< Advanced Mathematics for Engineers and Scientists
Nomenclature
ODE | Ordinary Differential Equation |
PDE | Partial Differential Equation |
BC | Boundary Condition |
IVP | Initial Value Problem |
BVP | Boundary Value Problem |
IBVP | Initial Boundary Value Problem |
Common Operators
Operators are shown applied to the scalar or the vector field .
Notation | Common names and other notation | Description and notes | Definition in Cartesian coordinates |
---|---|---|---|
Partial derivative, | The rate of change of with respect to , holding the other independent variables constant. | ||
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. | ||
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. | ||
Laplacian, Scalar Laplacian, Laplace operator, | A measure of the concavity of , equivalently a comparison of the value of at some point to neighboring values. | ||
Divergence, | A measure of "generation", in other words how much the vector field acts as a source or sink at a point. | ||
Curl, rotor, circulation density, | A vector that describes the rate of rotation of a (normally 3D) vector field and the corresponding axis of rotation. | ||
Vector Laplacian | Similar to the (scalar) Laplacian. Note however, that it is generally not equal to the element-by-element Laplacian of a vector. |
3D Operators in Different Coordinate Systems
Cartesian representations appear in the table above. The convention is used for spherical coordinates.
Operator | Cylindrical | Spherical |
---|---|---|
This article is issued from
Wikibooks.
The text is licensed under Creative
Commons - Attribution - Sharealike.
Additional terms may apply for the media files.