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Single operators (summary) This section explicitly lists what some symbols mean for clarity.
## Divergence
## Divergence of a vector fieldFor a vector field , divergence is generally written as and is a scalar .
## Divergence of a tensorFor a tensor , divergence is generally written as and is a vector.
## CurlFor a vector field , curl is generally written as and is a vector field.
## Gradient
## Gradient of a vector fieldFor a vector field , gradient is generally written as and is a tensor.
## Gradient of a scalar fieldFor a scalar field, ψ, the gradient is generally written as and is a vector.
## Combinations of multiple operators
## Curl of the gradientThe curl of the gradient of 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 where the right hand side is a determinant, and ∂ / ∂ x etc. For example, the x-component of the above equation is:where the left-hand side evaluates as zero assuming the order of differentiation is immaterial.
## Divergence of the curlThe divergence of the curl of
## Divergence of the gradientThe Laplacian of a scalar field is defined as the divergence of the gradient: Note that the result is a scalar quantity.
## Curl of the curl
## Properties
## Distributive property
## Vector dot productIn simpler form, using Feynman subscript notation: where the notation A less general but similar idea is used in 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
## Vector cross productwhere the Feynman subscript notation
## Product of a scalar and a vector
## Product rule for the gradientThe gradient of the product of two scalar fields ψ and φ follows the same form as the Product rule in single variable Calculus. |