Advertisements
Advertisements
Question
If U(x, y, z) = `(x^2 + y^2)/(xy) + 3z^2y`, find `(del"U")/(delx), (del"U")/(dely)` and `(del"U")/(del"z)`
Advertisements
Solution
U(x, y, z) = `(x^2 + y^2)/(xy) + 3z^2y`
`(del"U")/(delx) = ((xy)(2x) - (x^2 + y^2)(y))/(xy)^2` + 0
= `(2x^2y - x^2y - y^3)/(xy)^2`
= `(x^2y - y^3)/(xy)^2`
= `(y(x^2 - y^2))/(x^2y^2)`
= `(x^2 - y^2)/(x^2y)`
`(del"U")/(dely) = ((xy)(2y) - (x^2 + y^2)(x))/(xy)^2 + 3z^2`
= `(2xy^2 - x^3 - y^2x)/(xy)^2 + 3z^2`
= `(xy^2 - x^3)/(xy)^2 + 3z^2`
= `(x(y^2 - x^2))/(x^2y^2)`
= `(y^2 - x^2)/(y^2x) + 3z^2`
`(del"U")/(delz)` = 0 + 6zy = 6zy
APPEARS IN
RELATED QUESTIONS
If u = exy, then show that `(del^2"u")/(delx^2) + (del^2"u")/(del"y"^2)` = u(x2 + y2).
Let u = x cos y + y cos x. Verify `(del^2"u")/(delxdely) = (del^"u")/(del"y"del"x")`
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.
Find the partial derivatives of the following functions at indicated points.
h(x, y, z) = x sin (xy) + z2x, `(2, pi/4, 1)`
For the following functions find the fx, and fy and show that fxy = fyx
f(x, y) = `(3x)/(y + sinx)`
For the following functions find the fx, and fy and show that fxy = fyx
f(x, y) = `tan^-1 (x/y)`
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)`
A from produces two types of calculates each week, x number of type A and y number of type B. The weekly revenue and cost functions = (in rupees) are R(x, y) = 80x + 90y + 0.04xy – 0.05x2 – 0.05y2 and C (x, y) = 8x + 6y + 2000 respectively. Find the profit function P(x, y)
A from produces two types of calculates each week, x number of type A and y number of type B. The weekly revenue and cost functions = (in rupees) are R(x, y) = 80x + 90y + 0.04xy – 0.05x2 – 0.05y2 and C(x, y) = 8x + 6y + 2000 respectively. Find `(del"P")/(delx)` (1200, 1800) and `(del"P")/(dely)` (1200, 1800) and interpret these results
If v(x, y) = `x^2 - xy + 1/4 y^2 + 7, x, y ∈ "R"`, find the differential dv
Let V (x, y, z) = xy + yz + zx, x, y, z ∈ R. Find the differential dV
