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

到上任何点的距离都样。

That's also of great value scientifically.

这外区域的科学研究意义很大。

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

然后你会看到另来自短波岸的波。

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

dA的积分是A；即同的积。

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

现在，我们想谈谈两的乘积。

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

所以，电通量与同的半径关？

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

当从上的油线中滚回时， 你可以看到这点。

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

假设我们有具有的镜头 两侧，称为凸透镜。

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

上完全相对的点称为“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，有叫黎曼的东西，它涉及到通过不同的方法来除以 0，但今天我们且不谈这。

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

作为例子，博苏克-乌拉姆定理也适用于将4维空间中的超映射到3维空间中。

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

它建立了上的点与项链分割之间的对应关系。

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

如果上的点 (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.

需要提到的主要点是， 有更般的博苏克-乌拉姆定理版本，适用于更高维度的。

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.

对于上的每点，从南极的-1点画条直线穿过该点，看看它与平相交的位置。

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.

星星和其他天体的方位可通过它们在这上的位置来定义，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将上的某点映射到二维空间的原点(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.

如果论我们考虑哪区域补丁， 这情况都成立，那就是我们在上所说的均匀分布。

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的方法都会给我们上的唯点。

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，到达上的对跖点会导致输出空间中的输出关于原点反射。