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

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

Published $\text {1968}$

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$22 \quad$ Formulas from Vector Analysis

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

$22.16$: Equivalence of Definitions of Scalar Triple Product

$22.17$: 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 on Scalar Field

$22.33$: Laplacian on Vector Field

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

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