What Is the Triple Point?

Suppose the ternary function f (x, y, z) has a first-order continuous partial derivative on the region , is arbitrarily divided into n small regions, and the diameter of each small region is recorded as r (i = 1,2 ,. .., n), the volume is denoted as , || T || = max {r}, taking a point f (, , ) in each small area, and formulating a sum f (, , ) If the limit of the sum when || T || 0 exists and is unique (that is, has nothing to do with the division of and the selection of points), then the limit is called the function f (x, y, z) on the region The triple integral of is denoted as f (x, y, z) dV, where dV = dxdydz.

The triple integral is the quality of the solid.

When the integral function is 1, the density distribution is uniform and is 1, and the mass is equal to its volume value.

When the integral function is not 1, the density distribution is not uniform.

Let be the bounded closed region in space, and f (x, y, z) is continuous on

• (1) If is symmetric with respect to xOy (or xOz or yOz) and f (x, y, z) is an odd function with respect to z (or y or x), then:

• (2) If is symmetrical about xOy (or xOz or yOz), 1 is the part of on one side of the corresponding coordinate plane, and f (x, y, z) is an even function about z (or y or x), :

• (3) If and 'are symmetrical about the plane y = x, then: