SUMMARY
The position vector in a general local basis, such as spherical coordinates, is defined as ##\vec r = r \hat {\mathbf e_r}##, where ##r## represents the distance from the origin and ##\hat {\mathbf e_r}(\theta,\phi)## is the unit vector pointing in that direction. This derivation is rooted in the definition of the coordinate system, emphasizing that the local basis vectors are specific to the coordinate system being used. In contrast, a global Cartesian basis allows for expressing Cartesian coordinates and basis vectors in terms of curvilinear coordinates.
PREREQUISITES
- Understanding of spherical coordinates and their properties
- Familiarity with unit vectors and their significance in vector representation
- Knowledge of Cartesian coordinates and their relationship to curvilinear coordinates
- Basic grasp of vector calculus and coordinate transformations
NEXT STEPS
- Study the derivation of position vectors in different coordinate systems
- Learn about the transformation between Cartesian and spherical coordinates
- Explore the concept of curvilinear coordinates in depth
- Investigate applications of local and global bases in physics and engineering
USEFUL FOR
Students and professionals in mathematics, physics, and engineering who are working with coordinate systems and vector analysis will benefit from this discussion.