熹華 asked in 科學數學 · 1 decade ago

工數問題~~~10 點

show that div(curl v)=0

show that div(curl v)=0

show that div(curl v)=0

show that div(curl v)=0

show that div(curl v)=0

show that div(curl v)=0

show that div(curl v)=0

show that div(curl v)=0

1 Answer

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  • 1 decade ago
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    let V = (V1, V2, V3) be a vector field that is continuously differentiable

    i   j   k

    ∂/∂x ∂/∂y ∂/∂z

    V1  V2  V3

    this is a 3x3 matrix denoted it by M

    take det(M)

    then curl(V) = det(M)

    = i*(∂V3/∂y - ∂V2/∂z) + j*(∂V1/∂z - ∂V3/∂x) + k*(∂V2/∂x - ∂V1/∂y)

    and div(curl(V)) = ∂/∂x(∂V3/∂y - ∂V2/∂z) + ∂/∂y(∂V1/∂z - ∂V3/∂x) + ∂/∂z(∂V2/∂x - ∂V1/∂y)

    = (∂^2V3/∂y∂x - ∂^2V2/∂z∂x) + (∂^2V1/∂z∂y - ∂^2V3/∂x∂y) + (∂^2V2/∂x∂z - ∂^2V1/∂y∂z) ≡ 0.........Ans

    Source(s): me
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