Anyone interested in a visual exploration should checkout Geometrical Vectors by Gabriel Weinreich. https://www.maa.org/press/maa-reviews/geometrical-vectors
…study of smooth manifolds and differential forms. His exercises are approachable, and his book is the most expedient to the full theory of differential forms. As a quirky intuition pump, I recommend Geometrical Vectors by Gabriel Weinreich. The Fortney book mentioned in another comment is a nice, visual book, and there are other references in the replies to the comment I linked.
The book Geometrical Vectors does exactly that. In fact, it considers the gradient vector to be a different kind of vector than a normal distance vector. If I have a vector between two points, and I compress space so that the points…
对视觉化探索感兴趣的人，强烈推荐看看加布里埃尔·温赖希（Gabriel Weinreich）写的《几何矢量》（Geometrical Vectors）。https://www.maa.org/press/maa-reviews/geometrical-vectors
Anyone interested in a visual exploration should checkout Geometrical Vectors by Gabriel Weinreich. https://www.maa.org/press/maa-reviews/geometrical-vectors
……光滑流形和微分形式的研究。他的习题很平易近人，他的书是通往完整微分形式理论最快捷的捷径。作为一种奇巧的直觉泵（intuition pump），我推荐加布里埃尔·温赖希写的《几何矢量》（Geometrical Vectors）。另一条评论里提到的 Fortney 的书是一本很棒的视觉化教材，而在我链接的那条评论的回复里还有其他参考书。
…study of smooth manifolds and differential forms. His exercises are approachable, and his book is the most expedient to the full theory of differential forms. As a quirky intuition pump, I recommend Geometrical Vectors by Gabriel Weinreich. The Fortney book mentioned in another comment is a nice, visual book, and there are other references in the replies to the comment I linked.
《几何矢量》（Geometrical Vectors）这本书讲的恰恰就是这个。事实上，它认为梯度向量和普通的距离向量是完全不同类型的向量。如果我在两点之间有一个向量，然后我压缩空间使这两个点……
The book Geometrical Vectors does exactly that. In fact, it considers the gradient vector to be a different kind of vector than a normal distance vector. If I have a vector between two points, and I compress space so that the points…