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show that the vector F = (y+z)i +(z+x)j +(x+y)k is solenoidal

topic : VECTOR CALCULUS

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Given,

F = (y+z)i + (z+x)j + (x+y)k


To Prove :- F is solenoidal


Proof :-

F is solenoidal if ∇.F = 0

$\therefore\ F = (i\frac{∂}{∂x}\ +\ j\frac{∂}{∂y}\ +\ k\frac{∂}{∂z}). [(y+z)i + (z+x)j + (x+y)k]$

$\implies F =\ \frac{∂}{∂x}(y+z)\ +\ \frac{∂}{∂y}(z+x)\ +\ \frac{∂}{∂z}(x+y)$

$\implies F =\ 0$

$So,\ \therefore\ F\ is\ solenoidal.$

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