# 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.
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

$22.29$: 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

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