Physics[Vectors][Nabla] - the differential operator nabla
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Calling Sequence
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Nabla(A)
Nabla(A, differential_operation)
Nabla(A, differential_operation, useComponent)
Remark: these calling sequences are also valid with the inert %Nabla command
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Parameters
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A
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any algebraic expression
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differential_operation
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any of Curl, Divergence, Gradient, or Laplacian
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useComponent
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optional, to request representing the components of a non-projected vector using the Component command
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Description
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Nabla is a command representation for the nabla differential operator. Thus, is can be used to calculate the gradient, divergence, curl or Laplacian of a function as well.
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%Nabla is the inert form of Nabla, that is: it represents the same mathematical operation while holding the operation and checking of arguments unperformed. The expansion rules defined for Nabla, however, also work (are implemented) for the inert %Nabla. To activate the operation use value.
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Nabla works in a context sensitive manner. For instance, Nabla(F) returns the gradient of or the divergence of according to whether is identified as a scalar or a vector function (see ?Identify). So when called with only one argument Nabla(F) = Nabla . F. When called with two arguments, Nabla(F, Curl), Nabla(F, Divergence), Nabla(F, Gradient) and Nabla(F,Laplacian) respectively return the curl the divergence, the gradient and the Laplacian of . The result is expressed in the specific coordinates and orthonormal vector basis system related to the functional dependence detected in . When does not depend on the geometrical coordinates , Nabla(F, ...) returns 0; and when depends on coordinates of more than one system, the calculations are interrupted an error message is displayed on the screen.
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Since Nabla performs differentiation using the standard diff command, all differentiation knowledge of diff is used when computing with Nabla.
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For the conventions about the geometrical coordinates and vectors see Identify.
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Examples
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Nabla can be used to represent a gradient
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Note the "application" of Nabla as in the above can be performed as well using the . operator:
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The Laplacian
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To distinguish a non-projected vector from a scalar variable a postfix identified is used. You can use the default identified, that is the underscore , or set any valid Maple sequence of characters as identifier. For illustration purposes let's set again this default identifier:
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So the divergence of a non-projected vector can be computed in these three equivalent manners
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The curl
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To actually compute the operation instead of just representing it, in the case of non-projected vectors use the optional argument useComponent, so that the unknown components are represented using the Component command
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