Advertisements
Advertisements
Question
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)`
Solution
`(del"W")/(delx) = y + z, (delx)/(del"u") = 1, (delx)/(del"v") = - 1`
`(del"W")/(dely) = x + z, (dely)/(delu) = "v" = (dely)/(delv) = "u"`
`(del"W")/(del"z") = y + x, (delz)/(del"u") = 1, (delz)/(delv) = 1`
`(del"W")/(delu) = (del"W")/(delx) (delx)/(del"u") + (del"W")/(dely) (dely)/(del"u") + (del"W")/(delz) (delz)/(del"u")`
= (y + z) × 1 + (x + z) × v + (y + x) × 1
= uv + u + v + (u – v + u + v) v+ (uv + u – v)
= uv + u + v + uv + uv + uv + u – v
= 4 uv + 2u
`(delw)/(delu) (1/2, 1) = 4 xx 1/2 xx 1 + 2 xx 1/2`
= 2 + 1
= 3
`(del"w")/(delv) = (del"W")/(delx) (delx)/(del"v") + (del"W")/(dely) (dely)/(del"v") + (del"W")/(delz) (delz)/(del"v")`
= (y + z) (– 1) + (x + z) u + (y + x) × 1
= – y – z + xu + zu + y + x
= –u – v + (u – v) u + (u + v) u + u – v
= – u – v + u2 – vu + u2 + vu + u – v
= 2u2 – 2v
`(del"W")/(delv) (1/2, 1) = 2 xx 1/4 - 2 xx 1`
= `1/2 - 2`
= `- 3/2`
APPEARS IN
RELATED QUESTIONS
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 z(x, y) = x tan–1(xy), x = t², y = s et, s, t ∈ R. Find `(delz)/(del"s")` and `(delz)/(del"t")` 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)`
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.
U(x, y, z) = `xy + sin((y^2 - 2z^2)/(xy))`
Prove that g(x, y) = `x log(y/x)` is homogeneous What is the degree? Verify Eulers Theorem for g
If `"u"(x , y) = (x^2 + y^2)/sqrt(x + y)`, prove that `x (del"v")/(delx) + y (del"u")/(dely) = 3/2 "u"`
If v(x, y) = `log((x^2 + y^2)/(x + y))`, prove that `x (del"v")/(delx) + y (del"u")/(dely) = 1`
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 f(x, y) = exy, then `(del^2"f")/(delxdely)` 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
