Soluzione Per definizione, curl ⟨ cos ( x y ) , e x y z , sin ( x y ) ⟩ = ∇ × ⟨ cos ( x y ) , e x y z , sin ( x y ) ⟩ \operatorname{curl} \left\langle \cos{\left(x y \right)}, e^{x y z}, \sin{\left(x y \right)}\right\rangle = \nabla\times \left\langle \cos{\left(x y \right)}, e^{x y z}, \sin{\left(x y \right)}\right\rangle curl ⟨ cos ( x y ) , e x yz , sin ( x y ) ⟩ = ∇ × ⟨ cos ( x y ) , e x yz , sin ( x y ) ⟩ , o, equivalentemente, curl ⟨ cos ( x y ) , e x y z , sin ( x y ) ⟩ = ∣ i ⃗ j ⃗ k ⃗ ∂ ∂ x ∂ ∂ y ∂ ∂ z cos ( x y ) e x y z sin ( x y ) ∣ \operatorname{curl} \left\langle \cos{\left(x y \right)}, e^{x y z}, \sin{\left(x y \right)}\right\rangle = \left|\begin{array}{ccc}\mathbf{\vec{i}} & \mathbf{\vec{j}} & \mathbf{\vec{k}}\\\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z}\\\cos{\left(x y \right)} & e^{x y z} & \sin{\left(x y \right)}\end{array}\right| curl ⟨ cos ( x y ) , e x yz , sin ( x y ) ⟩ = ∣ ∣ i ∂ x ∂ cos ( x y ) j ∂ y ∂ e x yz k ∂ z ∂ sin ( x y ) ∣ ∣ , dove × \times × è l'operatore prodotto incrociato .
Pertanto, curl ⟨ cos ( x y ) , e x y z , sin ( x y ) ⟩ = ⟨ ∂ ∂ y ( sin ( x y ) ) − ∂ ∂ z ( e x y z ) , ∂ ∂ z ( cos ( x y ) ) − ∂ ∂ x ( sin ( x y ) ) , ∂ ∂ x ( e x y z ) − ∂ ∂ y ( cos ( x y ) ) ⟩ . \operatorname{curl} \left\langle \cos{\left(x y \right)}, e^{x y z}, \sin{\left(x y \right)}\right\rangle = \left\langle \frac{\partial}{\partial y} \left(\sin{\left(x y \right)}\right) - \frac{\partial}{\partial z} \left(e^{x y z}\right), \frac{\partial}{\partial z} \left(\cos{\left(x y \right)}\right) - \frac{\partial}{\partial x} \left(\sin{\left(x y \right)}\right), \frac{\partial}{\partial x} \left(e^{x y z}\right) - \frac{\partial}{\partial y} \left(\cos{\left(x y \right)}\right)\right\rangle. curl ⟨ cos ( x y ) , e x yz , sin ( x y ) ⟩ = ⟨ ∂ y ∂ ( sin ( x y ) ) − ∂ z ∂ ( e x yz ) , ∂ z ∂ ( cos ( x y ) ) − ∂ x ∂ ( sin ( x y ) ) , ∂ x ∂ ( e x yz ) − ∂ y ∂ ( cos ( x y ) ) ⟩ .
Trovare le derivate parziali:
∂ ∂ y ( sin ( x y ) ) = x cos ( x y ) \frac{\partial}{\partial y} \left(\sin{\left(x y \right)}\right) = x \cos{\left(x y \right)} ∂ y ∂ ( sin ( x y ) ) = x cos ( x y ) (per i passaggi, vedere calcolatrice delle derivate ).
∂ ∂ z ( e x y z ) = x y e x y z \frac{\partial}{\partial z} \left(e^{x y z}\right) = x y e^{x y z} ∂ z ∂ ( e x yz ) = x y e x yz (per i passaggi, vedere calcolatrice delle derivate ).
∂ ∂ z ( cos ( x y ) ) = 0 \frac{\partial}{\partial z} \left(\cos{\left(x y \right)}\right) = 0 ∂ z ∂ ( cos ( x y ) ) = 0 (per i passaggi, vedere calcolatrice delle derivate ).
∂ ∂ x ( sin ( x y ) ) = y cos ( x y ) \frac{\partial}{\partial x} \left(\sin{\left(x y \right)}\right) = y \cos{\left(x y \right)} ∂ x ∂ ( sin ( x y ) ) = y cos ( x y ) (per i passaggi, vedere calcolatrice delle derivate ).
∂ ∂ x ( e x y z ) = y z e x y z \frac{\partial}{\partial x} \left(e^{x y z}\right) = y z e^{x y z} ∂ x ∂ ( e x yz ) = yz e x yz (per i passaggi, vedere calcolatrice delle derivate ).
∂ ∂ y ( cos ( x y ) ) = − x sin ( x y ) \frac{\partial}{\partial y} \left(\cos{\left(x y \right)}\right) = - x \sin{\left(x y \right)} ∂ y ∂ ( cos ( x y ) ) = − x sin ( x y ) (per i passaggi, vedere calcolatrice delle derivate ).
Ora basta inserire le derivate parziali trovate per ottenere il curl: curl ⟨ cos ( x y ) , e x y z , sin ( x y ) ⟩ = ⟨ x ( − y e x y z + cos ( x y ) ) , − y cos ( x y ) , x sin ( x y ) + y z e x y z ⟩ . \operatorname{curl} \left\langle \cos{\left(x y \right)}, e^{x y z}, \sin{\left(x y \right)}\right\rangle = \left\langle x \left(- y e^{x y z} + \cos{\left(x y \right)}\right), - y \cos{\left(x y \right)}, x \sin{\left(x y \right)} + y z e^{x y z}\right\rangle. curl ⟨ cos ( x y ) , e x yz , sin ( x y ) ⟩ = ⟨ x ( − y e x yz + cos ( x y ) ) , − y cos ( x y ) , x sin ( x y ) + yz e x yz ⟩ .
Risposta curl ⟨ cos ( x y ) , e x y z , sin ( x y ) ⟩ = ⟨ x ( − y e x y z + cos ( x y ) ) , − y cos ( x y ) , x sin ( x y ) + y z e x y z ⟩ \operatorname{curl} \left\langle \cos{\left(x y \right)}, e^{x y z}, \sin{\left(x y \right)}\right\rangle = \left\langle x \left(- y e^{x y z} + \cos{\left(x y \right)}\right), - y \cos{\left(x y \right)}, x \sin{\left(x y \right)} + y z e^{x y z}\right\rangle curl ⟨ cos ( x y ) , e x yz , sin ( x y ) ⟩ = ⟨ x ( − y e x yz + cos ( x y ) ) , − y cos ( x y ) , x sin ( x y ) + yz e x yz ⟩ A