Identify the Vector Field

de-CH

utf-8

math math-format graphie

va-00-02

radio

196

randRangeExclude(-8,8,[-1,0,1]) randRangeExclude(-8,8,[-1,0,1])

Which vector field K : \mathbb R^2 \to \mathbb R^2, (x,y) \mapsto K(x,y) fits?

style({ stroke: "black", strokeWidth: 2 }); graphInit({ range: [[-11, 11], [-9, 9]], scale: [22, 22], axisArrows: "->", tickStep: 2, labelStep: 1 }); label([-2,0], "\\llap{-}2", "below"); label([0,-2], "\\llap{-}2", "left"); // Vektorfeld for(var i = -10; i <= 10; i+=2) { for(var j = -8; j <= 8; j+=2) { var kx = A * i / 35; var ky = B * j / 35; line([i - kx/2, j - ky/2], [i + kx/2, j + ky/2], { arrows: "->", strokeWidth: 1, stroke: "gray" }); } }

K(x,y) = \begin{pmatrix}  
			negParens(A) \cdot x \\ negParens(B) \cdot y 
			\end{pmatrix}

K(x,y) = \begin{pmatrix} negParens(-A) \cdot x \\ negParens(B) \cdot y \end{pmatrix}
K(x,y) = \begin{pmatrix} negParens(-A) \cdot x \\ negParens(-B) \cdot y \end{pmatrix}
K(x,y) = \begin{pmatrix} negParens(A) \cdot x \\ negParens(-B) \cdot y \end{pmatrix}

We read the sign in the coordinates of K(x,y) in each of the quadrants.

For example, in the first quadrant, x,y >0 .

Thus, it must be K(x,y) = \begin{pmatrix} negParens(A) \cdot x \\ negParens(B) \cdot y \end{pmatrix}.