All points on the surface of a sphere are equally distant from the center.

球心到球面上任何一点距离都一样。

That's also of great value scientifically.

这个外球面区域科研究意义很大。

And then you have another spherical wave coming from the short bank.

然后你会看到另一个来自短波岸球面波。

The integral of dA is A; the area of the concentric spherical surface.

dA分是A；即同心球面面。

Now, we wanna talk about the product of two spheres.

现在，我们想谈谈两个球面乘。

So, the electric flux is independent of the  radius of the concentric spherical surface?

所以，电通量与同心球面半径关？

And you can see this when the ball back by the spread-out lines of oil on the ball surface.

当球从球面上油线中滚回时， 你可以看到这一点。

Say we have a lens that has a spherical face on both sides, known as a convex lens.

假设我们有一个具有球面 两侧，称为凸透。

Points on the exact opposite side of a sphere are called " antipodes" , or " antipodal points" .

球面上完全相对点称为“对跖点”或“对跖点”。

There actually is something called the Riemann sphere that involves dividing by zero by a different method, but that's a story for another day.

事实上，有个叫黎曼球面东西，它涉及到通过不同方法来除以 0，但今天我们且不谈这个。

As an example, Borsuk-Ulam also applies to mapping hyperspheres in 4d space into 3d space.

作为一个例子，博苏克-乌拉姆定理也适用于将4维空间中超球面映射到3维空间中。

It gives a correspondence between points on the sphere and necklace divisions.

它建立了球面上点与项链分割之间对应关系。

If the point (x, y, z) on the sphere corresponds to some necklace allocation, what does the point (-x, -y, -z) correspond to?

如果球面上点 (x, y, z) 对应某种项链分配，那么点 (-x, -y, -z) 对应什么？

The main thing to mention is that there's a more general version of the Borsuk-Ulam theorem, one which applies to higher-dimensional spheres.

需要提到主要一点是， 有一个更一般博苏克-乌拉姆定理版本，适用于更高维度球面。

For each point on the sphere, draw a line from -1 at the south pole through that point and see where it intersects the plane.

对于球面上每一点，从南极-1点画一条直线穿过该点，看看它与平面相交位置。

The positions of stars and other celestial bodies can be defined by their places on this sphere, if they were imagined to be attached to it.

星星和其他天体方位可通过它们在这个球面上位置来定义，假设它们被想象为附着于其上。

This way, we need to show that g maps some point of the sphere to the origin of 2d space, (0, 0).

这样，我们需要证明g将球面上某一点映射到二维空间原点(0, 0)。

If that's true no matter what patch of area we're considering, that's what we  mean by a uniform distribution on the sphere.

如果论我们考虑哪个区域补丁， 这个情况都成立，那就是我们在球面上所说均匀分布。

And you could go the other way around; any way to divide up the necklace and divvy up the pieces gives us a unique point on the sphere.

你也可以反过来思考；任何分割项链并分配碎片方法都会给我们球面上一个唯一点。

In other words, going to the antipodal point on the sphere results in reflecting the output through the origin in the output space.

换句话说，到达球面上对跖点会导致输出空间中输出关于原点反射。