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एकल ऑपरेटर वाली सर्वसमिकाएँ

किसी सदिश क्षेत्र का डाइवरजेंस (Divergence of a vector field)

किसी सदिश क्षेत्र $\mathbf{𝐯}$ के लिये, डाइवर्जेंस प्रायः निम्नलिखित प्रकार से लिखा जाता है:

$\mathrm{div}(\mathbf{𝐯})=\mathrm{\nabla }\cdot \mathbf{𝐯}$

किसी सदिश क्षेत्र का डाइवरजेंस एक अदिश राशि होती है।

किसी टेंसर का डाइवर्जेंस (Divergence of a tensor)

किसी टेंसर $\stackrel{\mathfrak{𝔗}}{}$ के लिये, डाइवर्जेंस प्रायः निम्नलिखित प्रकार से लिखा जाता है:

$\mathrm{div}(\mathfrak{𝔗})=\mathrm{\nabla }\cdot \mathfrak{𝔗}$

जो कि एक सदिश राशि है।

कर्ल (Curl)

किसी सदिश क्षेत्र $\mathbf{𝐯}$ के लिये, कर्ल इस प्रकार लिखा जाता है:

$\mathrm{curl}(\mathbf{𝐯})=\mathrm{\nabla }\times \mathbf{𝐯}$

किसी सदिश क्षेत्र का कर्ल, एक सदिश राशि होती है।

ग्रेडिएन्ट (Gradient)

किसी सदिश क्षेत्र $\mathbf{𝐯}$ के लिये, ग्रेडिएन्ट प्रायः इस प्रकार लिखा जाता है:

$\mathrm{grad}(\mathbf{𝐯})=\mathrm{\nabla }\mathbf{𝐯}$

जो कि एक टेंसर है।

किसी अदिश क्षेत्र का ग्रेडिएंट (Gradient of a scalar field)

किसी अदिश क्षेत्र, $\psi$ के लिये, ग्रेडिएन्ट इस प्रकार लिखा जाता है:

$\mathrm{grad}(\psi )=\mathrm{\nabla }\psi$

किसी अदिश क्षेत्र का ग्रेडिएन्ट, एक सदिश राशि होती है।

Combinations of multiple operators

Curl of the gradient

The curl of the gradient of any scalar field $\varphi$ is always zero:

$\mathrm{\nabla }\times (\mathrm{\nabla }\varphi )=0$

One way to establish this identity (and most of the others listed in this article) is to use three-dimensional Cartesian coordinates. According to the article on curl,

$\mathrm{\nabla }\times \mathrm{\nabla }\varphi =\left[\begin{array}{ccc}\mathbf{𝐢}& \mathbf{𝐣}& \mathbf{𝐤}\\ \\ {\partial }_x& {\partial }_y& {\partial }_z\\ \\ {\partial }_x\varphi & {\partial }_y\varphi & {\partial }_z\varphi \end{array}\right]$

where the right hand side is a determinant, and i, j, k are unit vectors pointing in the positive axes directions, and ∂_x = ∂ / ∂ x etc. For example, the x-component of the above equation is:

$\mathbf{𝐢}({\partial }_y{\partial }_z-{\partial }_z{\partial }_y)\varphi =0$

where the left-hand side evaluates as zero assuming the order of differentiation is immaterial.

Divergence of the curl

The divergence of the curl of any vector field A is always zero:

$\mathrm{\nabla }\cdot (\mathrm{\nabla }\times \mathbf{𝐀})=0$

Divergence of the gradient

The Laplacian of a scalar field is defined as the divergence of the gradient:

$\mathrm{\nabla }\cdot (\mathrm{\nabla }\psi )={\mathrm{\nabla }}^2\psi$

Note that the result is a scalar quantity.

Curl of the curl

$\mathrm{\nabla }\times (\mathrm{\nabla }\times \mathbf{𝐀})=\mathrm{\nabla }(\mathrm{\nabla }\cdot \mathbf{𝐀})-{\mathrm{\nabla }}^2\mathbf{𝐀}$

Properties

Distributive property

$\mathrm{\nabla }\cdot (\mathbf{𝐀}+\mathbf{𝐁})=\mathrm{\nabla }\cdot \mathbf{𝐀}+\mathrm{\nabla }\cdot \mathbf{𝐁}$

$\mathrm{\nabla }\times (\mathbf{𝐀}+\mathbf{𝐁})=\mathrm{\nabla }\times \mathbf{𝐀}+\mathrm{\nabla }\times \mathbf{𝐁}$

Vector dot product

$\mathrm{\nabla }(\mathbf{𝐀}\cdot \mathbf{𝐁})=(\mathbf{𝐀}\cdot \mathrm{\nabla })\mathbf{𝐁}+(\mathbf{𝐁}\cdot \mathrm{\nabla })\mathbf{𝐀}+\mathbf{𝐀}\times (\mathrm{\nabla }\times \mathbf{𝐁})+\mathbf{𝐁}\times (\mathrm{\nabla }\times \mathbf{𝐀})$

In simpler form, using Feynman subscript notation:

$\mathrm{\nabla }(\mathbf{𝐀}\cdot \mathbf{𝐁})={\mathrm{\nabla }}_A(\mathbf{𝐀}\cdot \mathbf{𝐁})+{\mathrm{\nabla }}_B(\mathbf{𝐀}\cdot \mathbf{𝐁})$

where the notation ∇_A means the subscripted gradient operates on only the factor A.

A less general but similar idea is used in geometric algebra where the so-called Hestenes overdot notation is employed. The above identity is then expressed as:

$\mathrm{\nabla }(\mathbf{𝐀}\cdot \mathbf{𝐁})=\dot{\mathrm{\nabla }}(\dot{\mathbf{𝐀}}\cdot \mathbf{𝐁})+\dot{\mathrm{\nabla }}(\mathbf{𝐀}\cdot \dot{\mathbf{𝐁}})$

where overdots define the scope of the vector derivative. In the first term it is only the first (dotted) factor that is differentiated, while the second is held constant. Likewise, in the second term it is the second (dotted) factor that is differentiated, and the first is held constant.

As a special case, when A = B:

$\frac{1}{2}\mathrm{\nabla }(\mathbf{𝐀}\cdot \mathbf{𝐀})=\mathbf{𝐀}\times (\mathrm{\nabla }\times \mathbf{𝐀})+(\mathbf{𝐀}\cdot \mathrm{\nabla })\mathbf{𝐀}.$

Vector cross product

$\mathrm{\nabla }\cdot (\mathbf{𝐀}\times \mathbf{𝐁})=\mathbf{𝐁}\cdot \mathrm{\nabla }\times \mathbf{𝐀}-\mathbf{𝐀}\cdot \mathrm{\nabla }\times \mathbf{𝐁}$

$\mathrm{\nabla }\times (\mathbf{𝐀}\times \mathbf{𝐁})=\mathbf{𝐀}(\mathrm{\nabla }\cdot \mathbf{𝐁})-\mathbf{𝐁}(\mathrm{\nabla }\cdot \mathbf{𝐀})+(\mathbf{𝐁}\cdot \mathrm{\nabla })\mathbf{𝐀}-(\mathbf{𝐀}\cdot \mathrm{\nabla })\mathbf{𝐁}$

$\mathbf{𝐀}\mathbf{\times }\left(\mathrm{\nabla }\mathbf{\times }\mathbf{𝐁}\right)={\mathrm{\nabla }}_B\left(\mathbf{𝐀}\mathbf{\cdot }\mathbf{𝐁}\right)-\left(\mathbf{𝐀}\mathbf{\cdot }\mathrm{\nabla }\right)\mathbf{𝐁}$

where the Feynman subscript notation ∇_B means the subscripted gradient operates on only the factor B. In overdot notation, explained above:

$\mathbf{𝐀}\mathbf{\times }\left(\mathrm{\nabla }\mathbf{\times }\mathbf{𝐁}\right)=\dot{\mathrm{\nabla }}\left(\mathbf{𝐀}\mathbf{\cdot }\dot{\mathbf{𝐁}}\right)-\left(\mathbf{𝐀}\mathbf{\cdot }\mathrm{\nabla }\right)\mathbf{𝐁}.$

Product of a scalar and a vector

$\mathrm{\nabla }\cdot (\psi \mathbf{𝐀})=\mathbf{𝐀}\cdot \mathrm{\nabla }\psi +\psi \mathrm{\nabla }\cdot \mathbf{𝐀}$

$\mathrm{\nabla }\times (\psi \mathbf{𝐀})=\psi \mathrm{\nabla }\times \mathbf{𝐀}-\mathbf{𝐀}\times \mathrm{\nabla }\psi$

Product rule for the gradient

The gradient of the product of two scalar fields $\psi$ and $\varphi$ follows the same form as the Product rule in single variable Calculus.

$\mathrm{\nabla }(\psi \varphi )=\varphi \mathrm{\nabla }\psi +\psi \mathrm{\nabla }\varphi$

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