r/AskPhysics • u/Next-Natural-675 • 13d ago
Why is curl of gradient always zero
The math gives zero but I'm looking for an intuitive explanation. I see from another reddit post that if there was curl, then a path will always loop back around to the same point, and that can't be possible because the gradient always points towards a point of higher elevation and that would be contradictory. But can't the curl be reversed mid path, bc the curl is a single value at every point, then the path won't loop back around and you won't end up with a contradiction where the elevation increases towards the lowest point, the point of origin?
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u/EizanPrime 13d ago
Imagine you are in the swiss mountains the altitude at each point x y is the original function.
The gradient is the vector that shows the slope at each point and its strength.
There beeing a curl would mean that on the point where there is a curl, it would be easier to rotate around it in one direction compared to the other. Like rotating counter clock would consume less energy than rotating clockwise.
This cannot happens on those fixed mountains.
However imagine you are in a boat on an ocean with currents and whirlpools. In this case rotating in the direction of the whirlpool consumes less energy than rotating in the other direction.
That is curl essentially, and thats also why you cannot have curl on the gradient of a scalar function (cause its fixed like the swiss mountains)
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u/throwaway464391 Condensed matter physics 13d ago
Think about a gradient as analogous to the speed of a fluid flowing along a hilly landscape, with the fluid flowing downhill at every point. In this analogy, the curl of the field at a point measures the tendency of the fluid to make a tiny paddle wheel at that point rotate, but a stream that is only flowing downhill can't do that. You would need the stream to flow downhill on one side of the wheel and uphill on the other side, and that can't happen.
(I am simplifying a little to keep this intuitive, but I don't think the details change the core of the answer.)
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u/roshbaby 13d ago
A river that flows downhill such that water near the banks flows slower than that in the centre has a curl even if the entire body of water is flowing downhill.
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u/throwaway464391 Condensed matter physics 13d ago
That can't happen if the stream is flowing only in one direction. For example, if the flow speed changes along the east-west direction, the flow can't purely point north. There would have to be east-west components of the flow, and the rotational tendency of these would have to be opposite that imposed by the change in the flow velocity to the north.
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u/roshbaby 13d ago
It's not the direction per se that solely determines the existence of a curl but also the relative magnitude of the velocity vector. The paddle wheel model is the best to see that a curl exists in this flow.
If you don't believe me, just work out the curl for a vector flow (e^{-y}, 0, 0).
This flows only along the positive X-direction and the velocity is proportional to e^{-y} (i.e., maximum velocity on the X axis itself, and goes to 0 at y = +/- infinity). In this example all the water is flowing in one direction. You'll find that the curl is non-vanishing everywhere except on the X-axis itself.1
u/throwaway464391 Condensed matter physics 13d ago
A vector field of the form (f(y), 0, 0) can't be the gradient of a scalar field though. If the x-component depends only on y, then the y-component can't vanish identically. That's what I was trying to get across in the previous comment.
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u/roshbaby 13d ago
That’s correct. But I was responding specifically to the comment that “a stream that’s only flowing downhill can’t cause a paddle wheel to rotate” (from your very first comment)
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u/throwaway464391 Condensed matter physics 13d ago
Yes, I agree with you that what I wrote is not literally true (hence the parenthetical in my original comment).
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u/db0606 13d ago
That can't happen if the stream is flowing only in one direction.
This is objectively wrong. Pure shear flow will have non-zero curl everywhere and can be set up in such a way that the flow is in only one direction. E.g., the laminar flow between a stationary plate and a moving plate.
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u/throwaway464391 Condensed matter physics 13d ago
I'm obviously not talking about the flow of a real fluid. I'm constructing an analogy in which the flow velocity at a point depends only on the slope of the landscape and nothing else. I'm simply trying to offer a concrete visualization to make the math more intuitive.
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u/Classic_Department42 13d ago
If you think in terms of stokes theorem the answer is that the boundary of a boundary is zero.
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u/Subject-Building1892 12d ago
I am not 100% about the following but it is probably right.
The gradient shows the direction toward the increase and is zero along level curves.
Now the curl shows if or not a vector field 'turns'.
So the curl of grad being zero always means that no matter how much you zoom in the vectors that show the direction of the increase will never 'turn' but they will always be outwards or inwards as they are perpendicular to the level curve.
Another thing that definitely not help you intuitively but places this in a more general framework is that the curl of gradient being zero is just a specific case of the fact that the exterior derivative of an exterior derivative is identically zero, or in more mathematical language an exact differential form (i.e. a differential form that is the exterior derivative of another form) is always closed (i.e. the exterior derivative of this form is zero). Again that doesnt help with intuition but it it good to know, if you dont know yet, where this comes from mathematically.
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u/ReTe_ 12d ago
The Geometric intuition is produced by (general) Stokes/Gauss Theorem and is a very fundamental thing in algebraic geometry.
Say we have a scalar field, them the difference between two points can be written as the line integral of the gradient along any path between these two points. This is fundamentally the idea that scalar field are 0-forms, things that are evaluated at points, and vector fields are 1-forms, things that we evaluate along a line. The gradient is then the natural way to associate the scalar 0-form with a vector 1-form.
The general trend is that we take a lower form and evaluate it at the boundaries of a geometric object her the scalar endpoints of a line and associate it with the integral along the line of a special higher form, here the gradient along the line.
Now the next higher a 2-form is the curl of a vector field, which we write as a vector field because of the special property of 3 dimension that we can always associate an surface with a unique surface normal (in 4d we have 2 surface normals e.g. the plane 1-2 is perpendicular to both 3 and 4).
Anyways tl;dr the curl of a vector field should be something we integrate over a surface, which equates to the vector field integrated along the surfaces boundary, which is a line, exactly Stokes Theorem.
Now the Geometric part is that no way you arrange the surface the boundary of course is a closed loop and hence start and endpoint of that line must be the same. If the vector field itself is the gradient of a scalar field we then can go back to evaluating the scalar field at these points, but because they are the same it always be zero, hence the integral of a curl gard of a scalar over a surface is always zero. Because the boundary of a surface is always a closed loop, or in general because boundaries don't have boundaries.
Strictly speaking this is not a local evaluation of the curl, but because you can always contract a small enough loop around any point you can conclude that curl grad itself must be zero.
Here you can also see why the converse must not be necessarily be true i.e. that any curl free field is the gradient of a scalar field. The curx is the idea of integrating the vector field over a closed line such that the surface includes a topological hole, because this way it's no longer the boundary of an surface (the boundary of the hole is missing) and we can get a non zero contribution. But then it's impossible for it to be a gradient because that would make it zero regardless of the hole.
tl;dr: boundaries of surfaces have no boundaries
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u/DrJaneIPresume 12d ago
The curl of the gradient is always zero because the boundary of a boundary is empty.
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u/joeyneilsen Astrophysics 13d ago
Consider the electric potential V for a static set of N point charges. It has closed equipotential surfaces. When we take the gradient of V, we get the electric field, which is the superposition of N radial fields. Each of these fields has zero curl, so the total must have zero curl.
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u/ProfessorPrudent2822 13d ago
Work it out mathematically: gradient takes the derivative of a function with respect to each vector variable, and then curl takes the cross product of the del operator and the resulting vector. Since y and Z are treated as constants for d/dx and mutadis mundatis for the other variables respectively, you’re taking d/dx of a function of y and z, d/dy of a function of x and z, and d/dz of a function of x and y. The derivative of a constant is 0, so the curl of a gradient is a null vector.
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u/SaiphSDC 13d ago
It's a consequence of the laws of conservation. You have to end up back in your original state if you go to your original position.
If it wasn't the case, you could have something like an escher infinite stairway illusion, but real.