The direction is tangent to the trajectory.
方向就是轨道的切线方向。
And that line is tangent to the surface.
一条与曲面相切的直线。
But, actually, that is not true because here the circle is tangent to the axis.
但实际上，这是不对的，因为这个圆与这个坐标轴相切。
They are just flowing along the circle around and around so the flux will be zero. F now is tangent to c.
只是沿着曲线流动，所以通量为零，F与C相切。
An algorithm for constructing rational spline closed curve which is tangent to the given polygon is described.
描述了一种与给定多边形相切的有理样条曲线的算法。
If you remember what we did several times, well, along the circle that vector field now is tangent to the circle.
我们做过若干次了，沿着圆，向量场与圆相切。
That means that you want to pick, ultimately, a point on this — this line is tangent to the efficient portfolio frontier with all the other assets in it.
那意味着你会最终会在这上面选一个点-，这条直线是与有效边界相切的，后者包含了所有的资产组合。
The access is tangent to the curve and the Spaces refer to the interior, while the "public space" of the living room, being transparent, sees outside with a view to the thicket.
访问是相切的曲线和空间是指内地，而“公共空间的客厅”，是透明的，看到外面的灌木丛，以查看。
The wedge angle bisector gives as intersection with the cutting edge profile point pint. (3) Draw a circle that intersects point pint and is tangent to both straight fitting lines.
这楔角的等分线给作为路口停止切割边缘轮廓点品脱的啤酒。 （3）画一个圆订交点，两品脱和相切直线拟合直线。
In particular, an easy case where you know you can get away without computing anything is, of course, if your vector field is tangent to the surface because then you know that there is no flux.
特别地，有一种简单的情况，是不用计算的，当向量场与曲面相切，没有通量。
And, in some cases, for example, if you know that the vector field is tangent to the curve or if a dot product is constant or things like that then that might actually give you a very easy answer.
例如，在某些情况下，如果已知向量场与曲线相切，或者内积是一个常数等等，那么结果将会很简单。
The green spiral is made from quarter-circles tangent to the interior of each square, while the red spiral is a Golden spiral, a special type of logarithmic spiral.
绿色螺旋线由内部每一个正方形和四分之一圆相切线构成，而红色的螺旋线是黄金螺旋线，它是对数螺旋线的一种特殊类型。
We know that the gradient is perpendicular to this vector that's tangent to the level surface What about other vectors tangent to the level surface?
我们知道梯度向量垂直于这个向量，而速度向量是相切于等值面的，其他相切于等值面的向量，又如何呢？
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。
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。
A tangent is a straight line that just touches a curve, so the concept of "running the tangents" is to run the shortest distance possible by running straight from one curve to the next.
切线是一条直线，刚好接触到曲线，所以跑切线是从一个弯道跑到另一个最短的距离。
Because so far I have shown you pictures and have said see they are tangent.But why is it that they have to be tangent in general?
因为一直到现在我们都是在看图像，而且知道它们，是相切的，但是为什么在一般情况下它们都相切？
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。
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。
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。
Then actually there are ways you can use basically differentials and constrained partials to figure out what the tangent vector to the curve is and so on.
然后，可以用微分关系,或受约束的偏微分来表示,曲线的切向量，就是这样了。
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。
Another way to say it, and I think that is probably closer to the truth, is that these things are somehow placeholders to put values and get a tangent approximation.
换种我认为更贴近事实的说法，它们是通过赋值而得到切线的近似值的，一种占位符。
That's actually pretty neat because there is a nice application of this which is to try to figure out, now we know, actually how to find the tangent plane to anything, pretty much.
这个性质很漂亮，因为它有一个很好的应用，可以用来，求算很多东西的切面。
The other way to think about it, the direction in which the value doesn't change is a direction that's tangent to the level surface.
另一种方法是，函数值不变的方向是,是与等值面相切的方向。
So, the velocity vector is going to be always tangent to the curve.
速度向量总是与曲线相切的。
That is a vector field in which the trajectories are going to be circles centered in the z-axis and our vector field is just going to be tangent to each of these circles.
这是一个向量场，其中旋转轨道是，以z轴为圆心的圆周，并且这个向量场与每个圆相切。
So you should be able to calculate what the friction coefficient is using the equation tangent alpha equals mu s.
你们则需要学会,计算摩擦系数,使用μ静=tanα这个公式。
The first of these integrals is seen to be a logarithmic form and the second the inverse tangent form.
这些积分中的第一个被看成是对数形式，而第二个被看成是反正切形式。
The first of these integrals is seen to be a logarithmic form and the second the inverse tangent form.
这些积分中的第一个被看成是对数形式，而第二个被看成是反正切形式。
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