Advanced Mathematics for Engineers and Scientists/The Front Cover

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