Functions of Several Variables. Their Derivatives and Differentials

Show that

Show that

while the quantity

nevertheless has no meaning.

Show that if $x \to \infty$ and $y \to \infty$ in such a way that the ratio $\frac{y}{x}$ has any given constant value, then $\frac{1}{1+(x^{2}-y)} \to 0$. At the same time, this quantity may fail to tend to any limit if the ratio $\frac{y}{x}$ is not constant.

Show that $f(x)$, defined by the equality

equals one if $x$ is a rational number, and zero if $x$ is an irrational number; $m$ and $n$ are natural numbers.

Show that in any neighborhood of the point $(0,0)$ the function $f(x,y)$, defined by the equalities

takes every value in the interval $(-1,1)$. Here, by a neighborhood of the point $(0,0)$ is meant any region (a disk, a rectangle, etc.) containing the point $(0,0)$ in its interior.

A function is defined by the equalities:

Show that this function is continuous with respect to each of the variables $x$ and $y$ separately, without being, in a neighborhood of the point $(0,0)$, a continuous function of both variables under their joint variation.

Show that the function, defined for $x^{2}+y^{2} > 0$ by the equality $f(x,y) = e^{x/(x^{2}+y^{2})}$, takes every positive value in any region containing the point $(0,0)$ in its interior.

Find the partial derivatives of the function $u = x^{3}+y^{3}-3xy$.

Find the partial derivatives of the function $u = 2x^{3}-3x^{2}y^{2}+3y^{3}$.

Find the partial derivatives of the function $u = xy+xz+yz$.

Find the partial derivatives of the function $u = x^{y}$.

Find the partial derivatives of the function $u = x^{3}+yz^{2}+3xy-x+z$.

Find the partial derivatives of the function $u = e^{xy}$.

Find the partial derivatives of the function $u = \operatorname {arctg} \frac{x}{y}$.

Find the partial derivatives of the function $u = xy-\frac{3}{x}+\frac{5}{y}$.

Find the partial derivatives of the function $u = x^{2}+y^{2}+z^{2}+xy+xz+yz$.

Find the partial derivatives of the function $u = (xy)^{z}$.

Find the partial derivatives of the function $u = z^{xy}$.

Find the total differential of the function $u = \sin (x^{2}+y^{2})$.

Find the total differential of the function $u = \operatorname {arctg} \frac{x}{y}$.

Find the total differential of the function $u = \ln \operatorname {tg} \frac{x}{y}$.

Find the total differential of the function $u = \sqrt{x^{2}+y^{2}+z^{2}}$.

Find the total differential of the function $u = \ln (x+y+z)$.

Find the total differential of the function $u = x^{y}$.

Verify Euler's theorem on an example by direct computation of the derivatives: $u = \sqrt{x^{2}+y^{2}+z^{2}} \ln \frac{x}{y}$.

Verify Euler's theorem by direct computation of the derivatives: $u = \frac{x}{y}e^{x/z}$.

Verify Euler's theorem by direct computation of the derivatives: $u = \sin \frac{x+y+z}{\sqrt{x^{2}+y^{2}+z^{2}}}$.

Verify Euler's theorem by direct computation of the derivatives: $u = \frac{x}{x^{2}+y^{2}+z^{2}}$.

Verify Euler's theorem by direct computation of the derivatives: $u = \sqrt{x^{2}+y^{2}+z^{2}}$.

Verify Euler's theorem by direct computation of the derivatives: $u = \operatorname {arctg} \frac{x}{y}$.

Find the derivative of the function $f = 4x^{3}-2xy+3y^{3}$ at the point $(1,2)$ in the direction $\frac{\pi }{3}$ to the $Ox$ axis.

Find the derivative of the function $f = 5x+10x^{2}y+y^{5}$ at the point $(1,2)$ in the direction toward the point $(5,-1)$.

Find the derivative of the function $f = x^{2}+y^{2}$ at the point $(1,2)$ in the direction of the line $3x+4y = 11$:

Write the expression for the derivative of the function $f(x,y)$:

Find the derivative of the function $f = x^{3}+2xy^{2}+3yz^{2}$ at the point $(3,3,1)$ in the direction of the vector $2\overrightarrow {i}+2\overrightarrow {j}+\overrightarrow {k}$.

Find the derivative of the function $F = \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}$ in the direction of the outward normal to the surface $F = \mathrm{const}$.

Find $\frac{\partial^{2} u}{\partial x^{2}}$, $\frac{\partial^{2} u}{\partial x \partial y}$, $\frac{\partial^{2} u}{\partial y^{2}}$; $u = \frac{1}{2}\ln (x^{2}+y^{2})$.

Find $\frac{\partial^{2} u}{\partial x^{2}}$, $\frac{\partial^{2} u}{\partial x \partial y}$, $\frac{\partial^{2} u}{\partial y^{2}}$; $u = \operatorname {arctg} \frac{x+y}{1-xy}$.

Find $\frac{\partial^{2} u}{\partial x^{2}}$, $\frac{\partial^{2} u}{\partial x \partial y}$, $\frac{\partial^{2} u}{\partial y^{2}}$; $u = x\sin (x+y)+y\cos (x+y)$.

Find $\frac{\partial^{2} u}{\partial x \partial y}$, if $u = x^{y}$.

Find $\frac{\partial^{3} u}{\partial x^{2} \partial y}$, if $u = \ln (x+y)$.

Find $\frac{\partial^{4} u}{\partial x^{2} \partial y^{2}}$, if $u = \ln (x^{2}+y^{2})$.

Find $\frac{\partial^{2} u}{\partial x^{2}}$, $\frac{\partial^{2} u}{\partial y^{2}}$, $\frac{\partial^{2} u}{\partial z^{2}}$, if $u = \sqrt{x^{2}+y^{2}+z^{2}}$.

Find $\frac{\partial^{3} u}{\partial x \partial y \partial z}$, if $u = e^{xyz}$.

Find $d^{2}u$, if $u = x^{2}-xy+2y^{2}+3x-5y+7$.

Find $d^{2}u$, if $u = x^{2}y^{2}$.

Find $d^{2}u$, if $u = e^{xy}$.

Find $d^{2}u$, if $u = \frac{1}{2}\ln (x^{2}+y^{2})$.

Find $d^{2}u$, if $u = \sin (x+y+z)$.

Find $d^{4}u$, if $u = x^{4}+4x^{3}y+2xy^{2}z-3xyz^{2}+z^{4}$.

Find $d^{4}u$, if $u = x^{4}-3x^{2}y^{2}+y^{4}+5xyz-x^{3}-y^{3}+x^{2}-xy+y^{2}+2x-5y$.

Find $d^{4}u$, if $u = x^{4}+3x^{3}y+z^{4}-x^{2}y+z^{3}$.

Find $d^{3}u$, if $u = xyz$.

Find $d^{4}u$, if $u = \ln (2x+3y-z)$.

Find the partial derivatives by first computing the total differentials of the needed order: $u = \sin (2x+y)$; find $\frac{\partial^{3} u}{\partial x^{3}}$ and $\frac{\partial^{3} u}{\partial x \partial y^{2}}$.

Find the partial derivatives by first computing the total differentials of the needed order: $u = \cos (x+y)$; find $\frac{\partial^{4} u}{\partial x^{2} \partial y^{2}}$.

Find the partial derivatives by first computing the total differentials of the needed order: $u = \ln (ax+by+cz)$; find $\frac{\partial^{4} u}{\partial x^{4}}$ and $\frac{\partial^{4} u}{\partial x^{2} \partial y^{2}}$.

Find the partial derivatives by first computing the total differentials of the needed order: $u = e^{x+2y+3z}$; find $\frac{\partial^{3} u}{\partial x^{3}}$ and $\frac{\partial^{3} u}{\partial x \partial y \partial z}$.

Find the total differentials of the first two orders for the composite function $u = \varphi (t)$; $t = xy$.

Find the total differentials of the first two orders for the composite function $u = \varphi (t)$; $t = x^{2}+y^{2}$.

Find the total differentials of the first two orders for the composite function $u = \varphi (t)$; $t = x^{2}+y^{2}+z^{2}$.

Find the total differentials of the first two orders for the composite function $u = \varphi (\xi ,\eta )$; $\xi = ax+by+cz$, $\eta = a_{1}x+b_{1}y+c_{1}z$.

Find the total differentials of the first two orders for the composite function $u = \varphi (\xi ,\eta ,\zeta )$; $\xi = ax$, $\eta = by$, $\zeta = cz$.

Find the derivatives of the first two orders of the function $u = \varphi (\xi ,\eta )$; $\xi = x+y$, $\eta = x-y$.

Find the derivatives of the first two orders of the function $u = \varphi (\xi ,\eta )$; $\xi = x^{2}+y^{2}$, $\eta = xy$.

Find the derivatives of order $n$ of $u = \varphi (t)$, where $t = ax+by+cz$.

Find $\frac{\partial^{n} u}{\partial x^{n}}$, if $u = \varphi (\xi ,\eta ,\zeta )$, $\xi = ax$, $\eta = by$, $\zeta = cz$.

Show that if $x^{2} = \eta \xi$, $y^{2} = \zeta \xi$, $z^{2} = \xi \eta$, then

Setting $x = \rho \cos \varphi$, $y = \rho \sin \varphi$, find

Setting $x = \rho \sin \varphi \cos \psi$, $y = \rho \sin \varphi \sin \psi$, $z = \rho \cos \varphi$, compute the value of the functional determinant (Jacobian)

Setting $r^{2} = a^{2}+\rho^{2}-2a\rho \cos \varphi$, find $\frac{D(\rho ,\varphi ,\psi )}{D(\rho ,r,\psi )}$.

Setting $x = \xi \eta \zeta$, $y = \xi \eta -\xi \eta \zeta$, $z = \eta -\xi \eta$, find the Jacobian $\frac{D(x,y,z)}{D(\xi ,\eta ,\zeta )}$.

Prove that for $x = \cos \varphi$, $y = \sin \varphi \cos \theta$, $z = \sin \varphi \sin \theta \cos \psi$ the Jacobian equals $-\sin^{3}\varphi \sin^{2}\theta \sin \psi$.

Prove that for

where $r^{2} = x_{1}^{2}+x_{2}^{2}+x_{3}^{2}$; the equality

holds.

Prove that

Here $v = v(x^{1},x^{2},\ldots ,x_{n};y^{1},y_{2},\ldots ,y_{n})$ is a function with continuous second derivatives.

Prove that for homogeneous functions $\varphi (x,y,z)$ of degree $n$ the equality

holds.

Verify that $\frac{\partial u}{\partial t} = a^{2}\frac{\partial^{2} u}{\partial x^{2}}$, if $u = \frac{1}{\sqrt{t}} e^{-x^{2}/(4a^{2}t)}$.

Verify that $x\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y}+z\frac{\partial u}{\partial z} = 0$, if $u = \left(\frac{x}{y}\right)^{z/y}$.

Compute $\frac{\partial^{2} u}{\partial x^{2}}-2\frac{\partial^{2} u}{\partial x \partial y}+\frac{\partial^{2} u}{\partial y^{2}}$, if $u = \varphi (x+y)$.

Compute $x^{2}\frac{\partial^{2} u}{\partial x^{2}}-2xy\frac{\partial^{2} u}{\partial x \partial y}+y^{2}\frac{\partial^{2} u}{\partial y^{2}}$, if $u = f\left(\sqrt{x^{2}+y^{2}}\right)$.

Compute $\frac{\partial^{2} u}{\partial x^{2}} \cdot \frac{\partial^{2} u}{\partial y^{2}}-\left(\frac{\partial^{2} u}{\partial x \partial y}\right)^{2}$, if $u = \varphi (xy)$.

The function called the potential of the ball $x^{2}+y^{2}+z^{2} = a^{2}$ is defined by the equalities:

Verify that $u$ and its first derivatives are continuous for any $x$, $y$, and $z$, and that the Laplacian

depending on whether the point $M(x,y,z)$ lies outside the ball $x^{2}+y^{2}+z^{2} = a^{2}$ or inside it.

Verify the equality

Verify the equality $\frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}+\frac{\partial u}{\partial z} = \frac{3}{x+y+z}$; $u = \ln (x^{3}+y^{3}+z^{3}-3xyz)$.

Verify the equality $\frac{1}{x}\frac{\partial z}{\partial x}+\frac{1}{y}\frac{\partial z}{\partial y} = \frac{z}{y^{2}}$; $z = y\varphi (x^{2}-y^{2})$.

Verify the equality $x\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y} = xy+u$; $u = xy+x\varphi \left(\frac{y}{x}\right)$.

Verify the equality $\frac{\partial^{2} \ln r}{\partial x^{2}}+\frac{\partial^{2} \ln r}{\partial y^{2}}+\frac{\partial^{2} \ln r}{\partial z^{2}} = \frac{1}{r^{2}}$; $r = \sqrt{x^{2}+y^{2}+z^{2}}$.

Verify the equality $\frac{\partial^{2} u}{\partial x^{2}}-\frac{\partial^{2} u}{\partial y^{2}}-2\alpha \frac{\partial u}{\partial y} = \alpha^{2}u$; $u = e^{-\alpha x}\varphi (x-y)$.

Verify the equality $\frac{\partial^{2} u}{\partial x^{2}}-\frac{\partial^{2} u}{\partial y^{2}} = -2\varphi ''$; $u = \varphi (y-x)-x\varphi '(y-x)$.

Verify the equality $(x^{2}-y^{2})\frac{\partial z}{\partial x}+xy\frac{\partial z}{\partial y} = xyz$; $z = e^{y}\varphi \left(ye^{x^{2}/(2y^{2})}\right)$.

Verify the equality $x\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y} = u-x^{2}-y^{2}$; $u = x\varphi \left(\frac{y}{x}\right)-x^{2}-y^{2}$.

Verify the equality $\frac{\partial^{2} u}{\partial t^{2}} = a^{2}\frac{\partial^{2} u}{\partial x^{2}}$; $u = \varphi (x+at)+\psi (x-at)$.

Verify the equality $\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}} = 0$; $u = \ln (x^{2}+y^{2})$.

Verify the equality $\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}+\frac{\partial^{2} u}{\partial z^{2}} = 0$; $u = \frac{1}{\sqrt{x^{2}+y^{2}+z^{2}}}$.

Verify the equality $\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}+\frac{\partial^{2} u}{\partial z^{2}} = a^{2}u$; $u = \frac{1}{r}(Ae^{-ar}+Be^{ar})$, where $r = \sqrt{x^{2}+y^{2}+z^{2}}$.

Verify the equality $\frac{\partial^{2} \ln z}{\partial x \partial y} = 2z$; $z = \frac{\varphi '(x)\psi '(x)}{(\varphi (x)+\psi (y))^{2}}$.

Verify the equality $x^{2}\frac{\partial^{2} u}{\partial x^{2}}+2xy\frac{\partial^{2} u}{\partial x \partial y}+y^{2}\frac{\partial^{2} u}{\partial y^{2}} = 0$, if $u = x\varphi \left(\frac{y}{x}\right)$.

Verify the equality $x^{2}\frac{\partial^{2} u}{\partial x^{2}}+2xy\frac{\partial^{2} u}{\partial x \partial y}+y^{2}\frac{\partial^{2} u}{\partial y^{2}}+x\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y} = n^{2}u$, where $u = x^{n}\varphi \left(\frac{y}{x}\right)+y^{-n}\psi \left(\frac{y}{x}\right)$.

Verify the equality $x^{2}\frac{\partial^{2} u}{\partial x^{2}}+2xy\frac{\partial^{2} u}{\partial x \partial y}+y^{2}\frac{\partial^{2} u}{\partial y^{2}} = n(n-1)u$, where $u = x^{n}\varphi \left(\frac{y}{x}\right)+x^{1-n}\psi \left(\frac{y}{x}\right)$.

Verify the equality $\frac{\partial u}{\partial x} \cdot \frac{\partial^{2} u}{\partial x \partial y} = \frac{\partial u}{\partial y} \cdot \frac{\partial^{2} u}{\partial x^{2}}$, if $u = f\left(x+\varphi (y)\right)$.

Verify the equality $\frac{\partial^{2} u}{\partial x^{2}}-2\frac{\partial^{2} u}{\partial x \partial y}+\frac{\partial^{2} u}{\partial y^{2}} = 0$, if $u = x\varphi (x+y)+y\psi (x+y)$.

Verify the equality

where $u = \varphi (ay+bx)\psi (bx-ay)$.