Multivariable calculus

Curl

How much a vector field rotates about a point.

• Curl of the Rotation Field (-y, x, 0)The curl of the field with components -y, x and 0 is the constant vector (0, 0, 2), twice the angular velocity of a rigid rotation about the z-axis.
• Curl of the Gradient Field (2xy, x^2 + z^2, 2yz)The curl of the field with components 2xy, x^2 + z^2 and 2yz is the zero vector, and the potential it comes from is x^2 y + y z^2.
• Curl of the Vector Field (xy, yz, zx)The curl of the field with components xy, yz and zx is (-y, -z, -x). In each difference the first partial is zero, so only the negative term survives.
• Curl of the Vector Field (x^2, y^2, z^2)The curl of the field with components x^2, y^2 and z^2 is zero, because each component depends only on its own variable. Its divergence is 2x + 2y + 2z.
• Curl of the Vector Field (z, x, y)The curl of the field with components z, x and y is the constant vector (1, 1, 1), so this field spins at the same rate everywhere about the (1, 1, 1) axis.
• Curl of the Gradient Field (yz, xz, xy)The curl of the field with components yz, xz and xy is zero. The field is the gradient of xyz, and the curl of any smooth gradient is the zero vector.
• Curl of the Vector Field (sin y, sin z, sin x)The curl of the field with components sin y, sin z and sin x is (-cos z, -cos x, -cos y), one negative cosine per component and no chain rule factors.
• Curl of the Shear Flow (z^2, 0, 0)The curl of the shear flow with components z^2, 0 and 0 is (0, 2z, 0): straight parallel flow lines, and a nonzero curl everywhere off the plane z = 0.
• Curl of the Vector Field (y e^z, xz, e^z)The curl of the field with components y e^z, xz and e^z is (-x, y e^z, z - e^z). All three components survive, and the last one is never zero.
• Curl of the Vortex Field Around the z-AxisThe curl of the vortex field with components -y/(x^2+y^2), x/(x^2+y^2) and 0 is zero everywhere off the z-axis, and yet the field is not conservative.