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Caso 2: En un flujo de calor, el campo de temperatura está dado por \( F^{-}(x, y)=-y i+x i \) ¿Cuál es el rotacional de F.?

Ask by Reeves Hammond. in Mexico
Mar 15,2025

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El rotacional de \( F \) es 2.

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Bonus Knowledge

Para encontrar el rotacional de un campo vectorial, debemos aplicar la fórmula del rotacional en coordenadas cartesianas. Para el campo de temperatura \( F^{-}(x, y) = -y \mathbf{i} + x \mathbf{j} \), el rotacional se define como: \[ \nabla \times \mathbf{F} = \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \mathbf{k} \] donde \( \mathbf{F} = P \mathbf{i} + Q \mathbf{j} \). En este caso, \( P = -y \) y \( Q = x \). Calculamos: \[ \frac{\partial Q}{\partial x} = \frac{\partial x}{\partial x} = 1 \] \[ \frac{\partial P}{\partial y} = \frac{\partial (-y)}{\partial y} = -1 \] Ahora, aplicamos estos valores en la fórmula del rotacional: \[ \nabla \times \mathbf{F} = \left( 1 - (-1) \right) \mathbf{k} = 2 \mathbf{k} \] Por lo tanto, el rotacional de \( F^{-} \) es \( 2 \mathbf{k} \).

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