Its really intersting that divergence and curl share a relationship. Precviously, I just applied the formula to calculate the values, but didn't understand what they meant conceptually.
Shell
The side by side pictures really make it more clear on how they relate to each other, as you can see the arrows more clearly and the shapes make themselves more apparent
penguin
Would the relationship between div(X) = curl(X^T) in 3D? just with extra dimension added
evannw
I was never taught the relationship between divergence and curl. So interesting to see how closely related the two operations are.
clam
I believe the relationship is that the ith component of the curl is equal to the divergence after rotating -pi/4 about the ith basis vector and then taking the projection onto the normal plane to the ith basis vector---since if A = [[0,1],[-1,0]] is the 2d -pi/4 rotation matrix, we have that curl(X) = div(AX), well we can expand that to 3d with the matrices that rotate -pi/4 about each basis vector then do the projection (e.g. for e1 it would be A[1]=[[0,0,0],[0,0,1],[0,-1,0]]) then just some algebra gets you that curl(X)[i] = div(A[i]X). Probably works better the other way around though but I'm too tired to figure that out
oadrian96
Something i really appreciate about this class is the intution it gives us about 3D calc and linear algebra concepts. While taking those classes i was mostly focused on applying some formulas to get answers but never really got the chance to stop and think about all the geometric interpretations. These last two lectures have helped further my intuition and its really awesome.
peanut
Thanks for the comments above, at first I found it hard to understand the pictures, but your thoughts provide me insights.
bepis
This reminds me a lot of E&M in physics! Seems like there's a lot of similar ideas that cross over into graphics.
zhenliz
The comments above about the relationship in 3D indeed help me understand it.
Its really intersting that divergence and curl share a relationship. Precviously, I just applied the formula to calculate the values, but didn't understand what they meant conceptually.
The side by side pictures really make it more clear on how they relate to each other, as you can see the arrows more clearly and the shapes make themselves more apparent
Would the relationship between div(X) = curl(X^T) in 3D? just with extra dimension added
I was never taught the relationship between divergence and curl. So interesting to see how closely related the two operations are.
I believe the relationship is that the ith component of the curl is equal to the divergence after rotating -pi/4 about the ith basis vector and then taking the projection onto the normal plane to the ith basis vector---since if A = [[0,1],[-1,0]] is the 2d -pi/4 rotation matrix, we have that curl(X) = div(AX), well we can expand that to 3d with the matrices that rotate -pi/4 about each basis vector then do the projection (e.g. for e1 it would be A[1]=[[0,0,0],[0,0,1],[0,-1,0]]) then just some algebra gets you that curl(X)[i] = div(A[i]X). Probably works better the other way around though but I'm too tired to figure that out
Something i really appreciate about this class is the intution it gives us about 3D calc and linear algebra concepts. While taking those classes i was mostly focused on applying some formulas to get answers but never really got the chance to stop and think about all the geometric interpretations. These last two lectures have helped further my intuition and its really awesome.
Thanks for the comments above, at first I found it hard to understand the pictures, but your thoughts provide me insights.
This reminds me a lot of E&M in physics! Seems like there's a lot of similar ideas that cross over into graphics.
The comments above about the relationship in 3D indeed help me understand it.