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# 3d curl intuition, part 2

Continuing the intuition for how three-dimensional curl represents rotation in three-dimensional fluid flow. Created by Grant Sanderson.

## Want to join the conversation?

• At the z component is added. Why isn't it z instead of 0? I thought with z = 0 all the time you just get a vectorfield in the xy-plane.
• Think of the 3rd component as of z*0. If you input z=1 or 2 or 3 you will have still have to plot y^3-9y and x^3-9x in a flat plane, but since z=1,2 or 3 each plotted vector that lied in xy plane will have to start higher. By adding z*0 as 3rd component you don't change the direction or magnitude of the plotted vector, but the starting position (the beginning of a vector communicates the input, the direction and magnitude is the output- which doesn't change )
• when imagining the curl in 3-D I was wondering if it is possible to associate the idea of curl with the rotation of ceiling fan and the air flow. the rotation of the fan can be considered the rotation in 2D i.e. in the plane of the fan whereas the flow of air can be considered as the Z -(normal vector) pointing out of the plane of the fan.
If I curl my fingers in direction of rotation of the fan, I get the downward flow of air and thus right-hand rule holds perfectly true here. similarly, if the rotation is reversed the flow is in the upward direction(which again follows right-hand thumb rule.
I am asking the question because I was wondering if the association of this idea is correct here, it can be further extended to turbines and windmills as well. of course, the engineering design of fans, windmills etc play an important role here but I would like to ask if such an association as at all possible in general?
• The curvature of the blades of the fan is what determines the airflow direction.
• Thank you!
• Thank you so much ilysm!
• I was 15 when I first started watching your (Grant Sanderson's) series on multivariate calculus. At first I was very sceptical about the content of your videos because it appeared as if you were assuming too much own knowledge / understanding. But now (16th birthday today!) things are finally beginning to make sense and hopefully I'll be able to review some line and surface integrals before the very hectic Year 12 starts in September !!
• They say that gravity is a conservative vector field, because at any point in the field, the curl is zero. So that must mean that gravity by itself would not cause a rotation. Why, then, do the planets rotate as they revolve around the Sun?

Maybe because the rotation due to gravity represents the torque, but not the angular velocity of the planet. So does this fact account for why the speed of rotation of planets is constant in the universe?
• I believe the conservation of inertia causes this to be. If you took a system of just a star and a planet on the xy plane, where the planet rotates at some speed clockwise, the planet and star must take on some rotation to conserve inertia. You can see this on a much smaller scale if you sit on a chair that can spin and hold a spinning bike wheel by the axle, so that the axle is horizontal to the ground. If you rotate the bike wheel so that its axle is vertically aligned, you will feel a torque applied and you will begin rotating (barring other forces).