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3d curl formula, part 1

How to compute a three-dimensional curl, imagined as a cross product of sorts. Created by Grant Sanderson.

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• In the video about the cross product, Sal said the cross products are only defined in 3-d, while here you said that it could be defined in 2-d.
So, did Sal originally meant that cross products are defined in 3-d `or less` ?
• I think he meant this:
If you are working in two dimensions, let's say in the x-y plane, the cross product will be a vector in the z direction. If you are in the x-z plane, the cross product will be a vector in the y direction. If you are in the y-z plane, the cross product will be a vector in the x direction. Hence the reason why Sal said that the cross products are only defined in 3-d.
Maybe he meant something else or just made a mistake but that's my understanding of it. :)
• Where would one find this cross product video Grant mentioned? I've done the matrices playlist and only the dot product was brought up.
• I think there are introductory videos in Precalculus, and there are more in Linear Algebra / Vectors and Spaces.
• I'm really confused with the use of the cross product to represent curl; we take the product of each partial derivative with its corresponding components and subtract them from each other, which gives us the 2-D formula for curl, but previously we have been taking the partial of P with respect to y for example. I thought that taking a partial derivative is an operation rather than a multiplication?
• So using the cross product, is curl generalizable to more than 3 dimensions?
• is the value of vector,V same in the whole 3d space? for example if we take v vector as velocity of air, sometimes it may flow at different speed at some locations. So how come a sine function represents the velocity of whole air?
• Why does computation go as nabla X vector, why not the other way around- vector X nabla?
(1 vote)
• For cross-products, order matters, so a X b is not the same as b X a (a X b = -(b X a)). The nabla X vector notation isn't really a calculation, it's more of a way to remember the formula for curl (a "notational trick", as Grant says).
Hope this helps!