## 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.

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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