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प्रश्न
Let w(x, y, z) = `1/sqrt(x^2 + y^2 + z^2)` = 1, (x, y, z) ≠ (0, 0, 0), show that `(del^2w)/(delx^2) + (del^2w)/(dely^2) + (del^2w)/(delz^2)` = 0
उत्तर
w(x, y, z) = `1/sqrt(x^2 + y^2 + z^2)`
`(delw)/(delx) = (- 1/2(2x))/(x^2 + y^2 + z^2)^(3/2)`
= `(-x)/(x^2 + y^2 + z^2)^(3/2)`
= `(delw)/(dely) = (- y)/(x^2 + y^2 + z^2)^(3/2)`
`(del^2w)/(delx^2) = ((x^2 + y^2 + z^2)^(3/2) (- 1) + x 3/2 (x^2 + y^2 + z^2)^(1/*2) (2x))/[(x^2 + y^2 + z^2)^(3/2)]^2`
= `((x^2 + y^2 + z^2)^(1/2) [- x^2 y^2 - z^2 + 3x^2])/(x^2 + y^2 + z^2)^2`
= `(2x^2 - y^2 - z^2)/(x^2 + y^2 + z^2)^(5/2)` ......(1)
`(del^2w)/(dely^2) = (-x^2 + 2y^2 - z^2)/(x^2 + y^2 + z^2)^(5/2)` ........(2)
`(del^2w)/(delz^2) = (-x^2 - y^2 + 2z^2)/(x^2 + y^2 + z^2)^(5/2)` ........(3)
(1) + (2) + (3)
⇒ `(del^2w)/(delx^2) + (del^2w)/(dely^2) + (del^2w)/(delz^2) = 0/(x^2 + y^2 + z^2)^(5/2)`
`(del^2w)/(delx^2) + (del^2w)/(dely^2) + (del^2w)/(delz^2)` = 0
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