# Advanced Mathematics for Engineers and Scientists/The Front Cover

From Wikibooks, open books for an open world

### Nomenclature[edit]

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 |
---|---|---|