Core - Physics

Chapter 2

Vectors and Scalars

2.1 : Introduction to Scalars

2.1 : 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 ...

2.2 : Introduction to Vectors

2.2 : 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 ...

2.3 : Vector Components in the Cartesian Coordinate System

2.3 : 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 ...

2.4 : Polar and Cylindrical Coordinates

2.4 : 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 ...

2.5 : Spherical Coordinates

2.5 : Spherical Coordinates

Spherical coordinate systems are preferred over Cartesian, polar, or cylindrical coordinates for systems with spherical symmetry. For example, to describe ...

2.6 : Vector Algebra: Graphical Method

2.6 : 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 ...

2.7 : Vector Algebra: Method of Components

2.7 : 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 ...

2.8 : Scalar Product (Dot Product)

2.8 : 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 ...

2.9 : Vector Product (Cross Product)

2.9 : 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 ...

2.10 : Scalar and Vector Triple Products

2.10 : 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 ...

2.11 : Gradient and Del Operator

2.11 : 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 ...

2.12 : Divergence and Curl

2.12 : 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 ...

2.13 : Second Derivatives and Laplace Operator

2.13 : 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 ...

2.14 : Line, Surface, and Volume Integrals

2.14 : 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 ...

2.15 : Divergence and Stokes' Theorems

2.15 : 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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