Advertisements
Advertisements
Question
If V(x, y) = ex (x cosy – y siny), then Prove that `(del^2"V")/(delx^2) + (del^2"V")/(dely^2)` = 0
Advertisements
Solution
V(x, y) = ex (x cosy – y siny)
`(del"V")/(delx)` = ex (x cosy – y siny) + ex cosy
`(del^2"V")/(delx^2)` = ex (x cosy – y siny) + ex cosy + ex cosy
`(del"V")/(dely)` = – xex (– siny) – ex (y cosy + siny)
`(del^2"V")/(dely^2)` = – xex cosy – ex (y(– siny) + cosy + cosy)
= – ex (x cosy – y siny) – 2ex cosy
`(del^2"V")/(delx^2) + (del^2"V")/(dely^2)` = ex [x cosy – y siny] + 2ex cosy
– ex (x cosy – y siny) – 2ex cosy
0
APPEARS IN
RELATED QUESTIONS
If z = (ax + b) (cy + d), then find `(∂z)/(∂x)` and `(∂z)/(∂y)`.
Let u = x2y3 cos`(x/y)`. By using Euler’s theorem show that `x*(del"u")/(delx) + y * (del"u")/(dely)`
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) = `(3x)/(y + sinx)`
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) = `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)
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
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)
Let z(x, y) = x2y + 3xy4, x, y ∈ R, Find the linear approximation for z at (2, –1)
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
Choose the correct alternative:
If g(x, y) = 3x2 – 5y + 2y2, x(t) = et and y(t) = cos t then `"dg"/"dt"` is equal to
