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
Famous quotes containing the word examples:
“It is hardly to be believed how spiritual reflections when mixed with a little physics can hold peoples attention and give them a livelier idea of God than do the often ill-applied examples of his wrath.”
—G.C. (Georg Christoph)
“Histories are more full of examples of the fidelity of dogs than of friends.”
—Alexander Pope (16881744)
“No rules exist, and examples are simply life-savers answering the appeals of rules making vain attempts to exist.”
—André Breton (18961966)