7.7. Tensor Calculus in Euclidean Spaces - Spehrical coordinates in \(E^3\)#
Using spherical coordinates \((q^1, q^2, q^3) = (r, \phi, \theta)\) and spherical base vectors (uniform in space, so that their derivatives are zero), a point in Euclidean vector space \(E^3\) can be represented as
\[\vec{r} = r \cos \theta \sin \phi \, \hat{x} + r \sin \theta \sin \phi \, \hat{y} + r \cos \phi \, \hat{z} \ .\]
7.7.1. Natural basis, reciprocal basis, metric tensor, and Christoffel symbols#
7.7.2. Differential operators#
7.7.2.1. Gradient#
Example 7.22 (Gradient of a scalar field)
Example 7.23 (Gradient of a vector field)
Example 7.24 (Gradient of a \(2^{nd}\)-order tensor field)
7.7.2.2. Directional derivative#
7.7.2.3. Divergence#
Example 7.25 (Divergence of a vector field)
Example 7.26 (Divergence of a \(2^{nd}\)-order tensor field)
7.7.2.4. Laplacian#
Example 7.27 (Laplacian of a scalar field)
Example 7.28 (Laplacian of a vector field)