Multivariable calculus

Divergence

How much a vector field spreads out from a point.

• Divergence of the Radial Field F = (x, y)The divergence of F = (x, y) is the constant 2. See both diagonal partials, why every point is a source of the same strength, and the slip to avoid.
• Divergence of the Rotation Field F = (-y, x)The divergence of F = (-y, x) is 0. Both diagonal partials vanish, so this spinning field is incompressible even though nothing about it is standing still.
• Divergence of F = (x^2 y, x y^2)The divergence of F = (x^2 y, x y^2) is 4xy. Each diagonal partial gives 2xy. See the work, the quadrant sign pattern, and the term students drop.
• Divergence of F = (e^x cos y, e^x sin y)The divergence of F = (e^x cos y, e^x sin y) is 2 e^x cos y. Both diagonal partials give e^x cos y. See the work and where the field turns into a sink.
• Divergence of F = (sin(xy), cos(xy))The divergence of F = (sin xy, cos xy) is y cos(xy) minus x sin(xy). Two chain rules with two different inner derivatives. See the work and the sign trap.
• Divergence of F = (x, y) / (x^2 + y^2 + 1)The divergence of (x, y) divided by x^2 + y^2 + 1 is 2 over (x^2 + y^2 + 1) squared. Two quotient rules, then a cancellation that clears every variable.
• Divergence of the Gradient Field of x^3 - 3x y^2The divergence of the gradient of x^3 - 3x y^2 is 0. Divergence of a gradient is the Laplacian, and this function is harmonic, so the two terms cancel.
• Divergence of F = (x^2 y, y^2 z, z^2 x)The divergence of F = (x^2 y, y^2 z, z^2 x) is 2xy + 2yz + 2zx. Each component contributes one term, and the cyclic pattern carries into the answer.
• Divergence of F = (x, y, z) / (x^2 + y^2 + z^2 + 1)The divergence of (x, y, z) over x^2 + y^2 + z^2 + 1 is (r^2 + 3) over (r^2 + 1) squared, with r^2 = x^2 + y^2 + z^2. Three quotient rules, one clean sum.
• Divergence of F = (xyz, y^2, z sin x)The divergence of F = (xyz, y^2, z sin x) is yz + 2y + sin x. Each component is differentiated with respect to its own variable and nothing else.