Advertisements
Advertisements
प्रश्न
If `"u"(x , y) = (x^2 + y^2)/sqrt(x + y)`, prove that `x (del"v")/(delx) + y (del"u")/(dely) = 3/2 "u"`
उत्तर
`"u"(x , y) = (x^2 + y^2)/sqrt(x + y)`
`"u"(lambdax , lambday) = (lambda^2x^2 + lambda^2y^2)/sqrt(lambdax + lambday)`
= `(lambda^2(x^2 + y^2))/(lambda^(1/2) sqrt(x + y))`
= `(lambda^(1/2) (x^2 + y^2))/sqrt(x + y)`
u is a homogeneous function of degree `3/2`
By Euler's Theorem,
`x (del"v")/(delx) + y (del"u")/(dely) = 3/2 "u"`
APPEARS IN
संबंधित प्रश्न
If w(x, y) = x3 – 3xy + 2y2, x, y ∈ R, find the linear approximation for w at (1, –1)
If u(x, y) = x2y + 3xy4, x = et and y = sin t, find `"du"/"dt"` and evaluate if at t = 0
Let u(x, y, z) = xy2z3 x = sin t, y = cos t, z = 1 + e2t, Find `"du"/"dt"`
If w(x, y, z) = x2 + y2 + z2, x = et, y = et sin t and z = et cos t, find `("d"w)/"dt"`
Let U(x, y, z) = xyz, x = e–t, y = e–t cos t, z – sin t, t ∈ R, find `"dU"/"dt"`
Let w(x, y) = 6x3 – 3xy + 2y2, x = es, y = cos s, s ∈ R. Find `("d"w)/"ds"` and evaluate at s = 0
Let U(x, y) = ex sin y where x = st2, y = s2t, s, t ∈ R. Find `(del"U")/(del"s"), (del"u")/(del"t")` and evaluate them at s = t = 1
Let z(x, y) = x3 – 3x2y3 where x = set, y = se–t, s, t ∈ R. Find `(delz)/(del"s")` and `(delz)/(delt)`
W(x, y, z) = xy + yz + zx, x = u – v, y = uv, z = u + v, u, v ∈ R. Find `(del"W")/(del"u"), (del"W")/(del"v")` and evaluate them at `(1/2, 1)`
In the following, determine whether the following function is homogeneous or not. If it is so, find the degree.
f(x, y) = x2y + 6x3 + 7
In the following, determine whether the following function is homogeneous or not. If it is so, find the degree.
h(x, y) = `(6x^3y^2 - piy^5 + 9x^4y)/(2020x^2 + 2019y^2)`
In the following, determine whether the following function is homogeneous or not. If it is so, find the degree.
g(x, y, z) = `sqrt(3x^2+ 5y^2+z^2)/(4x + 7y)`
Prove that g(x, y) = `x log(y/x)` is homogeneous What is the degree? Verify Eulers Theorem for g
If v(x, y) = `log((x^2 + y^2)/(x + y))`, prove that `x (del"v")/(delx) + y (del"u")/(dely) = 1`
If w(x, y, z) = `log((5x^3y^4 + 7y^2xz^4 - 75y^3zz^4)/(x^2 + y^2))`, find `x (del"w")/(delx) + y (del"w")/(dely) + z (del"w")/(delz)`
Choose the correct alternative:
If v(x, y) = log(ex + ey), then `(del"v")/(delx) + (del"u")/(dely)` is equal to
Choose the correct alternative:
If w(x, y) = xy, x > 0, then `(del"w")/(delx)` is equal to
Choose the correct alternative:
If f(x, y) = exy, then `(del^2"f")/(delxdely)` is equal to
Choose the correct alternative:
f u(x, y) = x2 + 3xy + y – 2019, then `(delu)/(delx) "|"_(((4 , - 5)))` is equal to
Choose the correct alternative:
If w(x, y, z) = x2(y – z) + y2(z – x)+ z2(x – y) then `(del"w")/(delz) + (del"w")/(dely) + (del"w")/(delz)` is
