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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