differential geometry Why do you have to include the Jacobian for every coordinate system, but

Spherical Coordinates Jacobian. differential geometry The jacobian and the change of coordinates Mathematics Stack Exchange The spherical coordinates are represented as (ρ,θ,φ) Just as we did with polar coordinates in two dimensions, we can compute a Jacobian for any change of coordinates in three dimensions

The Jacobian generalizes to any number of dimensions (again, the proof would lengthen an already long post), so we get, reverting to our primed and unprimed. We also used this idea when we transformed double integrals in rectangular coordinates to polar coordinates and transformed triple integrals in rectangular coordinates to cylindrical or spherical coordinates to make the computations simpler

Spherical Coordinates Equations

The (-r*cos(theta)) term should be (r*cos(theta)). 1 $\begingroup$ here, the determinant is indeed $-\rho^2\sin\phi$, so the absolute value (needed for integrals) is $\rho^2\sin\phi$ Recall that Hence, The Jacobian is Correction There is a typo in this last formula for J

Spherical Coordinates Equations. The determinant of a Jacobian matrix for spherical coordinates is equal to ρ 2 sinφ. Understanding the Jacobian is crucial for solving integrals and differential equations.

Free FullText An Improved 3D Inversion Based on Smoothness. It quantifies the change in volume as a point moves through the coordinate space The physics convention.Spherical coordinates (r, θ, φ) as commonly used: (ISO 80000-2:2019): radial distance r (slant distance to origin), polar angle θ (angle with respect to positive polar axis), and azimuthal angle φ (angle of rotation from the initial meridian plane).This is the convention followed in this article