Math Problems Vault

Problem 2.2.13 from de Souza P.N., Silva J. "Berkeley Problems in Mathematics" [deSi04]More problems in Section 2.2 "Differential Calculus"

Let the function $u$ on $\mathbb {R}^{2}$ be harmonic, not identically $0$ , and homogeneous of degree $d$ , where $d > 0$ .

To say that $u$ is homogeneous of degree $d$ means that $u(tx,ty) = t^d u(x,y)$ for $t > 0$ .

Prove that $d$ is an integer.

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Problem 3231 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 6.2 "Partial Derivatives. Differentials of Functions of Several Variables"

(a)(b)(c)

Let $u=f(x,y,z)$ be a homogeneous function of degree $n$ . Verify Euler's theorem on homogeneous functions for each of the following: if the homogeneous function $f(x,y,z)$ of degree $n$ has continuous partial derivatives with respect to all variables in a domain $A$ , then the following identity holds

xf'_{x}(x,y,z)+yf'_{y}(x,y,z)+zf'_{z}(x,y,z)=nf(x,y,z) .

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(a)

$u=(x-2y+3z)^{2}$ ;

(b)

$u=\dfrac {x}{\sqrt{x^{2}+y^{2}+z^{2}}}$ ;

(c)

$u=\left(\dfrac {x}{y}\right)^{\frac{z}{y}}$ .

Problem 3232 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 6.2 "Partial Derivatives. Differentials of Functions of Several Variables"

Prove that if the differentiable function $u=f(x,y,z)$ satisfies the equation

x\dfrac {\partial u}{\partial x}+y\dfrac {\partial u}{\partial y}+z\dfrac {\partial u}{\partial z}=nu ,

then it is a homogeneous function of degree $n$ .

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Problem 3233 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 6.2 "Partial Derivatives. Differentials of Functions of Several Variables"

Prove that if $f(x,y,z)$ is a differentiable homogeneous function of degree $n$ , then its partial derivatives $f'_{x}(x,y,z), f'_{y}(x,y,z), f'_{z}(x,y,z)$ are homogeneous functions of degree $(n-1)$ .

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Problem 3279 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 6.2 "Partial Derivatives. Differentials of Functions of Several Variables"

Let $P_{n}(x,y,z)$ be a homogeneous polynomial of degree $n$ . Prove that

d^{n}P_{n}(x,y,z)=n! P_{n}(dx,dy,dz) .

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Problem 3.2.34 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Функции многих переменных" [ККЧ+03]More problems in Subsection 3.2.2 "Problems"

Find the degree of homogeneity or positive homogeneity of the function given by the formula:

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(1)

$u = \sqrt{x^{2}+y^{2}} / x, x > 0$ ;

(2)

$u = \sqrt{x^{2}+y^{2}}$ ;

(3)

$u = x^{\sqrt{2}} \sin \frac{y}{x}+y^{\sqrt{2}} \cos \frac{x}{y}$ ;

(4)

$u = \sqrt[6]{x^{6} z^{2}+2 x^{3} y^{4} z+x y^{2} z^{5}}$ ;

(5)

$u = \sqrt{x^{2}+y^{2}+z^{2}-t^{2}}$ ;

(6)

$u = \frac{x y+z t}{x y z+y z t}$ ;

(7)

$u = \sum_{i = 1}^{n-1} x_{i} x_{i+1}$ ;

(8)

$u = \sum_{i, j = 1}^{n} x_{i} x_{j} / \sum_{i = 1}^{n} x_{i}^{2}$ ;

(9)

$u = \sum_{i = 1}^{n-1} x_{i}^{\pi }\left(\ln x_{i+1}-\ln x_{i}\right)$ ;

(10)

$u = \left|\prod_{i = 1}^{n} x_{i}\right|^{\gamma }, \gamma \in R$ .

Problem 3.2.35 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Функции многих переменных" [ККЧ+03]More problems in Subsection 3.2.2 "Problems"

Prove that every function $f\left(x_{1} ; x_{2} ; \ldots ; x_{n}\right), \quad x_{n} \neq 0$ , homogeneous of degree $\alpha$

f\left(x_{1} ; x_{2} ; \ldots ; x_{n}\right), \quad x_{n} \neq 0

can be represented in the form

f\left(x_{1} ; x_{2} ; \ldots ; x_{n}\right) = x_{n}^{\alpha } F\left(\frac{x_{1}}{x_{n}} ; \frac{x_{2}}{x_{n}} ; \ldots ; \frac{x_{n-1}}{x_{n}}\right).

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Problem 3.2.36 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Функции многих переменных" [ККЧ+03]More problems in Subsection 3.2.2 "Problems"

(1)(2)

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(1)

Let $G$ be the domain consisting of all points of the plane, except the ray $x = 2, y \geqslant 0$ . Prove that the function

u = \begin{cases} y^{4} / x, & \text{ if } x > 2, y > 0, \\ y^{3} & \text{ at the remaining points of the domain } G \end{cases}

is locally homogeneous, but not homogeneous, in the domain $G$ .

(2)

Prove that in a convex domain every locally homogeneous function is homogeneous.

Problem 3.3.54 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Функции многих переменных" [ККЧ+03]More problems in Subsection 3.3.2 "Problems"

Let $f(x), x \in G \subset R^{n}$ , be a differentiable homogeneous function of degree $\alpha$ $(\S 2,(1))$ . Prove that the partial derivatives of the function $f$ are homogeneous functions of degree $\alpha -1$ .

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Problem 3.3.55 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Функции многих переменных" [ККЧ+03]More problems in Subsection 3.3.2 "Problems"

(1)(2)

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(1)

Prove that a function $f$ differentiable in the domain $G \subset R^{n}$ satisfies Euler's identity $\sum_{k = 1}^{n} x_{k} \frac{\partial f}{\partial x_{k}} = \alpha f$ in $G$ if and only if it is locally homogeneous of degree $\alpha$ in the domain $G$ (see § 2);

(2)

construct a function that satisfies Euler's identity in some domain but is not a homogeneous function in this domain.

Problem 3.3.74 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Функции многих переменных" [ККЧ+03]More problems in Subsection 3.3.2 "Problems"

Prove that if the equation $f\left(\frac{x_{1}}{x_{n}} ; \frac{x_{2}}{x_{n}} ; \ldots ; \frac{x_{n-1}}{x_{n}} ; \frac{z}{x_{n}^{\alpha }}\right) = 0$ , where $f\left(u_{1} ; u_{2} ; \ldots ; u_{n}\right)$ is a differentiable function and $\alpha$ is a constant, defines a differentiable function $z\left(x_{1} ; x_{2} ; \ldots ; x_{n}\right)$ , then it satisfies the equation

\sum _{k = 1}^{n} x_{k} \frac{\partial z}{\partial x_{k}} = \alpha z

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Homogeneous functions

References