Calcula la divergència i el rotacional de la següent funció vectorial: $$$F(x,y,z)=(4\cdot x^3 \cdot y-z, y^3, \cos z \cdot 4 \cdot x )$$$
Desenvolupament:
$$$\displaystyle div(F)=\frac{\partial}{\partial x}(4\cdot x^3 \cdot y-z)+\frac{\partial}{\partial y}(y^3)+\frac{\partial}{\partial z}(\cos z \cdot 4 \cdot x)=$$$ $$$=12\cdot y\cdot x^2+3\cdot y^2-4\cdot x\cdot \sin z$$$
$$$\displaystyle rot (F)=\Bigg(\frac{\partial}{\partial y}(\cos z \cdot 4 \cdot x)-\frac{\partial}{\partial z}(y^3),\frac{\partial}{\partial z}(4\cdot x^3 \cdot y-z)-\frac{\partial}{\partial x}(\cos z \cdot 4 \cdot x),$$$ $$$\displaystyle ,\frac{\partial}{\partial x}(y^3)-\frac{\partial}{\partial y}(4\cdot x^3 \cdot y-z) \Bigg)=\Big( 0-0,-1-4\cos z, 4 \cdot x^3 \Big)$$$
Solució:
$$div(F)=12\cdot y\cdot x^2+3\cdot y^2-4\cdot x\cdot \sin z$$
$$rot(F)=\Big(0,-1-4\cos z, 4 \cdot x^3 \Big)$$