关于形式外积可解释为对应图表交集的技术疑问

Hey there! Let's unpack this intuitive explanation step by step—it's a really clever way to wrap your head around exterior products without getting bogged down in abstract math.

First, let's recap the quote from the paper you referenced:

"In order to represent the exterior product of two $n$-forms you find the intersection of the diagrams representing the forms. For example if we draw $dx$ as vertical lines and $dy$ as horizontal lines then $dx ∧ dy$ is the set of intersection points which will form a uniform grid."

Let's break this down with geometric intuition:

• dx is essentially a tool that measures changes along the x-axis. Drawing it as vertical lines makes sense because those lines run parallel to the x-direction—each line represents a slice of space where the x-value is fixed, and together they cover all possible x positions.
• dy does the same for the y-axis: horizontal lines represent slices where y is fixed, covering all possible y positions.

Now, the exterior product dx ∧ dy is what we use to measure 2-dimensional area in the plane. When you take the intersection of vertical and horizontal lines, you create tiny uniform squares—each intersection point is the corner of one of these squares. That grid of points (and the squares they define) is exactly the "space" that dx ∧ dy measures: each square is a tiny unit of area, and the intersection points act as reference markers for those units.

Another angle to consider: exterior products exist to combine lower-dimensional measurement tools into higher-dimensional ones. The intersection of the diagrams is a visual way to show the overlap of the directions each form focuses on—when you combine a vertical-slice form and a horizontal-slice form, their overlap is the set of points that exist in both types of slices, which naturally defines a 2D grid.

Also, don't forget that exterior products are antisymmetric (dx ∧ dy = -dy ∧ dx). If you swapped the diagrams (drew dx as horizontal and dy as vertical), the intersection grid would be flipped, which matches the sign change in the product—this is a nice little alignment between the visual intuition and the formal math.

备注：内容来源于stack exchange，提问作者Sam