Ejercicios de Gradiente de un campo escalar, divergencia y rotacional de un campo vectorial

Desarrollo:

$${\displaystyle div(F)=\frac{\partial }{\partial x}(4\cdot x^3\cdot y-z)+\frac{\partial }{\partial y}(y^3)+\frac{\partial }{\partial z}(\cos z\cdot 4\cdot x)=}$$ $$=12\cdot y\cdot x^2+3\cdot y^2-4\cdot x\cdot \sin z$$

$${\displaystyle rot(F)=(\frac{\partial }{\partial y}(\cos z\cdot 4\cdot x)-\frac{\partial }{\partial z}(y^3),\frac{\partial }{\partial z}(4\cdot x^3\cdot y-z)-\frac{\partial }{\partial x}(\cos z\cdot 4\cdot x),}$$ $${\displaystyle ,\frac{\partial }{\partial x}(y^3)-\frac{\partial }{\partial y}(4\cdot x^3\cdot y-z))=(0-0,-1-4\cos z,4\cdot x^3)}$$

Solución:

$div(F)=12\cdot y\cdot x^2+3\cdot y^2-4\cdot x\cdot \sin z$

$rot(F)=(0,-1-4\cos z,4\cdot x^3)$

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