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ODE |
Ordinary Differential Equation
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PDE |
Partial Differential Equation
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BC |
Boundary Condition
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IVP |
Initial Value Problem
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BVP |
Boundary Value Problem
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IBVP |
Initial Boundary Value Problem
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Operators are shown applied to the scalar or the vector field .
Notation |
Common names and other notation |
Description and notes |
Definition in Cartesian coordinates
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Partial derivative, |
The rate of change of with respect to , holding the other independent variables constant. |
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Derivative, total derivative, |
The rate of change of with respect to . If is multivariate, this derivative will typically depend on the other variables following a path. |
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Gradient, del operator, |
Vector that describes the direction and magnitude of the greatest rate of change of a function of more than one variable. The symbol is called nabla. |
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Laplacian, Scalar Laplacian, Laplace operator, |
A measure of the concavity of , equivalently a comparison of the value of at some point to neighboring values. |
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Divergence, |
A measure of "generation", in other words how much the vector field acts as a source or sink at a point. |
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Curl, rotor, circulation density, |
A vector that describes the rate of rotation of a (normally 3D) vector field and the corresponding axis of rotation. |
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Vector Laplacian |
Similar to the (scalar) Laplacian. Note however, that it is generally not equal to the element-by-element Laplacian of a vector. |
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Cartesian representations appear in the table above. The convention is used for spherical coordinates.
Operator |
Cylindrical |
Spherical
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