In the following cases, determine whether the following function is homogeneous or not. If it is so, find the degree. g(x, y, z) = x2+5y2+z24x+y - Mathematics

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)`

योग

g(x, y, z) = `sqrt(3x^2+ 5y^2+z^2)/(4x + 7y)`

`"g"(lambdax, lambday, lambdaz) = sqrt(3lambda^2x^2 + 5lambda^2y^2 + lambda^2z^2)/(4lambdax + 7lambday)`

= `(lambdasqrt(3x^2 + 5y^2 + z^2))/(lambda(4x + 7y))`

= `lambda^circ "g"(x, y, z)`

Thus g is homogeneous with degree 0.

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Linear Approximation and Differential of a Function of Several Variables

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

अध्याय 8 Differentials and Partial Derivatives
Exercise 8.7 | Q 1. (iii) | पृष्ठ ८६

Let u(x, y, z) = xy²z³ x = sin t, y = cos t, z = 1 + e^2t, Find `"du"/"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 e^t, s, t ∈ R. Find `(delz)/(del"s")` and `(delz)/(del"t")` at s = t = 1

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)`

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

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 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)`