If w(x, y, z) = log(5x3y4+7y2xz4-75y3zz4x2+y2), find wwwx∂w∂x+y∂w∂y+z∂w∂z - Mathematics

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)`

बेरीज

w(x, y, z) = `log((5x^3y^4 + 7y^2xz^4 - 75y^3zz^4)/(x^2 + y^2))`

Convert into exponential function

e^w = `((5x^3y^4 + 7y^2xz^4 75y^3z^4)/(x^2 + y^2))` = f(x, y)

`"f"(lambdax, lambday) = (5lambda^3x^3lambda^4y^4 + 7lambda^2y^lambdaxlambda^4z^4 - 75lambda^3y^3lambda^4z^4)/(lambda^2x^2 + lambda^2y^2)`

= `(lambda^7(5x^3y^4 + 7y^2xz^4 - 75y^3y^4))/(lambda^2(x^2 + y^2))`

f is a homogeneous function of degree 5

By Euler's Theorem,

`x (del"f")/(delx) + y (del"f")/(dely) + z (del"f")/(delz)` = 5f

`x del/(delx) + "e"^"w" + y del/(dely) "e"^"w" + z del/(delz) "e"^"w"` = 5e^w

`"e"^"w" x (del"w")/(delx) + "e"^"w" y (del"w")/(dely) + "e"^"w" z (del"w")/(delz)` = 5e^w

Divided by e^w

`x (del"w")/(delx) + y (del"w")/(dely) + z (del"w")/(delz) = (5"e"^"w")/"e"^"w"`

`x (del"w")/(delx) + y (del"w")/(dely) + z (del"w")/(delz)` = 5

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Linear Approximation and Differential of a Function of Several Variables

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पाठ 8: Differentials and Partial Derivatives - Exercise 8.7 [पृष्ठ ८६]

पाठ 8 Differentials and Partial Derivatives
Exercise 8.7 | Q 6 | पृष्ठ ८६

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 = e^t 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 + e^2t, Find `"du"/"dt"`

If w(x, y, z) = x² + y² + z², x = e^t, y = e^t sin t and z = e^t 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 = e^s, y = cos s, s ∈ R. Find `("d"w)/"ds"` and evaluate at s = 0

Let U(x, y) = e^x 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 = se^t, 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.

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 f(x, y) = x³ – 2x²y + 3xy² + y³ is homogeneous. What is the degree? Verify Euler’s Theorem for f

If v(x, y) = `log((x^2 + y^2)/(x + y))`, prove that `x (del"v")/(delx) + y (del"u")/(dely) = 1`