Chapter 2
Vectors and Scalars
Introduction to Scalars
Many familiar physical quantities can be specified completely by giving a single number and the appropriate unit. For example, "a class period lasts ...
Introduction to Vectors
To define some physical quantities, there is a need to specify both magnitude as well as direction. For example, when the U.S. Coast Guard dispatches a ...
Vector Components in the Cartesian Coordinate System
Vectors are usually described in terms of their components in a coordinate system. Even in everyday life, we naturally invoke the concept of orthogonal ...
Polar and Cylindrical Coordinates
The Cartesian coordinate system is a very convenient tool to use when describing the displacements and velocities of objects and the forces acting on ...
Spherical Coordinates
Spherical coordinate systems are preferred over Cartesian, polar, or cylindrical coordinates for systems with spherical symmetry. For example, to describe ...
Vector Algebra: Graphical Method
Vectors can be multiplied by scalars, added to other vectors, or subtracted from other vectors. The vector sum of two (or more) vectors is called the ...
Vector Algebra: Method of Components
It is cumbersome to find the magnitudes of vectors using the parallelogram rule or using the graphical method to perform mathematical operations like ...
Scalar Product (Dot Product)
The scalar multiplication of two vectors is known as the scalar or dot product. As the name indicates, the scalar product of two vectors results in a ...
Vector Product (Cross Product)
Vector multiplication of two vectors yields a vector product, with the magnitude equal to the product of the individual vectors multiplied by the sine of ...
Scalar and Vector Triple Products
Two vectors can be multiplied using a scalar product or a vector product. The resultant of a scalar product is scalar, while with vector products, the ...
Gradient and Del Operator
In mathematics and physics, the gradient and del operator are fundamental concepts used to describe the behavior of functions and fields in space. The ...
Divergence and Curl
The divergence of a vector field at a point is the net outward flow of the flux out of a small volume through a closed surface enclosing the volume, as ...
Second Derivatives and Laplace Operator
The first order operators using the del operator include the gradient, divergence and curl. Certain combinations of first order operators on a scalar or ...
Line, Surface, and Volume Integrals
A line integral for a vector field is defined as the integral of the dot product of a vector function with an infinitesimal displacement vector along a ...
Divergence and Stokes' Theorems
The divergence and Stokes' theorems are a variation of Green's theorem in a higher dimension. They are also a generalization of the fundamental ...
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