Question
Determine whether the following vector field is solenoidal, irrotational or both:
F = x2 yi + xyzj – x2 y2k
Answer :
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To determine whether the vector field is solenoidal, irrotational, or both, we need to analyze its divergence and curl. Given the vector field: \[ \mathbf{F} = x^2 y \hat{i} + xyz \hat{j} - x^2 y^2 \hat{k} \] ### 1. Irrotational (Curl of F = 0): A vector field is irrotational if the curl of the field is zero: \[ \text{curl}(\mathbf{F}) = \nabla \times \mathbf{F} \] To calculate the curl, we use the following formula: \[ \nabla \times \mathbf{F} = \left( \frac{\partial F_k}{\partial y} - \frac{\partial F_j}{\partial z} \right) \hat{i} - \left( \frac{\partial F_k}{\partial x} - \frac{\partial F_i}{\partial z} \right) \hat{j} + \left( \frac{\partial F_j}{\partial x} - \frac{\partial F_i}{\partial y} \right)
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