Advanced Mathematics for Engineers and Scientists/The Front Cover

Nomenclature

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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

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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
  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

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Cartesian representations appear in the table above. The   convention is used for spherical coordinates.

Operator Cylindrical Spherical