Book:Murray R. Spiegel/Mathematical Handbook of Formulas and Tables/Chapter 22

Vectors and Scalars
Various quantities in physics such as temperature, volume and speed can be specified by a real number. Such quantities are called scalars.

Other quantities such as force, velocity and momentum require for their specification a direction as well as a magnitude. Such quantities are called vectors. A vector is represented by an arrow or a directed line segment indicating direction. The magnitude of the vector is determined by the length of the arrow, using an appropriate unit.

Notation for Vectors
A vector is denoted by a bold faced letter such as $\mathbf A$. The magnitude is denoted by $\size {\mathbf A}$. The tail end of the arrow is called the initial point while the head is called the terminal point.

Fundamental Definitions

 * $1.$ Equality of vectors.


 * $2.$ Multiplication of a vector by a scalar.
 * Zero or Null Vector


 * $3.$ Sums of vectors.
 * Parallelogram law for vector addition
 * Difference of vectors


 * $4.$ Unit Vector

Laws of Vector Algebra

 * $22.1$: Commutative law for addition


 * $22.2$: Associative law for addition


 * $22.3$: Associative law for scalar multiplication


 * $22.4$: Distributive law


 * $22.5$: Distributive law

Components of a Vector

 * $22.6$: Component of Vector

Dot or Scalar Product

 * $22.7$: Dot or Scalar Product


 * $22.8$: Commutative law


 * $22.9$: Distributive law


 * $22.10$: Dot Product: Product of Components

Cross or Vector Product

 * $22.11$: Cross or Vector Product


 * $22.12$: Cross or Vector Product: Determinant Definition


 * $22.13$: Vector Cross Product is Anticommutative


 * $22.14$: Vector Cross Product Distributes over Addition


 * $22.15$: Magnitude of Vector Cross Product equals Area of Parallelogram Contained by Vectors

Miscellaneous Formulas involving Dot and Cross Products

 * $22.16$: Scalar Triple Product equals Determinant


 * $22.17$: Magnitude of Scalar Triple Product equals Volume of Parallelepiped Contained by Vectors


 * $22.18$: Lagrange's Formula


 * $22.19$: Lagrange's Formula (Corollary)


 * $22.20$: Dot Product of Vector Cross Products


 * $22.21$: Vector Cross Product of Vector Cross Products

Derivatives of Vectors

 * $22.22$: Derivative of Vector-Valued Function at Point

Formulas involving Derivatives

 * $22.23$: Derivative of Dot Product of Vector-Valued Functions


 * $22.24$: Derivative of Vector Cross Product of Vector-Valued Functions


 * $22.25$: Derivative of Scalar Triple Product of Vector-Valued Functions


 * $22.26$: Dot Product of Vector-Valued Function with its Derivative


 * $22.27$: Dot Product of Constant Magnitude Vector-Valued Function with its Derivative is Zero

The Del Operator

 * $22.28$: Del Operator

The Gradient

 * $22.29$: Gradient Operator

The Divergence

 * $22.30$: Divergence Operator

The Curl

 * $22.31$: Curl Operator

The Laplacian

 * $22.32$: Laplacian of Real-Valued Function


 * $22.33$: Laplacian of Vector-Valued Function

The Biharmonic Operator

 * $22.34$: Biharmonic Operator

Miscellaneous Formulas involving $\nabla$

 * $22.35$: Gradient Operator Distributes over Addition


 * $22.36$: Divergence Operator Distributes over Addition


 * $22.37$: Curl Operator Distributes over Addition


 * $22.38$: Product Rule for Divergence


 * $22.39$: Product Rule for Curl


 * $22.40$: Divergence of Vector Cross Product


 * $22.41$: Curl of Vector Cross Product


 * $22.42$: Gradient of Dot Product


 * $22.43$: Curl of Gradient is Zero


 * $22.44$: Divergence of Curl is Zero


 * $22.45$: Curl of Curl is Gradient of Divergence minus Laplacian

Integrals involving Vectors

 * $22.46$: Primitive of Vector-Valued Function


 * $22.47$: Definite Integral of Vector-Valued Function

Line Integrals

 * $22.48$: Line Integral: Definition 1


 * $22.49$: Line Integral: Definition 2

Properties of Line Integrals

 * $22.50$: Inversion of Limits of Line Integral


 * $22.51$: Sum of Line Integrals on Adjacent Paths

Independence of the Path

 * $22.52$: Independence of Path of Line Integral
 * $22.53$: Line Integral on Closed Curve

Multiple Integrals

 * $22.54$: Double Integral: Definition 1
 * $22.55$: Double Integral: Definition 2
 * $22.56$: Double Integral: Definition 3

Surface Integrals

 * $22.57$: Surface Integral

Relation between Surface and Double Integrals

 * $22.58$: Relation between Surface and Double Integral

The Divergence Theorem

 * $22.59$: Divergence Theorem

Stokes' Theorem

 * $22.60$: Stokes' Theorem

Green's Theorem in the Plane

 * $22.61$: Green's Theorem in the Plane

Green's First Identity

 * $22.62$: Green's First Identity

Green's Second Identity

 * $22.63$: Green's Second Identity

Miscellaneous Integral Theorems

 * $22.64$: Integral of Curl equals Integral over Surface of Cross Product


 * $22.64$: Integral of Scalar equals Integral over Surface of Gradient

Curvilinear Coordinates

 * $22.66$: Cartesian


 * $22.67$: Derivative of Position with respect to Coordinate Curves


 * $22.68$: Scale Factors
 * Orthogonal

Formulas involving Orthogonal Curvilinear Coordinates

 * $22.69$: Derivative of Radius Vector in Curvilinear Coordinates


 * $22.70$: Arc Length Element in Curvilinear Coordinates


 * $22.71$: Volume Element in Curvilinear Coordinates


 * $22.72$: Jacobian of Transformation to Curvilinear Coordinates

Transformation of Multiple Integrals

 * $22.73$: Transformation of Multiple Integral into Curvilinear Coordinates

Gradient, Divergence, Curl and Laplacian

 * $22.74$: Gradient in Curvilinear Coordinates


 * $22.75$: Divergence in Curvilinear Coordinates


 * $22.76$: Curl in Curvilinear Coordinates


 * $22.77$: Laplacian in Curvilinear Coordinates

Special Orthogonal Coordinate Systems

 * Cylindrical Coordinates $\tuple {r, \theta, z}$:
 * $22.78$: Cartesian
 * $22.79$: Scale Factors
 * $22.80$: Laplacian


 * Spherical Coordinates $\tuple {r, \theta, \phi}$:
 * $22.81$: Cartesian
 * $22.82$: Scale Factors
 * $22.83$: Laplacian


 * Parabolic Cylindrical Coordinates $\tuple {u, v, z}$:
 * $22.84$: Cartesian
 * $22.85$: Scale Factors
 * $22.86$: Laplacian


 * Paraboloidal Coordinates $\tuple {u, v, \phi}$:
 * $22.87$: Cartesian
 * $22.88$: Scale Factors
 * $22.89$: Laplacian


 * Elliptic Cylindrical Coordinates $\tuple {u, v, z}$:
 * $22.90$: Cartesian
 * $22.91$: Scale Factors
 * $22.92$: Laplacian


 * Prolate Spheroidal Coordinates $\tuple {\xi, \eta, \phi}$:
 * $22.93$: Cartesian
 * $22.94$: Scale Factors
 * $22.95$: Laplacian


 * Oblate Spheroidal Coordinates $\tuple {\xi, \eta, \phi}$:
 * $22.96$: Cartesian
 * $22.97$: Scale Factors
 * $22.98$: Laplacian


 * Bipolar Coordinates $\tuple {u, v, z}$:
 * $22.99$: Cartesian
 * $22.100$: Cartesian to Bipolar
 * $22.101$: Scale Factors
 * $22.102$: Laplacian


 * Toroidal Coordinates $\tuple {u, v, \phi}$:
 * $22.103$: Cartesian
 * $22.104$: Scale Factors
 * $22.105$: Laplacian


 * Conical Coordinates $\tuple {\lambda, \mu, \nu}$:
 * $22.106$: Cartesian
 * $22.107$: Scale Factors


 * Confocal Ellipsoidal Coordinates $\tuple {\lambda, \mu, \nu}$:
 * $22.108$: Cartesian
 * $22.109$: Scale Factors
 * $22.110$: Laplacian


 * Confocal Paraboloidal Coordinates $\tuple {\lambda, \mu, \nu}$:
 * $22.111$: Cartesian
 * $22.112$: Scale Factors
 * $22.113$: Laplacian