Advertisements
Advertisements
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
APPEARS IN
संबंधित प्रश्न
If z = (ax + b) (cy + d), then find `(∂z)/(∂x)` and `(∂z)/(∂y)`.
Let u = x cos y + y cos x. Verify `(del^2"u")/(delxdely) = (del^"u")/(del"y"del"x")`
Let u = x2y3 cos`(x/y)`. By using Euler’s theorem show that `x*(del"u")/(delx) + y * (del"u")/(dely)`
Let u = `log (x^4 - y^4)/(x - y).` Using Euler’s theorem show that `x (del"u")/(del"x") + y(del"u")/(del"y")` = 3.
If u = x3 + 3xy2 + y3 then `(del^2"u")/(del "y" del x)`is:
If u = `e^(x^2)` then `(del"u")/(delx)` is equal to:
Find the partial dervatives of the following functions at indicated points.
g(x, y) = 3x2 + y2 + 5x + 2, (2, – 5)
Find the partial derivatives of the following functions at the indicated points.
`"G"(x, y) = "e"^(x + 3y) log(x^2 + y^2), (- 1, 1)`
For the following functions find the fx, and fy and show that fxy = fyx
f(x, y) = `cos(x^2 - 3xy)`
If U(x, y, z) = `(x^2 + y^2)/(xy) + 3z^2y`, find `(del"U")/(delx), (del"U")/(dely)` and `(del"U")/(del"z)`
If U(x, y, z) = `log(x^3 + y^3 + z^3)`, find `(del"U")/(delx) + (del"U")/(dely) + (del"U")/(del"z)`
For the following functions find the gxy, gxx, gyy and gyx
g(x, y) = xey + 3x2y
For the following functions find the gxy, gxx, gyy and gyx
g(x, y) = log(5x + 3y)
For the following functions find the gxy, gxx, gyy and gyx
g(x, y) = x2 + 3xy – 7y + cos(5x)
If V(x, y) = ex (x cosy – y siny), then Prove that `(del^2"V")/(delx^2) + (del^2"V")/(dely^2)` = 0
If v(x, y, z) = x3 + y3 + z3 + 3xyz, Show that `(del^2"v")/(delydelz) = (del^2"v")/(delzdely)`
Let V (x, y, z) = xy + yz + zx, x, y, z ∈ R. Find the differential dV
