Advanced Mathematics for Engineers and Scientists/The Front Cover

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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[edit]

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[edit]

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

Operator Cylindrical Spherical