Examples
Example 1. The transformation from spherical coordinates (r, θ, φ) to Cartesian coordinates (x1, x2, x3), is given by the function F : R+ × × [0,2π) → R3 with components:
The Jacobian matrix for this coordinate change is
The determinant is r2 sin θ. As an example, since dV = dx1 dx2 dx3 this determinant implies that the differential volume element dV = r2 sin θ dr dθ dϕ. Nevertheless this determinant varies with coordinates. To avoid any variation the new coordinates can be defined as Now the determinant equals 1 and volume element becomes .
Example 2. The Jacobian matrix of the function F : R3 → R4 with components
is
This example shows that the Jacobian need not be a square matrix.
Example 3.
The Jacobian determinant is equal to . This shows how an integral in the Cartesian coordinate system is transformed into an integral in the polar coordinate system:
Example 4. The Jacobian determinant of the function F : R3 → R3 with components
is
From this we see that F reverses orientation near those points where x1 and x2 have the same sign; the function is locally invertible everywhere except near points where x1 = 0 or x2 = 0. Intuitively, if you start with a tiny object around the point (1,1,1) and apply F to that object, you will get an object set with approximately 40 times the volume of the original one.
Read more about this topic: Jacobian Matrix And Determinant
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