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# 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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- MichaelLv 41 decade agoFavorite Answer
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

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