Curl

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Curl

Based on “Curl” in section 5.5 of the textbook.

Key Ideas

Geometrically, curl is the tendency for a vector field to rotate around individual points, e.g., like leaves in a stream.

There are formulas, rooted at least in form in the vector cross product; there’s a determinant mnemonic that’s helpful for remembering the formula.

2-dimensional curls exist, and have a a much simpler formula than 3-dimensional ones do.

Examples

Find ∇×F if F(x,y,z) = ⟨ xy²z, sinx e^x, (x+y)/z ⟩

Identify the P, Q, and R parts of the field, then plug them into the 3-dimensional formula for curl:

Find ∇×G if G(x,y) = ⟨ sinx siny, cosx cosy ⟩

Like the previous problem, this is mostly an exercise in using the formula, but in 2 dimensions:

Does my velocity field from the brewing example (V(x,y) = ⟨ -y, x ⟩) have any twist to it?

Find the curl of the field, since curl measures the tendency of a vector field to twist:

What’s the geometric interpretation of this result?

The direction the vector points in (⟨0, 0, 2⟩ in this case) is the axis things would spin around at each point. The magnitude of the vector indicates the strength of the twist, but I don’t know the units. Positive, however, means that the rotation is counterclockwise.

Can you devise a 3-dimensional vector field whose curl is 0?

There were a number of good ideas. One of the first was to realize that any field in which every component has no variables with respect to which it will get differentiated in curl will always have a 0 curl, basically this means making the x component depend only on x, the y component only on y, and the z component only on z. Another approach is to think geometrically, and ask what vector field would exert no spin on object in it; this thinking leads to a constant vector field.

Next

Properties, and maybe some applications, of divergence and curl.

Please read “Using Divergence and Curl” in section 5.5 of the textbook.

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