Visualizing High-Order Symmetric Tensor Field Structure with Differential Operators

Table 6

Detecting saddles and helices in fourth-order tensor fields.

	Div(Reduce(𝐿(𝐷)))	Tr(Helicity(𝐷))
Source	2735(𝑥^2+𝑦^2)(7𝑥^4+1884𝑥^2𝑦^2+7𝑦^4)	0
Vortex	249(𝑥^2+𝑦^2)(45𝑥^4−56𝑥^2𝑦^2+45𝑦^4)	0
Saddle	−2147(𝑥^2+𝑦^2)(61𝑥^4−28𝑥^2𝑦^2+61𝑦^4)	0
Helix	2147(5+135𝑥^6+5𝑦^2+14𝑦^4+135𝑦^6+𝑥^4(14−33𝑦^2)+𝑥^2(5−3𝑦^2−33𝑦^4))	2(1+𝑥^6+𝑦^6)
Spiral	𝑐1𝑥^6−𝑐2𝑥^5𝑦+𝑐3𝑥^4𝑦^2+𝑐4𝑥^3𝑦^3+𝑐3𝑥^2𝑦^4+𝑐2𝑥𝑦^5+𝑐1𝑦^6	0