Learn how curl is really defined,which involves mathematically capturing the intuition of fluid rotation.This is good preparation for Green's theorem.
Log in Yingtong 6 years agoPosted 6 years ago. Direct link to Yingtong's post “Could you please explain ...” Could you please explain how this formal definition equals the del operator cross F? • (14 votes) parsanoori79 7 years agoPosted 7 years ago. Direct link to parsanoori79's post “Why the integral of F.dr ...” Why the integral of F.dr is proportional to the area ? • (4 votes) guilhem.escudero 6 years agoPosted 6 years ago. Direct link to guilhem.escudero's post “I think if I'm right it i...” I think if I'm right it is because the integral of F.dr depend on dr which is equal to to a tiny step along the curve which is equal to the derivative of the parametrize function times dt. Since you add all tiny steps of the curve, you wander all the curve. The bigger the curve is, the bigger the area in the curve is, so the integral of F.dr is proportional to the area (4 votes) Aneesh Vasudev 8 years agoPosted 8 years ago. Direct link to Aneesh Vasudev's post “OMG!! So, 2D divergence a...” OMG!! So, 2D divergence and curl are the same?? And the line integral around a closed curve of a vector function, = to 2D Flux?? which is used to interpret divergence and curl? • (0 votes) Alexander Wu 8 years agoPosted 8 years ago. Direct link to Alexander Wu's post “Divergence and curl are n...” Divergence and curl are not the same. (The following assumes we are talking about 2D.) Curl is a line integral and divergence is a flux integral. For curl, we want to see how much of the vector field flows along the path, tangent to it, while for divergence we want to see how much flow is through the path, perpendicular to it. Line integrals and flux are different for the same reason. But yes, they are used to interpret divergence and curl, as a limit. (12 votes) Mehmet Anil ASLIHAK 4 years agoPosted 4 years ago. Direct link to Mehmet Anil ASLIHAK's post “This definition of curl o...” This definition of curl only gives a scalar, should not it be a vector? I mean what comes out from the integral in the definition is just a number, but mustn't curl be represented by a vector? • (1 vote) 𝜏 Is Better Than 𝝅 4 years agoPosted 4 years ago. Direct link to 𝜏 Is Better Than 𝝅's post “Technically, curl should ...” Technically, curl should be a vector quantity, but the vectorial aspect of curl only starts to matter in 3 dimensions, so when you're just looking at 2d-curl, the scalar quantity that you're mentioning is really the magnitude of the curl vector. The curl vector will always be perpendicular to the instantaneous plane of rotation, but in 2 dimensions it's implicit that the plane of rotation is the x-y plane so you don't really bother with the vectorial nature of curl until you get to 3 dimensional space. Then it starts to matter. (4 votes) Sergii Sergii 4 years agoPosted 4 years ago. Direct link to Sergii Sergii's post “I guess the diameter of t...” I guess the diameter of the region should tend to zero in the defining limit, not area, as area can tend to zero without shrinking of the region to a point, e.g. by flattening it into a line. • (2 votes) Hexuan Sun 9th grade 7 months agoPosted 7 months ago. Direct link to Hexuan Sun 9th grade's post “I don't get how did the i...” I don't get how did the integral of f dot dr go from representing fluid flow to fluid rotation • (2 votes) Andrea Menozzi 6 years agoPosted 6 years ago. Direct link to Andrea Menozzi's post “where i can find the conn...” where i can find the connection between the formal definition to the "practical" definition? do wo I go from one to the other? • (2 votes) Michael Gorbunov 3 years agoPosted 3 years ago. Direct link to Michael Gorbunov's post “Imagine a point that is a...” Imagine a point that is at the center of a counter clockwise curl. On the right of that center point, the vector field points up, while on the left the vector field field points down. Above, the vector field points left, and below it points right. Let's call this vector field F = <f(x,y), g(x,y)> Speaking in derivatives, as we go left to right (dx), the vertical component of the vector field (f) should increase. On the left, vectors point down (negative) while on the right, vectors point up (positive). This is equivalent to a positive dg/dx. As you move vertically, bottom to top, the horizontal component should decrease. Below vectors point right (positive) and above the point left (negative). This is why curl (F) = dg/dx - df/dy makes sense. Another interesting bit is this seems connected to the idea that 90 degree rotations in 2D are (x, y) -> (-y, x), however I'm not 100% sure if there's some simple and nice connection or it's just happen chance. (1 vote) MustardManExtremeTurboEdition 3 months agoPosted 3 months ago. Direct link to MustardManExtremeTurboEdition's post “Is the shape of the regio...” Is the shape of the region always a circle? In the breakdown of the formal definition at the top of the page, it feels like it's implied that the region could be something other than a circle. I don't really understand how that would work, so am I interpreting this wrong or did I just not pick up on something? • (1 vote)Want to join the conversation?
