Let φ be a scalar field, and u be a vector field. Which of the following identities is true for div(φu)?

div(φu) = φ div(u) + u · grad(φ)
div(φu) = φ div(u) + u × grad(φ)
div(φu) = φ grad(u) + u · grad(φ)
div(φu) = φ grad(u) + u × grad(φ)

Correct answer: div(φu) = φ div(u) + u · grad(φ)

Solution

The correct option is true because it follows the product rule for divergence, which states that the divergence of a product of a scalar field and a vector field can be expressed as the scalar field times the divergence of the vector field plus the vector field dotted with the gradient of the scalar field.

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