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# 3d curl formula, part 2

This finishes the demonstration of how to compute three-dimensional curl using a certain determinant. Created by Grant Sanderson.

## Want to join the conversation?

• Another question: Before I watched these videos I tried to do some 3D curl exercises, and I got the formula wrong: instead of crossing nabla with the vector valued function, I crossed nabla with the divergence of the vector valued function. Now I realize that this is not what 3d curl is, that is, after watching this video, but still: does crossing nabla with the divergence of a 3d vector valued function has any mathematical/physical meaning at all? Thank you for your time.
• crossing nabla with the divergence of a 3d vector valued function is basically taking the curl of the divergence, which I don't think has any physical meaning on its own.
An interesting point: the reverse--taking the divergence of the curl--is always equal to zero. ∇⋅ (∇ x F) = 0
• Is there a difference between a dot product and a cross product?

If yes, then what is the difference?
• Yes, there's a big difference. The dot product of two vectors is a scalar. The cross product of two vectors is a vector.

Also, for two nonzero vectors, the dot product is zero if and only if the two vectors are perpendicular, but the cross product is the zero vector if and only if the two vectors are parallel or antiparallel.

Have a blessed, wonderful day!
• What are the units of measurement for curl? Is there a way to tell based on the units of measurement for the vector field? For example, what if the vector field measures velocity?
• why are we subtracting j?
• We are subtracting j because of how the determinant of a matrix is computed. You could as well write +(dP/dz-dR/dx)j instead of -(dR/dx-dP/dz)j and it would have the same meaning, it's just that in the second case you grouped a -1 that in the first case is instead distributed inside the parentheses, thus changing the order of the two partial derivatives.

To add to this, the method used in the video to compute the determinant is called the method of Laplace. If instead you used Sarrus' method, that only works for 3x3 matrices, you would find your j already tied up and with the -1 already distributed inside the parentheses. By the way you would get the same result with both methods, apart for the differently distributed -1.
• For the j column, why do dR/dx - dP/dz rather than dP/dz - dR/dx?
• you can distribute the minus sign before the j-hat inside the parenthesis and it's gonna turn into dP/dz-dR/dx
• I think it's worth pointing out that the reason why i j k terms are similar to the 2d formula is that when we try to represent 3d rotation with a vector, we are actually trying to represent it using a linear combination of rotations in the positive direction of x, y and z. And as we accept the that 3d rotations are very similar to 3d vectors, that we can add rotations the same way we add vectors is also an interesting and potentially puzzling fact.
• How do you extend cross product into higher dimensional Cartesian space? Like, if I wanted to calculate the curl of a function with an R4 output space, what exactly would I do to calculate that? And how exactly does the determinant relate to the theory behind the cross product? Thank you!
(1 vote)
• Curl, as defined here, is only defined in 3-dimensional Euclidean Space. It can be generalized to lower dimensions like how Grant did it for 2-dimensional space in a previous video on 2D curl, but it can't be done for higher dimensions.

The determinant is simply a way to easily remember how a cross product is computed.
• What does the magnitude of curl vector represents? And does it follows all the properties of cross product? Please Explain with some Physical example.
(1 vote)
• The magnitude of the curl is the magnitude of angular velocity (in a physical instance ). Remember the situation of an airflow in a room with a ping pond "magically suspended" at a point in the flow? The ball would rotate about an axis either slowly of fast. Computing the curl for the motion of the airflow at that point where the ball is suspended gives the "speed" at which the ball would be rotating about its axis (Speed of the airflow at that point). The axis (direction) about which it rotates is given by the x, y ,z components of the curl.

Reiterating his explanation on cross products.

Now the cross product(c) between two vectors in 2d lets say a(on the x-axis) and b(angle theta to the positive x-axis) is given be |c| = |a||b|sin( theta).
c is perpendicular to both a and b so on the 'z' axis.
Intuitively |c| = the 2d area between a and b.
Thus area = area of the parallelogram = base * height
base = |a| and height = bsin(theta)....the y component of b.

Therefor area = |a||b|sin(theta) = |c|.

The components of the a and b were used directly and the outcome is the area (physically speaking).

if a and b are differentiable functions and their derivatives used instead then the cross product is the rate of rotation. (physically it may be the angular velocity of the ping pong)

Things get quite 'messy' in 3d but i hope this 2d intuition was useful to build up to 3d... would be glad if someone could offer some help.

Maybe in 3d, the curl on the x-y plane is on the z plane(as explained above where the vectors only had a and y components) Since they have three components here then the curl for the x-z and y-z planes have to also be computed and then to find the resultant rotation, they are computed arithmetically.(since they are all vectors - cross products are vectors)

Happy learning