What do you mean by "properly". You have a definition of transformation (for example x'=x+a etc.). The fact that jacobian of the transformation is same for all translations is coming from equivalence principle (all gauges have same result independently of position or relative velocity). If I understand in good way, would you like to find out exact translation of coordinates of transformation from Jacobian? It is impossible and equivalence principle says it is generally imposible from same experiment on two different places find out their relative distance.Shadumu said:Vectors and covectors transform differently with Jacobian Matrices inverse of each other. However, what is the general coordinate transformation is a simple translation of coordinates, the Jacobian Matrix will be trivially a delta and contains no information of how much the translation is. How to describe such a translation of coordinates properly?
Shadumu said:Vectors and covectors transform differently with Jacobian Matrices inverse of each other.
Shadumu said:what is the general coordinate transformation is a simple translation of coordinates
A coordinate translation vector is a mathematical concept used in linear algebra to represent a displacement or movement from one point to another in a coordinate system. It is typically denoted by t and is expressed in terms of the basis vectors of the coordinate system.
A covector is a mathematical object that represents a linear functional on a vector space. It is like a vector in that it has both magnitude and direction, but it operates on vectors instead of points. A covector is typically denoted by a row vector or a differential form.
Coordinate translation vectors and covectors are related in that they both represent geometric transformations in a coordinate system. Coordinate translation vectors represent translations or displacements, while covectors represent rotations or reflections.
A coordinate translation vector and a basis vector are different in that a coordinate translation vector represents a specific movement or displacement in a coordinate system, while a basis vector is a fixed vector that serves as a building block for other vectors in the coordinate system.
Coordinate translation vectors and covectors are used in a variety of practical applications, including computer graphics, physics, and engineering. They are used to model and manipulate geometric transformations, such as translating and rotating objects in 3D space. They are also used in vector calculus to calculate derivatives and integrals in multiple dimensions.