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1·The locus of space tangent point and the arc segment of deck at side are obtained based on the above conditions.

在此基础上，求出首圆弧空间切点线和甲板边线的修正段。

2·In this paper, some relations of the invariants concerning the simplex and its tangent point simplex, are given and then, an inequality for the dihedral angles of a simplex is also presented.

本文首先给出单形的不变量和它的切点单形的不变量之间的若干关系式，然后再给出一个关于单形二面角的不等式。

3·That is how we reformulated it. That means we take our curve and we figure out at each point how big the tangent component I guess I should probably take the same vector field as before.

这是另一种形式，它意味着,要求出每点处向量在切向量方向的分量,我还是用之前那个向量场吧。

4·Say you have a minimum, well, the tangent plane at this point, at the bottom of the graph is going to be horizontal.

如果你说这是个极小值，那么这点的切平面应该是水平的。

5·And visually what we're accomplishing is somehow to take the hyperbola, and take a point on the hyperbola, and figure out some tangent line.

从视觉上看，我们要做的是，想办法，在这个反函数，在反函数上取一个点，设法找出切线。

6·See the point in the middle, at the origin, is a saddle point. If you look at the tangent plane to this graph, you will see that it is actually horizontal at the origin.

我们来看中间的那个点，在原点上，那就是一个鞍点，在这点做一个切平面,你可以看到这个切平面,是在原点上水平的。

7·Well, that means the gradient is actually perpendicular to the tangent plane or to the surface at this point.

那意味着，梯度向量在这点上，垂直于切平面或者是等值面。

8·And, at this point, I have the tangent plane to the level surface OK, so this is tangent plane to the level.

在这点上，我们有一个切于等值面的切面，这就是等值面的切平面。

9·What I do at any point is project F to the tangent direction, I figure out how much F is going along my curve and then I sum these things together.

我所做的就是把F投影到切向量方向上，得出F沿着曲线的值,然后再把这些加起来。

10·A derivative at a given point is just the slope of the tangent line that kisses that point.

一个给定点的导数不过就是一条切线的斜率轻吻了那个点。