectures on Physics, Vol. II, Basic Books, ISBN: 9780465079988]])][(Jiles>[[http://books.google.com/books?isbn=9780412798603|David C. Jiles, Introduction to Magnetism and Magnetic Materials, Second Edition, Chapman & Hall, CRC, 1998, ISBN 9780412798603]])][(Purcell>[[https://isbnsearch.org/isbn/9781107014022|E.M. Purcell, D.J. Morin, Electricity and magnetism, 3rd edition, Cambridge University Press, 2013, ISBN 9781107014022]])][(Griffiths>[[http://books.google.com/books?isbn=0321856562|David J. Griffiths, Introduction to electrodynamics, 4th ed., Pearson, Boston, 2013, ISBN 0321856562]])] This is dictated mostly by the simplicity and clarity of notation, because typically fewer clear symbols are easier to follow.

There is no specific way in which tensors are denoted, and thus vector-like form can be used, unless there is a need to make sure that confusion should be avoided.[(Fleisch_Tensors)]

===== Vector operations =====

Vectors can be manipulated mathematically in some sense similar to scalars, but the complexity of operations is increased due to the number of components involved in such calculations.


==== Addition, subtraction, multiplication ====
**Vector calculus** allows analysis of [[electromagnetism|electromagnetic]] phenomena in 3D space, as dictated by the [[Maxwell's equations]].[(Fleisch_Maxwell)]

Vectors can be manipulated algebraically (addition, subtraction, negation), with the processing similar to the scalar values, but of course by taking into account the angle between them.[(Fleisch_Tensors)] For instance, scalar addition means a simple arithmetic sum of the components, but the vector addition means a geometric sum. 

<box 50% right #f0f0f0>
Definition of left-handed and right-handed system of coordinates, with the **vector cross product** $\vec{c}=\vec{a} × \vec{b}$ expressed with fingers outstretched at right angles
[[file/right_left_hand_system_magnetica_png|{{right_left_hand_system_magnetica.png}}]]
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Vectors can be multiplied, but there are three operations: 
  * In the **scalar product** the vector is multiplied by a scalar, simply scaling the vector by the magnitude of a scalar: $A · \mathbf{B} = A \mathbf{B}$. To avoid confusion, in most publications the scalar multiplication dot is //not used// (simply omitted). However, this omission is not strictly necessary, because the operation is dictated by the type of input values.
  * In the **vector dot product** the result is proportional to the projection of one vector onto another, and the resulting value is a scalar, numerically equal to a scalar multiplication including the angle: $\mathbf{A}·\mathbf{B}=|\mathbf{A}| \, |\mathbf{B}| \cos{}(θ_{AB})$. The order of input vectors does not matter and thus $\mathbf{A}·\mathbf{B}=\mathbf{B}·\mathbf{A}$. \\ A dot product of a vector with a normal vector of some surface gives the component perpendicular to that surface. \\ A dot product of two perpendicular vectors is zero, so for example $\hat{\imath} · \hat{\jmath} = 0$. This simplifies outcomes of certain calculations, because some terms end up eliminated.
  * In the **vector cross product** the result is a vector pointing in a direction perpendicular to both input vectors, and is proportional to the area stretched between the two input vectors. The order of input vectors matters so that: $\mathbf{A} × \mathbf{B} = - \mathbf{B} × \mathbf{A} = \mathbf{C}$. Also, the chirality of the system of coordinates is important, because the result can change sign, if the calculations are performed in the right- or left-handed system. \\ A cross product of two parallel vectors is zero, so for example $\hat{\imath} × \hat{\imath} = 0$. This simplifies outcomes of certain calculations, because some terms end up eliminated. \\ The vector cross product can be written in a matrix notation, with calculations similar to the way a determinant of an array is computed (by multiplying the components along diagonals of the array):[(Fleisch_Tensors)] 
$$\mathbf{A} × \mathbf{B} = \left( \begin{array}{c} \hat{\imath} & \hat{\jmath} & \hat{k} \\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{array} \right) = ( A_y B_z - A_z B_y ) \hat{\imath} + ( A_z B_x - A_x B_z ) \hat{\jmath} + ( A_x B_y - A_y B_x ) \hat{k} $$

The dot and cross products can be combined, for example the **triple scalar product** returns a scalar value proportional to the volume (parallelepiped) stretched between all three input vectors:[(Fleisch_Tensors)] 
$$ \mathbf{A} · ( \mathbf{B} × \mathbf{C}) = \left( \begin{array}{c} A_x & A_y & A_z \\ B_x & B_y & B_z \\ C_x & C_y & C_z \end{array} \right) = A_x ( B_y C_z - B_z C_y ) + A_y ( B_z C_x - B_x C_z ) + A_z ( B_x C_y - B_y C_x ) $$

Also, a **triple vector product** can be calculated in a similar way, but involving more tedious equations due to the number of involved components, and thus it is typically easier to use the equivalent identity which uses dot products instead:
$$\mathbf{A} × ( \mathbf{B} × \mathbf{C}) = \left( \begin{array}{c} \hat{\imath} & \hat{\jmath} & \hat{k} \\ A_x & A_y & A_z \\ ( B_y C_z - B_z C_y ) & ( B_z C_x - B_x C_z ) & ( B_x C_y - B_y C_x ) \end{array} \right) =  \mathbf{B} (\mathbf{A} ·\mathbf{C}) - \mathbf{C}( \mathbf{A} · \mathbf{B})$$

Interestingly, **vector division** cannot be easily defined and thus is not used in vector calculus.[(Wrede>[[https://books.google.co.uk/books?id=wbjCAgAAQBAJ|Robert C. Wrede, Introduction to Vector and Tensor Analysis, Courier Corporation, 2013, ISBN 048661879X]])] Instead, in some cases, such as in 2D analysis, it is much more advantageous to change the representation to complex numbers, which allow division and hence arriving at a correct result with more straightforward calculations.[(Feynman)]

==== Vector calculus ====

|< 100% >|
| {{/wiki/logo.png?20&nolink}} //See also the main article: [[Vector calculus]].// |

Calculus involves derivatives and integrals, and **vector calculus** involves derivatives and integrals of vectors, which can represent functions or [[field|fields]] in 3D space.

The nabla operator ∇ allows significant simplification of notation.

|< 100% >|
^  Vector calculus operators [(Fleisch_Tensors)]  ^^^^
^  operator  ^  input function  ^  mathematical representation  ^  outcome  ^
|  \\ **gradient**  |  \\ scalar $F$  |  $$\text{grad} (F) ≡ \text{grad} \, F ≡ ∇F ≡ \hat{\imath} \frac{∂F}{∂x} + \hat{\jmath} \frac{∂F}{∂y} + \hat{k} \frac{∂F}{∂z}$$   |  \\ vector  |
|  \\ **divergence**  |  \\ vector $\mathbf{F}$  |  $$\text{div} (\mathbf{F}) ≡ \text{div} \, \mathbf{F} ≡ ∇·\mathbf{F} ≡ \frac{∂ F_x}{∂x} + \frac{∂ F_y}{∂y} + \frac{∂ F_z}{∂z} $$  |  \\ scalar  |
|  \\ **curl** \\ (also **rot**)  |  \\ vector $\mathbf{F}$  |  $$\text{curl} (\mathbf{F}) ≡ \text{curl} \, \mathbf{F} ≡ ∇ × \mathbf{F} ≡ \left( \frac{∂ F_z}{∂y} - \frac{∂ F_y}{∂z} \right) \hat{\imath} + \left( \frac{∂ F_x}{∂z} - \frac{∂ F_z}{∂x} \right) \hat{\jmath} + \left( \frac{∂ F_y}{∂x} - \frac{∂ F_x}{∂y} \right) \hat{k} $$  |  \\ vector  |
|  \\ **Laplacian**  |  \\ scalar $F$  |  $$ ∇· ∇F ≡ ∇^2F ≡ \frac{∂^2 F_x}{∂x^2} + \frac{∂^2 F_y}{∂y^2} + \frac{∂^2 F_z}{∂z^2} $$  |  \\ scalar  |
|  \\ **Laplacian**  |  \\ vector $\mathbf{F}$  |  $$ ∇^2\mathbf{F} ≡ (∇^2 F_x) \hat{\mathbf{x}} + (∇^2 F_y) \hat{\mathbf{y}} + (∇^2 F_z) \hat{\mathbf{z}} $$  |  \\ vector  |


=== Nabla ∇ operator ===

An operator such as square root $\sqrt{F}$ informs that an operation should be performed on the variable //F// that follows the operator. Similarly, a derivative operator $\frac{dF}{dx} = \frac{d}{dx}(F)$ informs that the derivative should be performed on the quantity //F// that follows.

In a 3D space, vectors can be split into orthogonal components, and partial derivatives can be calculated accordingly for each directional components. The special ∂ character is used to denote that the derivatives are partial, so for one component it could be $\frac{∂F}{∂x} = \frac{∂}{∂x}(F)$, and more components can be used accordingly.

Thus, in a 3D space, the operator can be defined with the three directional unit vector components ($\hat{\imath}, \hat{\jmath}, \hat{k}$), and for simplicity of notation just one character called **//nabla//** or **//del//** denoted with an upside-down triangle $\vec{∇} ≡ \mathbf{∇} ≡ ∇$ (with the arrow or bold notation typically omitted) can be used to represent the whole operation. In a Cartesian system:[(Fleisch_Tensors)]

$$ \vec{∇} ≡ \mathbf{∇} ≡ ∇ ≡ \hat{\imath} \frac{∂}{∂x} + \hat{\jmath} \frac{∂}{∂y} + \hat{k} \frac{∂}{∂z}  $$ 

Nabla can be used in the "Laplacian" operator, referred to sometimes as "nabla squared" or "del squared", denoting effectively double differentiation:

$$ \vec{∇}·\vec{∇} ≡ \mathbf{∇}·\mathbf{∇} ≡ ∇·∇ ≡ \vec{∇}^2 ≡ \mathbf{∇}^2 ≡ ∇^2 ≡ \frac{∂^2}{∂x^2} +  \frac{∂^2}{∂y^2} + \frac{∂^2}{∂z^2}  $$ 


===== Scalar and vector fields =====

<box 40% right #f0f0f0>
[[Electric flux density]] //**D**// increases if the area of the high-permittivity dielectric is reduced in the same [[electric field]] (vector field)
[[file/electric_flux_density_magnetica_png|{{electric_flux_density_magnetica.png}}]]
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In physics, a [[field]] is a such region of space in which some value is assigned to each point of such space. 

<box 30% left #f0f0f0>
[[Compass needle|Compass needles]] follow the direction of [[magnetic field]] (//**B**// or //**H**//) around the wire with current //**I**// - magnetic field "circulates" around a wire with current
[[file/magnetic_field_around_wire_e-m_jpg|{{magnetic_field_around_wire_e-m.jpg}}]]
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If the values are scalar then it is a **[[scalar field]]** (such distribution of temperature inside of a body). 

If the values are vectors then it is a **[[vector field]]** (such as distribution of velocity of gas inside some container).

Vector calculus can be used to perform calculations on fields, and to compute such vector and scalar quantities as: [[magnetic flux]], [[magnetic flux density]], [[electric flux]], [[electric flux density]], etc.





===== System of coordinates =====

Vectors and tensors can be defined or expressed in different systems of coordinates, which are used for reducing the complexity of calculations of certain problems. This is because the mathematical equations can take slightly different form in each of such systems, and therefore might be easier to solve analytically (or numerically) in a given system.

Three most important systems are: **Cartesian**, **cylindrical**, and **spherical**. If they are 3D then each of them requires three unique values to identify a unique vector, or a distance from some reference point, e.g. (0, 0, 0). 

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Example of representation of the same vector in three different systems of coordinates: Cartesian (//x//, //y//, //z//), cylindrical (//r//, //φ//, //z//), and spherical (//r//, //φ//, //θ//)
[[file/cartesian_cylindrical_spherical_system_magnetica_png|{{cartesian_cylindrical_spherical_system_magnetica.png}}]]
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In a **Cartesian system**, there are three orthogonal directions, for example (//x//, //y//, //z//) and therefore each vector can be combined from three components, each expressing a linear distance from a reference point (0, 0, 0). The distances can be also represented in a relative sense defining the length and angle of the vector, but not its location as such.

In a **cylindrical system**, there are also three components. But a location of a given point inside a cylinder can be expressed by a radius //r//, angle //φ//, and height //z// of a cylinder, so such set of values uniquely identifies a given point or vector. Such system is useful for performing calculations on a geometry which has a one axis of rotational symmetry.

In a **spherical system**, the components denote a radius //r// and two angles //φ// and //θ//. Such system is useful for performing calculations in spherical or elliptical geometries.

The specific components differ between the systems, but they can be converted between them through strict mathematical relationships.[(Fleisch_Tensors)]

Of course, the names of the component variables (for lengths and angles) can be chosen in an arbitrary way, depending on the preference in a given approach.

===== See also =====
  * [[Vector calculus]]
  * [[Scalar field]]
  * [[Vector field]]
  * [[Maxwell's equations]]

===== References =====
~~REFNOTES~~

{{tag> Scalar Vector Tensor Scalar_field Vector_field Tensor_field Tensor_rank_2 Tensor_rank_1 Tensor_rank_0 Counter}}