Math for Physics

An azimuth (from Arabic al-sumūt, meaning "the directions") is an angular measurement in a spherical coordinate system. The vector from an observer (origin) to a point of interest is projected perpendicularly onto a reference plane; the angle between the projected vector and a reference vector on the reference plane is called the azimuth. An example is the position of a star in the sky. The star is the point of interest, the reference plane is the horizon or the surface of the sea, and the reference vector points north. The azimuth is the angle between the north vector and the...

Spherical Coordinate System as commonly used in physics Spherical coordinates (r, θ, φ) as commonly used in physics: radial distance r, polar angle θ (theta), and azimuthal angle φ (phi). The symbol ρ (rho) is often used instead of r.     Coordinate system conversions As the spherical coordinate system is only one of many three-dimensional coordinate systems, there exist equations for converting coordinates between the spherical coordinate system and others. Cartesian coordinates The spherical coordinates of a point in the ISO convention (radius r, inclination θ, azimuth φ) can be obtained from its Cartesian coordinates (x, y, z) by...

Half-angle formulas can express trigonometric functions of an angle  in terms of functions of an angle θ.   Half-angle formulas list:       Corresponding hyperbolic function half-angle formulas list:    

When discussing a rotation, there are two possible conventions: rotation of the object relative to fixed axes rotation of the axes 1. Rotation of the object relative to fixed axes In R2 space, consider the matrix that rotates a given vector v0 by a counterclockwise angle θ in a fixed coordinate system as shown below. Then the rotation matrix is: So v' = Rθ v0   2.  Rotation of the Axes On the other hand, consider the matrix that rotates the coordinate system through a counterclockwise angle θ. The coordinates of the fixed vector v in the rotated coordinate system are...

Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that, for any real number φ: eiφ = cos(φ) + i sin(φ) where e is the base of the natural logarithm, i is the imaginary unit, and cos and sin are the trigonometric functions cosine and sine respectively, with the argument φ given in radians. This is best shown geometrically. This complex exponential function is sometimes denoted cis(φ) ("cosine plus i sine"). The formula is still valid if φ is a complex...