2. Symmetry and parity of surface integral;
Symmetry of region q:
(1) If (x, y, z)∈S, then (x, y, a z)∈Q, then 0 is symmetric about xoy. 8 about xox plane, yo plane is symmetrical.
(2)? If (x.y, z)∈Q is (an x, an? Y.z)∈Q Then 2 is symmetrical about z axis. Q is symmetrical about x.
(3) If (xy.2)∈ then (x-1) 2) (y1.12) and (--y2) are ∈2, then O is symmetrical about three coordinate planes.
(4) If (x.y.2)∈Q, then (a x-γ→∈Q, then 0 is symmetrical about the origin.
(5) If (x, y, z)∈Q, then (,r.2) and (x, z)∈2, then 0 is symmetric about x and y∞. 1.2 function parity.
(6) If f(x, y, z) satisfies f(-x, y.z)- dry (x, y.2) on 2, it is said that f is an odd or even function of x on o, and f has similar parity with respect to y or 2.
(7) If f(x.y.z) satisfies f (one x, one y, z)= dry f(x.y.c) on 2, then the factory is said to be an even-odd function of y. On the parity of the center and: or) it is similar to Z.
(8) If f(x.y, z) satisfies the F(-x, 2-2) element Ff(x.y.2) on 2, the plant is called an odd or even function about x, and:.
Extended data:
How to learn integral well:
Look at four kinds of integrals carefully, write down those integral formulas, and then try to understand them intuitively, such as curve integral of coordinates and curve integral of arc length. The former can be understood as the work done by force, and the latter can be understood as the known curve density, so as to find out the curve quality, which is helpful for the memory of the formula after understanding, otherwise it will be chaotic.
If you understand the formula, you can use some symmetries and understand those symmetries instead of rote learning. It is also important to know that X is an even function, Y is a odd function, and the integral is double or zero. Wendeng Chen's book seems to sum up all this.
Then, after understanding the formula, I went to the textbook to find corresponding examples to consolidate it. The fifth edition of Tongji Higher Mathematics is a very good textbook with simple examples and knowledge points.
The first is to understand the formula. Don't see that the formula doesn't know what it means, or you can't remember it. This is to understand and remember it intuitively according to its physical meaning. Find some related topics to do, and pay special attention to the direction of the curve or surface you are considering in the curve integral and surface integral of coordinates.
Generally, the surface is positive towards the Z axis, that is, it is positive when the angle with the positive direction of the Z axis is less than 90 degrees, and negative when it is negative. Find some typical problems to do and summarize them yourself. If the integral area is symmetrical, consider applying symmetry as much as possible.
Let σ be a smooth surface and the function f(x, y, z) is defined on σ. σ is arbitrarily divided into n facets Si, and its area is defined as δ Si. Take any point (Xi, yi, sub) on each small surface Si as the product f (xi, yi, sub) Δ si and sum σ f (xi, yi, sub) Δ si. Remember.
If the limit of σ f (Xi, yi, sub) δ si eXists at λ→0, and the limit value has nothing to do with the division of σ and the selection of points (xi, yi, sub), it is called the surface integral of f(x, y, z) on σ, also known as the first kind of surface integral. Is ∫∫ f (x，y，z)ds； Where f(x, y, z) is called integrand, σ is called integral surface, and dS is called infinitesimal area.