Chapter 6

Differential Calculus of Functions of Several Variables

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Section 6.1Limits and Continuity of Functions of Several Variables[84/89%]

Problem 3136

Determine and sketch the domain of existence of the following function: $u=x+\sqrt{y}$ .

Problem 3137

Determine and sketch the domain of existence of the following function: $u=\sqrt{1-x^{2}}+\sqrt{y^{2}-1}$ .

Problem 3138

Determine and sketch the domain of existence of the following function: $u=\sqrt{1-x^{2}-y^{2}}$ .

Problem 3139

Determine and sketch the domain of existence of the following function: $u=\dfrac {1}{\sqrt{x^{2}+y^{2}-1}}$ .

Problem 3140

Determine and sketch the domain of existence of the following function: $u=\sqrt{(x^{2}+y^{2}-1)(4-x^{2}-y^{2})}$ .

Problem 3141

Determine and sketch the domain of existence of the following function: $u=\sqrt{\dfrac {x^{2}+y^{2}-x}{2x-x^{2}-y^{2}}}$ .

Problem 3142

Determine and sketch the domain of existence of the following function: $u=\sqrt{1-(x^{2}+y)^{2}}$ .

Problem 3143

Determine and sketch the domain of existence of the following function: $u=\ln (-x-y)$ .

Problem 3144

Determine and sketch the domain of existence of the following function: $u=\arcsin \dfrac {y}{x}$ .

Problem 3145

Determine and sketch the domain of existence of the following function: $u=\arccos \dfrac {x}{x+y}$ .

Problem 3146

Determine and sketch the domain of existence of the following function: $u=\arcsin \dfrac {x}{y^{2}}+\arcsin (1-y)$ .

Problem 3147

Determine and sketch the domain of existence of the following function: $u=\sqrt{\sin (x^{2}+y^{2})}$ .

Problem 3148

Determine and sketch the domain of existence of the following function: $u=\arccos \dfrac {z}{\sqrt{x^{2}+y^{2}}}$ .

Problem 3149

Determine and sketch the domain of existence of the following function: $u=\ln (xyz)$ .

Problem 3150

Determine and sketch the domain of existence of the following function: $u=\ln (-1-x^{2}-y^{2}+z^{2})$ .

Problem 3151

Sketch the level curves of the following function: $z=x+y$ .

Problem 3152

Sketch the level curves of the following function: $z=x^{2}+y^{2}$ .

Problem 3153

Sketch the level curves of the following function: $z=x^{2}-y^{2}$ .

Problem 3154

Sketch the level curves of the following function: $z=(x+y)^{2}$ .

Problem 3155

Sketch the level curves of the following function: $z=\dfrac {y}{x}$ .

Problem 3156

Sketch the level curves of the following function: $z=\dfrac {1}{x^{2}+2y^{2}}$ .

Problem 3157

Sketch the level curves of the following function: $z=\sqrt{xy}$ .

Problem 3158

Sketch the level curves of the following function: $z=\left|x\right|+y$ .

Problem 3159

Problem 3159.1

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Problem 3159.1

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Problem 3159.2

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Problem 3159.2

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Problem 3159.3

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Problem 3159.3

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Problem 3160

Sketch the level curves of the following function: $z=e^{\frac{2x}{x^{2}+y^{2}}}$ .

Problem 3161

Sketch the level curves of the following function: $z=x^{y}$ $(x>0)$ .

Problem 3162

Sketch the level curves of the following function: $z=x^{y}e^{-x}$ $(x>0)$ .

Problem 3163

Sketch the level curves of the following function: $z=\ln \sqrt{\dfrac {(x-a)^{2}+y^{2}}{(x+a)^{2}+y^{2}}}$ $(a>0)$ .

Problem 3164

Sketch the level curves of the following function: $z=\arctan \dfrac {2ay}{x^{2}+y^{2}-a^{2}}$ $(a>0)$ .

Problem 3165

Sketch the level curves of the following function: $z=\sgn (\sin x \sin y)$ .

Problem 3166

Find the level surfaces of the following function: $u=x+y+z$ .

Problem 3167

Find the level surfaces of the following function: $u=x^{2}+y^{2}+z^{2}$ .

Problem 3168

Find the level surfaces of the following function: $u=x^{2}+y^{2}-z^{2}$ .

Problem 3169

Find the level surfaces of the following function: $u=(x+y)^{2}+z^{2}$ .

Problem 3170

Find the level surfaces of the following function: $u=\sgn \sin (x^{2}+y^{2}+z^{2})$ .

Problem 3171

Investigate the properties of the surface from its given equation: $z=f(y-ax)$ .

Problem 3172

Investigate the properties of the surface from its given equation: $z=f(\sqrt{x^{2}+y^{2}})$ .

Problem 3173

Investigate the properties of the surface from its given equation: $z=xf\left(\dfrac {y}{x}\right)$ .

Problem 3174

Investigate the properties of the surface from its given equation: $z=f\left(\dfrac {y}{x}\right)$ .

Problem 3175

Sketch the graph of the function

F(t)=f(\cos t, \sin t)

where

f(x,y)= \begin{cases} 1, & \text{if} \ y \geqslant x, \\ 0, & \text{if} \ y<x . \end{cases}

Problem 3176

f(x,y)=\dfrac {2xy}{x^{2}+y^{2}} ,

find $f\left(1, \dfrac {y}{x}\right)$ .

Problem 3177

f\left(\dfrac {y}{x}\right)=\dfrac {\sqrt{x^{2}+y^{2}}}{x} \quad (x>0) ,

find $f(x)$ .

Problem 3178

Let

z=\sqrt{y}+f(\sqrt{x}-1) .

If $z=x$ when $y=1$ , find the function $f$ and $z$ .

Problem 3179

Let

z=x+y+f(x-y) .

If $z=x^{2}$ when $y=0$ , find the function $f$ and $z$ .

Problem 3180

If $f\left(x+y, \dfrac {y}{x}\right)=x^{2}-y^{2}$ , find $f(x,y)$ .

Problem 3181

Prove that for the function

f(x,y)=\dfrac {x-y}{x+y}

we have

\lim _{x \to 0}\left\{ \lim _{y \to 0} f(x,y)\right\} =1 ; \quad \lim _{y \to 0}\left\{ \lim _{x \to 0} f(x,y)\right\} =-1 ,

so that $\lim \limits_{\substack {x \to 0 \\ y \to 0}} f(x,y)$ does not exist.

Problem 3182

Prove that for the function

f(x,y)=\dfrac {x^{2}y^{2}}{x^{2}y^{2}+(x-y)^{2}}

we have

\lim _{x \to 0}\left\{ \lim _{y \to 0} f(x,y)\right\} =\lim _{y \to 0}\left\{ \lim _{x \to 0} f(x,y)\right\} =0 ,

yet $\lim \limits_{\substack {x \to 0 \\ y \to 0}} f(x,y)$ does not exist.

Problem 3183

Prove that for the function

f(x,y)=(x+y)\sin \dfrac {1}{x}\sin \dfrac {1}{y}

the iterated limits $\lim \limits_{x \to 0}\left\{ \lim \limits_{y \to 0} f(x,y)\right\}$ and $\lim \limits_{y \to 0}\left\{ \lim \limits_{x \to 0} f(x,y)\right\}$ do not exist, yet $\lim \limits_{\substack {x \to 0 \\ y \to 0}} f(x,y)=0$ .

Problem 3183.1

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Problem 3183.2

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Problem 3184

Find $\lim \limits_{x \to a}\left\{ \lim \limits_{y \to b} f(x,y)\right\}$ and $\lim \limits_{y \to b}\left\{ \lim \limits_{x \to a} f(x,y)\right\}$ , where:

(a)

$f(x,y)=\dfrac {x^{2}+y^{2}}{x^{2}+y^{4}}$ , $a=\infty$ , $b=\infty$ ;

(b)

$f(x,y)=\dfrac {x^{y}}{1+x^{y}}$ , $a=+\infty$ , $b=+0$ ;

(c)

$f(x,y)=\sin \dfrac {\pi x}{2x+y}$ , $a=\infty$ , $b=\infty$ ;

(d)

$f(x,y)=\dfrac {1}{xy}\tan \dfrac {xy}{1+xy}$ , $a=0$ , $b=\infty$ ;

(e)

$f(x,y)=\log_{x}(x+y)$ , $a=1$ , $b=0$ .

Problem 3185

Find the following limit: $\lim \limits_{\substack {x \to \infty \\ y \to \infty }} \dfrac {x+y}{x^{2}-xy+y^{2}}$ .

Problem 3186

Find the following limit: $\lim \limits_{\substack {x \to \infty \\ y \to \infty }} \dfrac {x^{2}+y^{2}}{x^{4}+y^{4}}$ .

Problem 3187

Find the following limit: $\lim \limits_{\substack {x \to 0 \\ y \to a}} \dfrac {\sin xy}{x}$ .

Problem 3188

Find the following limit: $\lim \limits_{\substack {x \to +\infty \\ y \to +\infty }} (x^{2}+y^{2})e^{-(x+y)}$ .

Problem 3189

Find the following limit: $\lim \limits_{\substack {x \to +\infty \\ y \to +\infty }} \left(\dfrac {xy}{x^{2}+y^{2}}\right)^{x^{2}}$ .

Problem 3190

Find the following limit: $\lim \limits_{\substack {x \to 0 \\ y \to 0}} (x^{2}+y^{2})^{x^{2}y^{2}}$ .

Problem 3191

Find the following limit: $\lim \limits_{\substack {x \to \infty \\ y \to a}} \left(1+\dfrac {1}{x}\right)^{\frac{x^{2}}{x+y}}$ .

Problem 3192

Find the following limit: $\lim \limits_{\substack {x \to 1 \\ y \to 0}} \dfrac {\ln (x+e^{y})}{\sqrt{x^{2}+y^{2}}}$ .

Problem 3193

(a)(b)

If $x=\rho \cos \varphi$ , $y=\rho \sin \varphi$ , along which directions $\varphi$ do the following limits have a definite limiting value:

(a)

$\lim \limits_{\rho \to +0} e^{\frac{x}{x^{2}+y^{2}}}$ ;

(b)

$\lim \limits_{\rho \to +\infty } e^{x^{2}-y^{2}} \cdot \sin 2xy$ .

Problem 3194

Find the points of discontinuity of the following function: $u=\dfrac {1}{\sqrt{x^{2}+y^{2}}}$ .

Problem 3195

Find the points of discontinuity of the following function: $u=\dfrac {xy}{x+y}$ .

Problem 3196

Find the points of discontinuity of the following function: $u=\dfrac {x+y}{x^{3}+y^{3}}$ .

Problem 3197

Find the points of discontinuity of the following function: $u=\sin \dfrac {1}{xy}$ .

Problem 3198

Find the points of discontinuity of the following function: $u=\dfrac {1}{\sin x \sin y}$ .

Problem 3199

Find the points of discontinuity of the following function: $u=\ln (1-x^{2}-y^{2})$ .

Problem 3200

Find the points of discontinuity of the following function: $u=\dfrac {1}{xyz}$ .

Problem 3201

Find the points of discontinuity of the following function: $u=\ln \dfrac {1}{\sqrt{(x-a)^{2}+(y-b)^{2}+(z-c)^{2}}}$ .

Problem 3202

Prove that the function

f(x,y)= \begin{cases} \dfrac {2xy}{x^{2}+y^{2}}, & \text{if} \ x^{2}+y^{2} \neq 0 ; \\ 0, & \text{if} \ x^{2}+y^{2}=0 , \end{cases}

is continuous separately in each of the variables $x$ or $y$ (when the value of the other variable is fixed), but is not continuous jointly in these variables.

Problem 3203

Prove that the function

f(x,y)= \begin{cases} \dfrac {x^{2}y}{x^{4}+y^{2}}, & \text{if} \ x^{2}+y^{2} \neq 0 , \\ 0, & \text{if} \ x^{2}+y^{2}=0 , \end{cases}

is continuous at the point $O(0,0)$ along every ray through this point

x=t\cos \alpha , \quad y=t\sin \alpha \quad (0 \leqslant t<+\infty )

that is,

\lim _{t \to 0} f(t\cos \alpha , t\sin \alpha )=f(0,0) ;

but this function is not continuous at the point $(0,0)$ .

Problem 3203.1

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Problem 3203.2

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Problem 3203.2

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Problem 3203.3

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Problem 3203.4

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Problem 3203.4

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Problem 3204

Prove that the function

f(x,y)=x\sin \dfrac {1}{y}, \text{if} \ y \neq 0 \text{and} \ f(x,0)=0

has a set of discontinuities that is not closed.

Problem 3205

Prove that if the function $f(x,y)$ is continuous in the variable $x$ within some domain $G$ , and is uniformly continuous in the variable $y$ with respect to $x$ , then this function is continuous in the domain under consideration.

Problem 3206

Prove that if, within some domain $G$ , the function $f(x,y)$ is continuous in the variable $x$ and satisfies a Lipschitz condition in the variable $y$ , that is,

\left|f(x,y')-f(x,y'')\right| \leqslant L\left|y'-y''\right| ,

where $(x,y') \in G$ , $(x,y'') \in G$ and $L$ is a constant, then this function is continuous in the given domain.

Problem 3207

Prove that if the function $f(x,y)$ is continuous separately in each of the variables $x$ and $y$ , and is monotone in one of them, then this function is continuous jointly in the two variables (Young's theorem).

Problem 3208

Let the function $f(x,y)$ be continuous on the domain $a \leqslant x \leqslant A$ , $b \leqslant y \leqslant B$ , and let the sequence of functions $\varphi_{n}(x)$ $(n=1,2,\ldots )$ converge uniformly on $\left[a,A\right]$ and satisfy the condition $b \leqslant \varphi_{n}(x) \leqslant B$ . Prove that the sequence of functions

F_{n}(x)=f\left[x,\varphi _{n}(x)\right] \quad (n=1,2,\ldots )

also converges uniformly on $\left[a,A\right]$ .

Problem 3209

Let: 1) the function $f(x,y)$ be continuous in the domain $R(a<x<A; b<y<B)$ ; 2) the function $\varphi (x)$ be continuous on the interval $(a,A)$ and have values belonging to the interval $(b,B)$ . Prove that the function

F(x)=f\left[x,\varphi (x)\right]

is continuous on the interval $(a,A)$ .

Problem 3210

Let: 1) the function $f(x,y)$ be continuous in the domain $R(a<x<A; b<y<B)$ ; 2) the functions $x=\varphi (u,v)$ and $y=\psi (u,v)$ be continuous in the domain $R'(a'<u<A'; b'<v<B')$ and have values belonging respectively to the intervals $(a,A)$ and $(b,B)$ . Prove that the function

F(u,v)=f\left[\varphi (u,v),\psi (u,v)\right]

is continuous in the domain $R'$ .

Section 6.4Change of Variables[97/100%]

Problem 3431

Taking $y$ as the new independent variable, transform the equation

y'y-3y''^{2}=x .

Problem 3432

Using the same method, transform the equation

(y')^{2}y^{(4)}-10y'y''y'''+15(y'')^{3}=0 .

Problem 3433

Taking $x$ as the function and $t=xy$ as the independent variable, transform the equation

y''+\dfrac {2}{x}y'+y=0 .

Problem 3434

Introduce a new variable and transform the following ordinary differential equation: $x^{2}y''+xy'+y=0$ , if $x=e^{t}$ .

Problem 3435

Introduce a new variable and transform the following ordinary differential equation: $y'''=\dfrac {6y}{x^{3}}$ , if $t=\ln \left|x\right|$ .

Problem 3436

Introduce a new variable and transform the following ordinary differential equation: $(1-x^{2})y''-xy'+n^{2}y=0$ , if $x=\cos t$ .

Problem 3437

Introduce a new variable and transform the following ordinary differential equation: $y''+y'\tanh x+\dfrac {m^{2}}{\cosh^{2}x}y=0$ , if $x=\ln \tan \dfrac {t}{2}$ .

Problem 3438

Introduce a new variable and transform the following ordinary differential equation: $y''+p(x)y'+q(x)y=0$ , let $y=ue^{-\frac{1}{2}\int_{x_{0}}^{x} p(\xi )d\xi }$ .

Problem 3439

Introduce a new variable and transform the following ordinary differential equation: $x^{4}y''+xyy'-2y^{2}=0$ . Let

x=e^{t} , \quad y=ue^{2t} ,

where $u=u(t)$ .

Problem 3440

Introduce a new variable and transform the following ordinary differential equation: $(1+x^{2})^{2}y''=y$ , if

x=\tan t , \quad y=\dfrac {u}{\cos t} ,

where $u=u(t)$ .

Problem 3441

Introduce a new variable and transform the following ordinary differential equation: $(1-x^{2})^{2}y''=-y$ , if

x=\tanh t , \quad y=\dfrac {u}{\cosh t} ,

where $u=u(t)$ .

Problem 3442

Introduce a new variable and transform the following ordinary differential equation: $y''+(x+y)(1+y')^{3}=0$ , if $x=u+t$ , $y=u-t$ , where $u=u(t)$ .

Problem 3443

Introduce a new variable and transform the following ordinary differential equation: $y'''-x^{3}y''+xy'-y=0$ , if $x=\dfrac {1}{t}$ and $y=\dfrac {u}{t}$ , where $u=u(t)$ .

Problem 3444

Suppose

u=\dfrac {y}{x-b} , \quad t=\ln \left|\dfrac {x-a}{x-b}\right| ,

and take $u$ as a function of the variable $t$ , to transform the Stokes equation

y''=-\dfrac {Ay}{(x-a)^{2}(x-b)^{2}} .

Problem 3445

Prove: if the equation

\dfrac {d^{2}y}{dx^{2}}+p(x)\dfrac {dy}{dx}+q(x)y=0 ,

is transformed by the substitution $x=\varphi (\xi )$ into the equation

\dfrac {d^{2}y}{d\xi ^{2}}+P(\xi )\dfrac {dy}{d\xi }+Q(\xi )y=0 ,

then

\begin{align} & \left[2P(\xi )Q(\xi )+Q'(\xi )\right]\left[Q(\xi )\right]^{-\frac{3}{2}} \\ & =\left[2p(x)q(x)+q'(x)\right]\left[q(x)\right]^{-\frac{3}{2}} . \end{align}

Problem 3446

In the equation

\Phi (y, y', y'')=0

(where $\Phi$ is a homogeneous function of the variables $y$ , $y'$ , $y''$ ), let $y=e^{\int_{x_{0}}^{x} u\, dx}$ .

Problem 3447

In the equation

F(x^{2}y'', xy', y)=0

(where $F$ is a homogeneous function of its variables), let $u=x \cdot \dfrac {y’}{y}$ .

Problem 3448

Prove: under the projective transformation

x=\dfrac {a_{1}\xi +b_{1}\eta +c_{1}}{a\xi +b\eta +c} , \quad y=\dfrac {a_{2}\xi +b_{2}\eta +c_{2}}{a\xi +b\eta +c} ,

the equation

y''(1+y'^{2})-3y'y''^{2}=0

does not change its form.

Problem 3449

Prove:

S\left[x(t)\right]=\dfrac {x”’(t)}{x’(t)}-\dfrac {3}{2}\left[\dfrac {x”(t)}{x’(t)}\right]^{2}

is invariant under the linear fractional transformation

y=\dfrac {ax(t)+b}{cx(t)+d} \quad (ad-bc \neq 0) .

Problem 3450

Transform the following equation into an equation expressed in polar coordinates $r$ and $\varphi$ , i.e. let $x=r\cos \varphi$ , $y=r\sin \varphi$ :

\dfrac {dy}{dx}=\dfrac {x+y}{x-y} .

Problem 3451

Transform the following equation into an equation expressed in polar coordinates $r$ and $\varphi$ :

(xy'-y)^{2}=2xy(1+y'^{2}) .

Problem 3452

Transform the following equation into an equation expressed in polar coordinates $r$ and $\varphi$ :

(x^{2}+y^{2})^{2}y''=(x+yy')^{3} .

Problem 3453

Transform the expression

\dfrac {x+yy’}{xy’-y}

into an expression in polar coordinates.

Problem 3454

Express the curvature of a plane curve

K=\dfrac {\left|y_{xx}”\right|}{(1+y_{x}’^{2})^{\frac{3}{2}}}

in terms of polar coordinates $r$ and $\varphi$ .

Problem 3455

Transform the system of equations

\dfrac {dx}{dt}=y+kx(x^{2}+y^{2}) ,

\dfrac {dy}{dt}=-x+ky(x^{2}+y^{2})

into a system of equations in polar coordinates.

Problem 3456

Introduce the new functions $r=\sqrt{x^{2}+y^{2}}$ , $\varphi =\arctan \dfrac {y}{x}$ , and transform the expression

W=x\dfrac {d^{2}y}{dt^{2}}-y\dfrac {d^{2}x}{dt^{2}} .

Problem 3457

In the Legendre transformation, each point $(x,y)$ of the curve $y=y(x)$ corresponds to a point $(X,Y)$ , where

X=y' , \quad Y=xy'-y .

Find $Y'$ , $Y''$ and $Y'''$ .

Problem 3458

Introduce new variables $\xi$ and $\eta$ , and solve the following equation: $\dfrac {\partial z}{\partial x}=\dfrac {\partial z}{\partial y}$ , let $\xi =x+y$ , $\eta =x-y$ .

Problem 3459

Introduce new variables $\xi$ and $\eta$ , and solve the following equation: $y\dfrac {\partial z}{\partial x}-x\dfrac {\partial z}{\partial y}=0$ , let $\xi =x$ , $\eta =x^{2}+y^{2}$ .

Problem 3460

Introduce new variables $\xi$ and $\eta$ , and solve the following equation: $a\dfrac {\partial z}{\partial x}+b\dfrac {\partial z}{\partial y}=1$ $(a \neq 0)$ , let $\xi =x$ , $\eta =y-bz$ .

Problem 3461

Introduce new variables $\xi$ and $\eta$ , and solve the following equation: $x\dfrac {\partial z}{\partial x}+y\dfrac {\partial z}{\partial y}=z$ , let $\xi =x$ and $\eta =\dfrac {y}{x}$ .

Problem 3462

Take $u$ and $v$ as new independent variables, and transform the following equation: $x\dfrac {\partial z}{\partial x}+\sqrt{1+y^{2}}\dfrac {\partial z}{\partial y}=xy$ , if $u=\ln x$ , $v=\ln (y+\sqrt{1+y^{2}})$ .

Problem 3463

Take $u$ and $v$ as new independent variables, and transform the following equation: $(x+y)\dfrac {\partial z}{\partial x}-(x-y)\dfrac {\partial z}{\partial y}=0$ , if $u=\ln \sqrt{x^{2}+y^{2}}$ , $v=\arctan \dfrac {y}{x}$ .

Problem 3464

Take $u$ and $v$ as the new independent variables, and transform the following equation: $x\dfrac {\partial z}{\partial x}+y\dfrac {\partial z}{\partial y}=z+\sqrt{x^{2}+y^{2}+z^{2}}$ , if $u=\dfrac {y}{x}$ , $v=z+\sqrt{x^{2}+y^{2}+z^{2}}$ .

Problem 3465

Take $u$ and $v$ as the new independent variables, and transform the following equation: $x\dfrac {\partial z}{\partial x}+y\dfrac {\partial z}{\partial y}=\dfrac {x}{z}$ , if $u=2x-z^{2}$ , $v=\dfrac {y}{z}$ .

Problem 3466

Take $u$ and $v$ as the new independent variables, and transform the following equation: $(x+z)\dfrac {\partial z}{\partial x}+(y+z)\dfrac {\partial z}{\partial y}=x+y+z$ , if $u=x+z$ , $v=y+z$ .

Problem 3467

Take

\xi =y+ze^{-x} , \quad \eta =x+ze^{-y}

as the new independent variables, and transform the expression

(z+e^{x})\dfrac {\partial z}{\partial x}+(z+e^{y})\dfrac {\partial z}{\partial y}-(z^{2}-e^{x+y}) .

Problem 3468

Suppose

x=uv , \quad y=\dfrac {1}{2}(u^{2}-v^{2})

transform the expression

\left(\dfrac {\partial z}{\partial x}\right)^{2}+\left(\dfrac {\partial z}{\partial y}\right)^{2} .

Problem 3469

In the equation

\dfrac {\partial u}{\partial x}+\dfrac {\partial u}{\partial y}+\dfrac {\partial u}{\partial z}=0

set $\xi =x$ , $\eta =y-x$ , $\zeta =z-x$ .

Problem 3470

Take $x$ as the function, and $y$ and $z$ as the independent variables, and transform the equation

(x-z)\dfrac {\partial z}{\partial x}+y\dfrac {\partial z}{\partial y}=0 .

Problem 3471

Take $x$ as the function, and $u=y-z$ , $v=y+z$ as the independent variables, and transform the equation

(y-z)\dfrac {\partial z}{\partial x}+(y+z)\dfrac {\partial z}{\partial y}=0 .

Problem 3472

Take $x$ as the function, and $u=xz$ , $v=yz$ as the independent variables, and transform the expression

A=\left(\dfrac {\partial z}{\partial x}\right)^{2}+\left(\dfrac {\partial z}{\partial y}\right)^{2} .

Problem 3473

In the equation

\begin{align} (y+z+u)\dfrac {\partial u}{\partial x} & +(x+z+u)\dfrac {\partial u}{\partial y} \\ & +(x+y+u)\dfrac {\partial u}{\partial z}=x+y+z \end{align}

set $e^{\xi }=x-u$ , $e^{\eta }=y-u$ , $e^{\zeta }=z-u$ .

Problem 3474

In the following equation, substitute the new variables $u$ , $v$ , $w$ , where $w=w(u,v)$ : $y\dfrac {\partial z}{\partial x}-x\dfrac {\partial z}{\partial y}=(y-x)z$ , set $u=x^{2}+y^{2}$ , $v=\dfrac {1}{x}+\dfrac {1}{y}$ , $w=\ln z-(x+y)$ .

Problem 3475

In the following equation, substitute the new variables $u$ , $v$ , $w$ , where $w=w(u,v)$ : $x^{2}\dfrac {\partial z}{\partial x}+y^{2}\dfrac {\partial z}{\partial y}=z^{2}$ , set $u=x$ , $v=\dfrac {1}{y}-\dfrac {1}{x}$ , $w=\dfrac {1}{z}-\dfrac {1}{x}$ .

Problem 3476

In the following equation, substitute the new variables $u$ , $v$ , $w$ , where $w=w(u,v)$ : $(xy+z)\dfrac {\partial z}{\partial x}+(1-y^{2})\dfrac {\partial z}{\partial y}=x+yz$ , let $u=yz-x$ , $v=xz-y$ , $w=xy-z$ .

Problem 3477

In the following equation, substitute the new variables $u$ , $v$ , $w$ , where $w=w(u,v)$ : $\left(x\dfrac {\partial z}{\partial x}\right)^{2}+\left(y\dfrac {\partial z}{\partial y}\right)^{2}=z^{2}\dfrac {\partial z}{\partial x}\dfrac {\partial z}{\partial y}$ , set $x=ue^{w}$ , $y=ve^{w}$ , $z=we^{w}$ .

Problem 3478

Suppose $u=\ln \sqrt{x^{2}+y^{2}}$ , $v=\arctan z$ , $w=x+y+z$ , where $w=w(u,v)$ , transform the expression

(x-y):\left(\dfrac {\partial z}{\partial x}-\dfrac {\partial z}{\partial y}\right) .

Problem 3479

Suppose $u=xe^{z}$ , $v=ye^{z}$ , $w=ze^{z}$ , where $w=w(u,v)$ , transform the expression

A=\dfrac {\partial z}{\partial x} \cdot \dfrac {\partial z}{\partial y} .

Problem 3480

In the equation

x\dfrac {\partial u}{\partial x}+y\dfrac {\partial u}{\partial y}+z\dfrac {\partial u}{\partial z}=u+\dfrac {xy}{z}

set $\xi =\dfrac {x}{z}$ , $\eta =\dfrac {y}{z}$ , $\zeta =z$ , $w=\dfrac {u}{z}$ , where $w=w(\xi ,\eta ,\zeta )$ .

Problem 3481

Given $x=r\cos \varphi$ , $y=r\sin \varphi$ , transform the following expression into polar coordinates $r$ and $\varphi$ : $w=x\dfrac {\partial u}{\partial y}-y\dfrac {\partial u}{\partial x}$ .

Problem 3482

Given $x=r\cos \varphi$ , $y=r\sin \varphi$ , transform the following expression into polar coordinates $r$ and $\varphi$ : $w=x\dfrac {\partial u}{\partial x}+y\dfrac {\partial u}{\partial y}$ .

Problem 3483

Given $x=r\cos \varphi$ , $y=r\sin \varphi$ , transform the following expression into polar coordinates $r$ and $\varphi$ : $w=\left(\dfrac {\partial u}{\partial x}\right)^{2}+\left(\dfrac {\partial u}{\partial y}\right)^{2}$ .

Problem 3484

Given $x=r\cos \varphi$ , $y=r\sin \varphi$ , transform the following expression into polar coordinates $r$ and $\varphi$ : $w=\dfrac {\partial^{2}u}{\partial x^{2}}+\dfrac {\partial^{2}u}{\partial y^{2}}$ .

Problem 3485

Given $x=r\cos \varphi$ , $y=r\sin \varphi$ , transform the following expression into polar coordinates $r$ and $\varphi$ : $w=x^{2}\dfrac {\partial^{2}u}{\partial x^{2}}+2xy\dfrac {\partial^{2}u}{\partial x \partial y}+y^{2}\dfrac {\partial^{2}u}{\partial y^{2}}$ .

Problem 3486

Given $x=r\cos \varphi$ , $y=r\sin \varphi$ , transform the following expression into polar coordinates $r$ and $\varphi$ :

w=y^{2}\dfrac {\partial ^{2}z}{\partial x^{2}}-2xy\dfrac {\partial ^{2}z}{\partial x \partial y}+x^{2}\dfrac {\partial ^{2}z}{\partial y^{2}}-\left(x\dfrac {\partial z}{\partial x}+y\dfrac {\partial z}{\partial y}\right) .

Problem 3487

In the expression

I=\dfrac {\partial u}{\partial x}\dfrac {\partial v}{\partial y}-\dfrac {\partial u}{\partial y}\dfrac {\partial v}{\partial x}

let $x=r\cos \varphi$ , $y=r\sin \varphi$ .

Problem 3488

Introduce the new independent variables

\xi =x-at , \quad \eta =x+at

and solve the equation

\dfrac {\partial ^{2}u}{\partial t^{2}}=a^{2} \cdot \dfrac {\partial ^{2}u}{\partial x^{2}} .

Problem 3489

Take $u$ and $v$ as new independent variables, and transform the following equation: $2\dfrac {\partial^{2}z}{\partial x^{2}}+\dfrac {\partial^{2}z}{\partial x \partial y}-\dfrac {\partial^{2}z}{\partial y^{2}}+\dfrac {\partial z}{\partial x}+\dfrac {\partial z}{\partial y}=0$ , where $u=x+2y+2$ and $v=x-y-1$ .

Problem 3490

Take $u$ and $v$ as new independent variables, and transform the following equation: $(1+x^{2})\dfrac {\partial^{2}z}{\partial x^{2}}+(1+y^{2})\dfrac {\partial^{2}z}{\partial y^{2}}+x\dfrac {\partial z}{\partial x}+y\dfrac {\partial z}{\partial y}=0$ , where $u=\ln (x+\sqrt{1+x^{2}})$ and $v=\ln (y+\sqrt{1+y^{2}})$ .

Problem 3491

Take $u$ and $v$ as new independent variables, and transform the following equation: $ax^{2}\dfrac {\partial^{2}z}{\partial x^{2}}+2bxy\dfrac {\partial^{2}z}{\partial x \partial y}+cy^{2}\dfrac {\partial^{2}z}{\partial y^{2}}=0$ ( $a$ , $b$ , $c$ are constants), where $u=\ln x$ , $v=\ln y$ .

Problem 3492

Take $u$ and $v$ as new independent variables, and transform the following equation: $\dfrac {\partial^{2}z}{\partial x^{2}}+\dfrac {\partial^{2}z}{\partial y^{2}}=0$ , where $u=\dfrac {x}{x^{2}+y^{2}}$ , $v=-\dfrac {y}{x^{2}+y^{2}}$ .

Problem 3493

Take $u$ and $v$ as new independent variables, and transform the following equation: $\dfrac {\partial^{2}z}{\partial x^{2}}+\dfrac {\partial^{2}z}{\partial y^{2}}+m^{2}z=0$ , where $x=e^{u}\cos v$ , $y=e^{u}\sin v$ .

Problem 3494

Take $u$ and $v$ as new independent variables, and transform the following equation: $\dfrac {\partial^{2}z}{\partial x^{2}}-y\dfrac {\partial^{2}z}{\partial y^{2}}=\dfrac {1}{2}\dfrac {\partial z}{\partial y}$ $(y>0)$ , where $u=x-2\sqrt{y}$ and $v=x+2\sqrt{y}$ .

Problem 3495

Take $u$ and $v$ as new independent variables, and transform the following equation: $x^{2}\dfrac {\partial^{2}z}{\partial x^{2}}-y^{2}\dfrac {\partial^{2}z}{\partial y^{2}}=0$ , where $u=xy$ , $v=\dfrac {x}{y}$ .

Problem 3496

Take $u$ and $v$ as new independent variables, and transform the following equation: $x^{2}\dfrac {\partial^{2}z}{\partial x^{2}}-(x^{2}+y^{2})\dfrac {\partial^{2}z}{\partial x \partial y}+y^{2}\dfrac {\partial^{2}z}{\partial y^{2}}=0$ , where $u=x+y$ , $v=\dfrac {1}{x}+\dfrac {1}{y}$ .

Problem 3497

Take $u$ and $v$ as new independent variables, and transform the following equation:

xy\dfrac {\partial ^{2}z}{\partial x^{2}}-(x^{2}+y^{2})\dfrac {\partial ^{2}z}{\partial x \partial y}+xy\dfrac {\partial ^{2}z}{\partial y^{2}}+y\dfrac {\partial z}{\partial x}+x\dfrac {\partial z}{\partial y}=0 ,

where $u=\dfrac {1}{2}(x^{2}+y^{2})$ and $v=xy$ .

Problem 3498

Take $u$ and $v$ as new independent variables, and transform the following equation: $x^{2}\dfrac {\partial^{2}z}{\partial x^{2}}-2x\sin y\dfrac {\partial^{2}z}{\partial x \partial y}+\sin^{2}y\dfrac {\partial^{2}z}{\partial y^{2}}=0$ , where $u=x\tan \dfrac {y}{2}$ , $v=x$ .

Problem 3499

Take $u$ and $v$ as the new independent variables and transform the following equation: $x\dfrac {\partial^{2}z}{\partial x^{2}}-y\dfrac {\partial^{2}z}{\partial y^{2}}=0$ $(x>0, y>0)$ , where $x=(u+v)^{2}$ and $y=(u-v)^{2}$ .

Problem 3500

Take $u$ and $v$ as the new independent variables and transform the following equation: $\dfrac {\partial^{2}z}{\partial x \partial y}=\left(1+\dfrac {\partial z}{\partial y}\right)^{3}$ , where $u=x$ , $v=y+z$ .

Problem 3501

Using the linear transformation

\xi =x+\lambda _{1}y , \quad \eta =x+\lambda _{2}y

transform the equation

A\dfrac {\partial ^{2}u}{\partial x^{2}}+2B\dfrac {\partial ^{2}u}{\partial x \partial y}+C\dfrac {\partial ^{2}u}{\partial y^{2}}=0 , \tag {1}

(where $A$ , $B$ and $C$ are constants and $C \neq 0$ , $AC-B^{2}<0$ ) into the form

\dfrac {\partial ^{2}u}{\partial \xi \partial \eta }=0 .

Find the general form of the function satisfying equation (1).

Problem 3502

Prove that Laplace's equation

\Delta z=\dfrac {\partial ^{2}z}{\partial x^{2}}+\dfrac {\partial ^{2}z}{\partial y^{2}}=0

keeps its form unchanged under the nondegenerate change of variables

x=\varphi (u,v) , \quad y=\psi (u,v)

satisfying the conditions $\dfrac {\partial \varphi }{\partial u}=\dfrac {\partial \psi }{\partial v}$ , $\dfrac {\partial \varphi }{\partial v}=-\dfrac {\partial \psi }{\partial u}$ .

Problem 3503

(a)(b)

Suppose $u=f(r)$ , where $r=\sqrt{x^{2}+y^{2}}$ , transform the equations

(a)

$\Delta u \equiv \dfrac {\partial^{2}u}{\partial x^{2}}+\dfrac {\partial^{2}u}{\partial y^{2}}=0$ ;

(b)

$\Delta (\Delta u)=0$ .

Problem 3504

If we set

w=f(u) ,

where

u=(x-x_{0})(y-y_{0}) ,

into what form does the equation

\dfrac {\partial ^{2}w}{\partial x \partial y}+cw=0

transform?

Problem 3505

Suppose

x+y=X , \quad y=XY ,

transform the expression $A=x\dfrac {\partial^{2}u}{\partial x^{2}}+y\dfrac {\partial^{2}u}{\partial x \partial y}+\dfrac {\partial u}{\partial x}$ .

Problem 3506

Prove that the equation

\dfrac {\partial ^{2}z}{\partial x^{2}}+2xy^{2}\dfrac {\partial z}{\partial x}+2(y-y^{3})\dfrac {\partial z}{\partial y}+x^{2}y^{2}z^{2}=0

keeps its form unchanged under the transformation $x=uv$ and $y=\dfrac {1}{v}$ .

Problem 3507

Prove that the equation

\dfrac {\partial ^{2}z}{\partial x^{2}}+2\dfrac {\partial ^{2}z}{\partial x \partial y}+\dfrac {\partial ^{2}z}{\partial y^{2}}=0

keeps its form unchanged under the transformation $u=x+z$ and $v=y+z$ .

Problem 3508

Suppose

x=\eta \zeta , \quad y=\xi \zeta , \quad z=\xi \eta ,

transform the equation $xy\dfrac {\partial^{2}u}{\partial x \partial y}+yz\dfrac {\partial^{2}u}{\partial y \partial z}+xz\dfrac {\partial^{2}u}{\partial x \partial z}=0$ .

Problem 3509

Suppose

y_{1}=x_{2}+x_{3}-x_{1} , \quad y_{2}=x_{1}+x_{3}-x_{2} ,

y_{3}=x_{1}+x_{2}-x_{3} ,

transform the equation

\begin{align} & \dfrac {\partial ^{2}z}{\partial x_{1}^{2}}+\dfrac {\partial ^{2}z}{\partial x_{2}^{2}}+\dfrac {\partial ^{2}z}{\partial x_{3}^{2}}+\dfrac {\partial ^{2}z}{\partial x_{1} \partial x_{2}} \\ & \quad +\dfrac {\partial ^{2}z}{\partial x_{1} \partial x_{3}}+\dfrac {\partial ^{2}z}{\partial x_{2} \partial x_{3}}=0 . \end{align}

Problem 3510

Given

\xi =\dfrac {y}{x} , \quad \eta =\dfrac {z}{x} , \quad \zeta =y-z ,

transform the equation

\begin{align} & x^{2}\dfrac {\partial ^{2}u}{\partial x^{2}}+y^{2}\dfrac {\partial ^{2}u}{\partial y^{2}}+z^{2}\dfrac {\partial ^{2}u}{\partial z^{2}}+2xy\dfrac {\partial ^{2}u}{\partial x \partial y} \\ & \quad +2xz\dfrac {\partial ^{2}u}{\partial x \partial z}+2yz\dfrac {\partial ^{2}u}{\partial y \partial z}=0 . \end{align}

Problem 3511

Given

x=r\sin \theta \cos \varphi , \quad y=r\sin \theta \sin \varphi , \quad z=r\cos \theta ,

express the quantities $\Delta_{1}u=\left(\dfrac {\partial u}{\partial x}\right)^{2}+\left(\dfrac {\partial u}{\partial y}\right)^{2}+\left(\dfrac {\partial u}{\partial z}\right)^{2}$ and $\Delta_{2}u=\dfrac {\partial^{2}u}{\partial x^{2}}+\dfrac {\partial^{2}u}{\partial y^{2}}+\dfrac {\partial^{2}u}{\partial z^{2}}$ in terms of spherical coordinates.

Problem 3512

In the equation

z\left(\dfrac {\partial ^{2}z}{\partial x^{2}}+\dfrac {\partial ^{2}z}{\partial y^{2}}\right)=\left(\dfrac {\partial z}{\partial x}\right)^{2}+\left(\dfrac {\partial z}{\partial y}\right)^{2}

introduce a new function $w$ , given by $w=z^{2}$ .

Problem 3513

Take $u$ and $v$ as the new independent variables and $w=w(u,v,z)$ as the new function, transform the following equation: $y\dfrac {\partial^{2}z}{\partial y^{2}}+2\dfrac {\partial z}{\partial y}=\dfrac {2}{x}$ , given $u=\dfrac {x}{y}$ , $v=x$ , $w=xz-y$ .

Problem 3514

Take $u$ and $v$ as the new independent variables and $w=w(u,v,z)$ as the new function, transform the following equation: $\dfrac {\partial^{2}z}{\partial x^{2}}-2\dfrac {\partial^{2}z}{\partial x \partial y}+\dfrac {\partial^{2}z}{\partial y^{2}}=0$ , given $u=x+y$ , $v=\dfrac {y}{x}$ , $w=\dfrac {z}{x}$ .

Problem 3515

Take $u$ and $v$ as the new independent variables and $w=w(u,v,z)$ as the new function, transform the following equation: $\dfrac {\partial^{2}z}{\partial x^{2}}+2\dfrac {\partial^{2}z}{\partial x \partial y}+\dfrac {\partial^{2}z}{\partial y^{2}}=0$ , given $u=x+y$ , $v=x-y$ , $w=xy-z$ .

Problem 3516

Take $u$ and $v$ as the new independent variables and $w=w(u,v,z)$ as the new function, transform the following equation: $\dfrac {\partial^{2}z}{\partial x^{2}}+\dfrac {\partial^{2}z}{\partial x \partial y}+\dfrac {\partial z}{\partial x}=z$ , given $u=\dfrac {x+y}{2}$ , $v=\dfrac {x-y}{2}$ , $w=ze^{x}$ .

Problem 3517

Take $u$ and $v$ as the new independent variables and $w=w(u,v,z)$ as the new function, transform the following equation: $\dfrac {\partial^{2}z}{\partial x^{2}}-2\dfrac {\partial^{2}z}{\partial x \partial y}+\left(1+\dfrac {y}{x}\right)\dfrac {\partial^{2}z}{\partial y^{2}}=0$ , given $u=x$ , $v=x+y$ , $w=x+y+z$ .

Problem 3518

Take $u$ and $v$ as the new independent variables and $w=w(u,v,z)$ as the new function, transform the following equation: $(1-x^{2})\dfrac {\partial^{2}z}{\partial x^{2}}+(1-y^{2})\dfrac {\partial^{2}z}{\partial y^{2}}=x\dfrac {\partial z}{\partial x}+y\dfrac {\partial z}{\partial y}$ , given $x=\sin u$ , $y=\sin v$ , $z=e^{w}$ .

Problem 3519

Take $u$ and $v$ as the new independent variables and $w=w(u,v,z)$ as the new function, and transform the following equation: $(1-x^{2})\dfrac {\partial^{2}z}{\partial x^{2}}-\dfrac {\partial^{2}z}{\partial y^{2}}-2x\dfrac {\partial z}{\partial x}-\dfrac {1}{4}z=0$ $(\left|x\right|<1)$ , where $u=\dfrac {1}{2}(y+\arccos x)$ , $v=\dfrac {1}{2}(y-\arccos x)$ , $w=z\sqrt[4]{1-x^{2}}$ .

Problem 3520

Take $u$ and $v$ as the new independent variables and $w=w(u,v,z)$ as the new function, and transform the following equation: $\dfrac {\partial^{2}z}{\partial x^{2}}+\dfrac {\partial^{2}z}{\partial y^{2}}=2\dfrac {x\dfrac {\partial z}{\partial x}-y\dfrac {\partial z}{\partial y}}{x^{2}-y^{2}}-\dfrac {3(x^{2}+y^{2})z}{(x^{2}-y^{2})^{2}}$ $(\left|x\right|>\left|y\right|)$ , where $u=x+y$ , $v=x-y$ , $w=\dfrac {z}{\sqrt{x^{2}-y^{2}}}$ .

Problem 3521

Prove that any equation

\dfrac {\partial ^{2}z}{\partial x \partial y}+a\dfrac {\partial z}{\partial x}+b\dfrac {\partial z}{\partial y}+cz=0

(with $a$ , $b$ , $c$ constants) can be reduced, by the substitution

z=ue^{\alpha x+\beta y}

(where $\alpha$ and $\beta$ are constants and $u=u(x,y)$ ), to the form

\dfrac {\partial ^{2}u}{\partial x \partial y}+c_{1}u=0 \quad (c_{1}=\text{a constant}) .

Problem 3522

Prove that the equation

\dfrac {\partial ^{2}u}{\partial x^{2}}=\dfrac {\partial u}{\partial y}

keeps its form unchanged under the change of variables

x'=\dfrac {x}{y} , \quad y'=-\dfrac {1}{y} , \quad u=\dfrac {u’}{\sqrt{y}}e^{-\frac{x^{2}}{4y}}

(where $u'$ is a function of the variables $x'$ and $y'$ ).

Problem 3523

In the equation

\begin{align} & q(1+q)\dfrac {\partial ^{2}z}{\partial x^{2}}-(1+p+q+2pq)\dfrac {\partial ^{2}z}{\partial x \partial y} \\ & \quad +p(1+p)\dfrac {\partial ^{2}z}{\partial y^{2}}=0 \end{align}

(where $p=\dfrac {\partial z}{\partial x}$ , $q=\dfrac {\partial z}{\partial y}$ ), let $u=x+z$ , $v=y+z$ , $w=x+y+z$ , and suppose $w=w(u,v)$ .

Problem 3524

In the equation

\begin{align} & x^{2}\dfrac {\partial ^{2}u}{\partial x^{2}}+y^{2}\dfrac {\partial ^{2}u}{\partial y^{2}}+z^{2}\dfrac {\partial ^{2}u}{\partial z^{2}}=\left(x\dfrac {\partial u}{\partial x}\right)^{2} \\ & \quad +\left(y\dfrac {\partial u}{\partial y}\right)^{2}+\left(z\dfrac {\partial u}{\partial z}\right)^{2} \end{align}

let $x=e^{\xi }$ , $y=e^{\eta }$ , $z=e^{\zeta }$ , $u=e^{w}$ , where $w=w(\xi ,\eta ,\zeta )$ .

Problem 3525

Prove that the form of the equation

\dfrac {\partial ^{2}z}{\partial x^{2}}\dfrac {\partial ^{2}z}{\partial y^{2}}-\left(\dfrac {\partial ^{2}z}{\partial x \partial y}\right)^{2}=0

does not depend on which roles the variables $x$ , $y$ , and $z$ respectively play.

Problem 3526

Take $x$ as a function of the variables $y$ and $z$ , and solve the equation

\begin{align} & \left(\dfrac {\partial z}{\partial y}\right)^{2}\dfrac {\partial ^{2}z}{\partial x^{2}}-2\dfrac {\partial z}{\partial x}\dfrac {\partial z}{\partial y}\dfrac {\partial ^{2}z}{\partial x \partial y} \\ & \quad +\left(\dfrac {\partial z}{\partial x}\right)^{2}\dfrac {\partial ^{2}z}{\partial y^{2}}=0 . \end{align}

Problem 3527

Using the Legendre transform

X=\dfrac {\partial z}{\partial x} , \quad Y=\dfrac {\partial z}{\partial y} , \quad Z=x\dfrac {\partial z}{\partial x}+y\dfrac {\partial z}{\partial y}-z ,

where $Z=Z(X,Y)$ , transform the equation

\begin{align} & A\left(\dfrac {\partial z}{\partial x}, \dfrac {\partial z}{\partial y}\right)\dfrac {\partial ^{2}z}{\partial x^{2}}+2B\left(\dfrac {\partial z}{\partial x}, \dfrac {\partial z}{\partial y}\right)\dfrac {\partial ^{2}z}{\partial x \partial y} \\ & \quad +C\left(\dfrac {\partial z}{\partial x}, \dfrac {\partial z}{\partial y}\right)\dfrac {\partial ^{2}z}{\partial y^{2}}=0 . \end{align}

Section 6.5Applications in Geometry[53/100%]

Problem 3528

Write the tangent line and normal plane equations at the given point for the following curve: $x=a\cos \alpha \cos t$ , $y=a\sin \alpha \cos t$ , $z=a\sin t$ ; at the point $t=t_{0}$ .

Problem 3529

Write the tangent line and normal plane equations at the given point for the following curve: $x=a\sin^{2}t$ , $y=b\sin t\cos t$ , $z=c\cos^{2}t$ ; at the point $t=\dfrac {\pi }{4}$ .

Problem 3530

Write the tangent line and normal plane equations at the given point for the following curve: $y=x$ , $z=x^{2}$ ; at the point $M(1,1,1)$ .

Problem 3531

Write the tangent line and normal plane equations at the given point for the following curve: $x^{2}+z^{2}=10$ , $y^{2}+z^{2}=10$ ; at the point $M(1,1,3)$ .

Problem 3532

Write the tangent line and normal plane equations at the given point for the following curve: $x^{2}+y^{2}+z^{2}=6$ , $x+y+z=0$ ; at the point $M(1,-2,1)$ .

Problem 3533

Find a point on the curve $x=t$ , $y=t^{2}$ , $z=t^{3}$ at which the tangent line is parallel to the plane $x+2y+z=4$ .

Problem 3534

Prove that the tangent line to the helix $x=a\cos t$ , $y=a\sin t$ , $z=bt$ makes a constant angle with the $Oz$ axis.

Problem 3535

Prove that the curve

x=ae^{t}\cos t , \quad y=ae^{t}\sin t , \quad z=ae^{t}

intersects all the generating lines of the cone $x^{2}+y^{2}=z^{2}$ at the same angle.

Problem 3536

Prove that the loxodrome

\tan \left(\dfrac {\pi }{4}+\dfrac {\psi }{2}\right)=e^{k\varphi } \quad (k=\text{constant}) ,

(where $\varphi$ is the longitude of a point on the Earth, and $\psi$ is the latitude of a point on the Earth) intersects all meridians of the Earth at a constant angle.

Problem 3537

Given the curve

z=f(x,y) , \quad \dfrac {x-x_{0}}{\cos \alpha }=\dfrac {y-y_{0}}{\sin \alpha } ,

where $f$ is a differentiable function. Find the tangent of the angle formed by the tangent line to the curve at the point $M_{0}(x_{0}, y_{0})$ with the $Oxy$ plane.

Problem 3538

Find the derivative of the function

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

at the point $M(1,2,-2)$ along the direction of the tangent line, at this point, to the curve $x=t$ , $y=2t^{2}$ , $z=-2t^{4}$ .

Problem 3539

Write the tangent plane and normal line equations at the given point for the following surface: $z=x^{2}+y^{2}$ ; at the point $M_{0}(1,2,5)$ .

Problem 3540

Write the tangent plane and normal line equations at the given point for the following surface: $x^{2}+y^{2}+z^{2}=169$ ; at the point $M_{0}(3,4,12)$ .

Problem 3541

Write the tangent plane and normal line equations at the given point for the following surface: $z=\operatorname {arctg}\dfrac {y}{x}$ ; at the point $M_{0}\left(1, 1, \dfrac {\pi }{4}\right)$ .

Problem 3542

Write the tangent plane and normal line equations at the given point for the following surface: $ax^{2}+by^{2}+cz^{2}=1$ ; at the point $M_{0}(x_{0}, y_{0}, z_{0})$ .

Problem 3543

Write the tangent plane and normal line equations at the given point for the following surface: $z=y+\ln \dfrac {x}{z}$ ; at the point $M_{0}(1,1,1)$ .

Problem 3544

Write the tangent plane and normal line equations at the given point for the following surface: $2^{\frac{x}{z}}+2^{\frac{y}{z}}=8$ ; at the point $M_{0}(2,2,1)$ .

Problem 3545

Write the tangent plane and normal line equations at the given point for the following surface: $x=a\cos \psi \cos \varphi$ , $y=b\cos \psi \sin \varphi$ , $z=c\sin \psi$ ; at the point $M_{0}(\varphi_{0}, \psi_{0})$ .

Problem 3546

Write the tangent plane and normal line equations at the given point for the following surface: $x=r\cos \varphi$ , $y=r\sin \varphi$ , $z=r\operatorname {ctg}\alpha$ ; at the point $M_{0}(\varphi_{0}, r_{0})$ .

Problem 3547

Write the tangent plane and normal line equations at the given point for the following surface: $x=u\cos v$ , $y=u\sin v$ , $z=av$ ; at the point $M_{0}(u_{0}, v_{0})$ .

Problem 3548

Find the limiting position of the tangent plane to the surface

x=u+v , \quad y=u^{2}+v^{2} , \quad z=u^{3}+v^{3}

at the point of tangency $M(u,v)$ $(u \neq v)$ , as it approaches without bound a point $M_{0}(u_{0}, v_{0})$ on the boundary curve $u=v$ of the surface.

Problem 3549

Find on the surface $x^{2}+2y^{2}+3z^{2}+2xy+2xz+4yz=8$ all the points of tangency at which the tangent plane is parallel to a coordinate plane.

Problem 3550

At which points on the ellipsoid

\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}+\dfrac {z^{2}}{c^{2}}=1

does the normal line to the ellipsoid make equal angles with the coordinate axes?

Problem 3551

Find the tangent planes to the surface $x^{2}+2y^{2}+3z^{2}=21$ that are parallel to the plane

x+4y+6z=0 .

Problem 3552

Prove that the tangent planes to the surface $xyz=a^{3}$ $(a>0)$ form, with the coordinate planes, a tetrahedron of constant volume.

Problem 3553

Prove that the sum of the segments cut off on the coordinate axes by the tangent plane to the surface

\sqrt{x}+\sqrt{y}+\sqrt{z}=\sqrt{a} \quad (a>0)

is a constant.

Problem 3554

Prove that the tangent plane of the conical surface

z=xf\left(\dfrac {y}{x}\right)

passes through its vertex.

Problem 3555

Prove that the normal of the surface of revolution

z=f\left(\sqrt{x^{2}+y^{2}}\right) \quad (f' \neq 0)

meets the axis of rotation.

Problem 3556

Find the projections of the ellipsoid

x^{2}+y^{2}+z^{2}-xy=1

onto the coordinate planes.

Problem 3557

Divide the square $\left\{ 0 \leqslant x \leqslant 1, 0 \leqslant y \leqslant 1\right\}$ into finitely many parts $\sigma$ of diameter $\leqslant \delta$ . If the surface

z=1-x^{2}-y^{2}

is such that at any two points $P(x,y)$ and $P_{1}(x_{1}, y_{1})$ belonging to the same part $\sigma$ the normal directions differ by less than $1^{\circ }$ , find an upper bound for the number $\delta$ .

Problem 3558

Let

z=f(x,y) , \quad (x,y) \in D

be the equation of a surface, and let $\varphi (P_{1}, P)$ be the angle between the normals of the surface at the two points $P(x,y) \in D$ and $P_{1}(x_{1}, y_{1}) \in D$ .

Prove that if the domain $D$ is bounded and closed, and the function $f(x,y)$ has bounded second-order derivatives in $D$ , then the Lyapunov inequality

\varphi (P_{1}, P)<C\rho (P_{1}, P)

holds, where $C$ is a constant and $\rho (P_{1}, P)$ is the distance between the points $P$ and $P_{1}$ .

Problem 3559

At what angle does the cylinder $x^{2}+y^{2}=a^{2}$ intersect the surface $bz=xy$ at a common point $M_{0}(x_{0}, y_{0}, z_{0})$ ?

Problem 3560

Prove that the coordinate surfaces of spherical coordinates $x^{2}+y^{2}+z^{2}=r^{2}$ , $y=x\operatorname {tg}\varphi$ , $x^{2}+y^{2}=z^{2}\operatorname {tg}^{2}\theta$ are pairwise orthogonal.

Problem 3561

Prove that the spheres $x^{2}+y^{2}+z^{2}=2ax$ , $x^{2}+y^{2}+z^{2}=2by$ , $x^{2}+y^{2}+z^{2}=2cz$ form a triply orthogonal system.

Problem 3562

When $\lambda =\lambda_{1}$ , $\lambda =\lambda_{2}$ , $\lambda =\lambda_{3}$ , through each point $M(x,y,z)$ there pass three quadric surfaces:

\dfrac {x^{2}}{a^{2}-\lambda ^{2}}+\dfrac {y^{2}}{b^{2}-\lambda ^{2}}+\dfrac {z^{2}}{c^{2}-\lambda ^{2}}=-1 \quad (a>b>c>0) .

Prove that these surfaces are orthogonal.

Problem 3563

(a)(b)(c)

Find the derivative of the function $u=x+y+z$ along the outward normal direction at the point $M_{0}(x_{0}, y_{0}, z_{0})$ on the sphere $x^{2}+y^{2}+z^{2}=1$ .

At which points on the sphere does this derivative attain:

(a)

its maximum value,

(b)

its minimum value,

(c)

the value zero?

Problem 3564

Find the derivative of the function $u=x^{2}+y^{2}+z^{2}$ along the outward normal direction at the point $M_{0}(x_{0}, y_{0}, z_{0})$ on the ellipsoid $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}+\dfrac {z^{2}}{c^{2}}=1$ .

Problem 3565

Let $\dfrac {\partial u}{\partial n}$ and $\dfrac {\partial v}{\partial n}$ be the derivatives of the functions $u$ and $v$ along the normal direction at points of the surface $F(x,y,z)=0$ ; prove that

\dfrac {\partial }{\partial n}(uv)=u\dfrac {\partial v}{\partial n}+v\dfrac {\partial u}{\partial n} .

Problem 3566

Find the envelope of the one-parameter family of plane curves: $x\cos \alpha +y\sin \alpha =p$ $(p=\text{constant})$ .

Problem 3567

Find the envelope of the one-parameter family of plane curves: $(x-a)^{2}+y^{2}=\dfrac {a^{2}}{2}$ .

Problem 3568

Find the envelope of the one-parameter family of plane curves: $y=kx+\dfrac {a}{k}$ $(a=\text{constant})$ .

Problem 3569

Find the envelope of the one-parameter family of plane curves: $y^{2}=2px+p^{2}$ .

Problem 3570

A segment of length $l$ has its two endpoints sliding along the coordinate axes; find the envelope of the family of segments so produced.

Problem 3571

Find the envelope of the family of ellipses $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}=1$ , given that the area $S$ of each ellipse in the family is constant.

Problem 3572

A shell is fired in a vacuum with initial speed $v_{0}$ ; as the angle of elevation $\alpha$ varies in the vertical plane, find the envelope of the shell's trajectories.

Problem 3573

Prove that the envelope of the normals of a plane curve is the evolute of that curve.

Problem 3574

(a)(b)(c)(d)

Investigate the properties of the discriminant curve of each of the following families of curves ( $c$ being the parameter):

(a)

the cubic parabola $y=(x-c)^{3}$ ;

(b)

the semicubical parabola $y^{2}=(x-c)^{3}$ ;

(c)

the Neile semicubical parabola $y^{3}=(x-c)^{2}$ ;

(d)

the conchoid $(y-c)^{2}=x^{2}\dfrac {a-x}{a+x}$ .

Problem 3575

Find the envelope surface of the family of spheres of radius $r$ with centers on the circle $x=R\cos t$ , $y=R\sin t$ , $z=0$ ( $t$ being the parameter, $R>r$ ).

Problem 3576

Find the envelope surface of the family of spheres

(x-t\cos \alpha )^{2}+(y-t\cos \beta )^{2}+(z-t\cos \gamma )^{2}=1

(where $\cos^{2}\alpha +\cos^{2}\beta +\cos^{2}\gamma =1$ , and $t$ is the parameter).

Problem 3577

Find the envelope surface of the family of ellipsoids $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}+\dfrac {z^{2}}{c^{2}}=1$ , where the volume $V$ of each ellipsoid is constant.

Problem 3578

Find the envelope surface of the family of spheres of radius $\rho$ with centers on the cone $x^{2}+y^{2}=z^{2}$ .

Problem 3579

A luminous point is located at the origin of coordinates. If $x_{0}^{2}+y_{0}^{2}+z_{0}^{2}>R^{2}$ , find the shadow cone produced by the projection of the sphere

(x-x_{0})^{2}+(y-y_{0})^{2}+(z-z_{0})^{2} \leqslant R^{2}

Problem 3580

If the parameters $p$ and $q$ are subject to the equation

p^{2}+q^{2}=1

find the envelope surface of the family of planes

z-z_{0}=p(x-x_{0})+q(y-y_{0})

Section 6.6Taylor's Formula[40/100%]

Problem 3581

Expand the function

f(x,y)=2x^{2}-xy-y^{2}-6x-3y+5

by Taylor's formula in a neighborhood of the point $A(1,-2)$ .

Problem 3582

Expand the function

f(x,y,z)=x^{3}+y^{3}+z^{3}-3xyz

by Taylor's formula in a neighborhood of the point $A(1,1,1)$ .

Problem 3583

When going from $x=1$ , $y=-1$ to $x_{1}=1+h$ , $y_{1}=-1+k$ , find the increment of the function $f(x,y)=x^{2}y+xy^{2}-2xy$ .

Problem 3584

Let

f(x,y,z)=Ax^{2}+By^{2}+Cz^{2}+2Dxy+2Exz+2Fyz ,

expand $f(x+h, y+k, z+l)$ in positive integer powers of the numbers $h,k$ and $l$ .

Problem 3585

Write the expansion of the function

f(x,y)=x^{y}

in a neighborhood of the point $A(1,1)$ , up to and including the quadratic terms.

Problem 3586

Using Maclaurin's formula, expand the function

f(x,y)=\sqrt{1-x^{2}-y^{2}}

up to and including the fourth-order terms.

Problem 3587

(a)(b)

If $\left|x\right|$ and $\left|y\right|$ are small quantities compared to $1$ , derive approximation formulas accurate to the quadratic terms for the following two expressions

(a)

$\dfrac {\cos x}{\cos y}$ ;

(b)

$\operatorname {arctg}\dfrac {1+x+y}{1-x+y}$

Problem 3588

Assuming that the absolute values of $x,y,z$ are small quantities, simplify the expression

\cos (x+y+z)-\cos x\cos y\cos z .

Problem 3589

Expand the function

F(x,y)=\dfrac {1}{4}\left[f(x+h,y)+f(x,y+h)+f(x-h,y)+f(x,y-h)\right]-f(x,y)

in powers of $h$ , accurate to $h^{4}$ .

Problem 3590

Consider a circle centered at the point $P(x,y)$ with radius $\rho$ , let $f(P)=f(x,y)$ and let $P_{i}(x_{i},y_{i})$ $(i=1,2,3)$ be the vertices of an equilateral triangle inscribed in this circle, with $x_{1}=x+\rho$ , $y_{1}=y$ . Expand the function

F(\rho )=\dfrac {1}{3}\left[f(P_{1})+f(P_{2})+f(P_{3})\right]

in positive integer powers of $\rho$ , accurate to $\rho^{2}$ .

Problem 3591

Expand the function

\begin{align} \Delta _{xy}f(x,y) & =f(x+h,y+k)-f(x+h,y) \\ & \quad -f(x,y+k)+f(x,y) \end{align}

in powers of $h$ and $k$ .

Problem 3592

Expand the function

F(\rho )=\dfrac {1}{2\pi }\int _{0}^{2\pi } f(x+\rho \cos \varphi , y+\rho \sin \varphi )\, d\varphi

in powers of $\rho$ .

Problem 3593

Expand the following function into a Maclaurin series: $f(x,y)=(1+x)^{m}(1+y)^{n}$ .

Problem 3594

Expand the following function into a Maclaurin series: $f(x,y)=\ln (1+x+y)$ .

Problem 3595

Expand the following function into a Maclaurin series: $f(x,y)=e^{x}\sin y$ .

Problem 3596

Expand the following function into a Maclaurin series: $f(x,y)=e^{x}\cos y$ .

Problem 3597

Expand the following function into a Maclaurin series: $f(x,y)=\sin x \operatorname {sh}y$ .

Problem 3598

Expand the following function into a Maclaurin series: $f(x,y)=\cos x \operatorname {ch}y$ .

Problem 3599

Expand the following function into a Maclaurin series: $f(x,y)=\sin (x^{2}+y^{2})$ .

Problem 3600

Expand the following function into a Maclaurin series: $f(x,y)=\ln (1+x)\ln (1+y)$ .

Problem 3601

Write the first three nonzero terms of the Maclaurin series of the function

f(x,y)=\int _{0}^{1}(1+x)^{t^{2}y}\, dt

Problem 3602

Expand the function $e^{x+y}$ into a power series in positive integer powers of the binomials $x-1$ and $y+1$ .

Problem 3603

Write the Taylor series expansion of the function $f(x,y)=\dfrac {x}{y}$ in a neighborhood of the point $M(1,1)$ .

Problem 3604

Let $z$ be the implicit function of $x$ and $y$ defined by the equation $z^{3}-2xz+y=0$ , whose value is $z=1$ when $x=1$ and $y=1$ .

Write several terms of the expansion of the function $z$ arranged in ascending powers of the binomials $x-1$ and $y-1$ .

Problem 3605

Investigate the type of singular point of the following curve and roughly sketch the curve: $y^{2}=ax^{2}+x^{3}$ .

Problem 3606

Investigate the type of singular point of the following curve and roughly sketch the curve: $x^{3}+y^{3}-3xy=0$ .

Problem 3607

Investigate the type of singular point of the following curve and roughly sketch the curve: $x^{2}+y^{2}=x^{4}+y^{4}$ .

Problem 3608

Investigate the types of singular points of the following curves and roughly sketch these curves: $x^{2}+y^{4}=x^{6}$ .

Problem 3609

Investigate the types of singular points of the following curves and roughly sketch these curves: $(x^{2}+y^{2})^{2}=a^{2}(x^{2}-y^{2})$ .

Problem 3610

Investigate the types of singular points of the following curves and roughly sketch these curves: $(y-x^{2})^{2}=x^{5}$ .

Problem 3611

Investigate the types of singular points of the following curves and roughly sketch these curves: $(a+x)y^{2}=(a-x)x^{2}$ .

Problem 3612

Investigate the relationship between the values of the parameters $a,b,c$ $(a \leqslant b \leqslant c)$ and the shape of the curve $y^{2}=(x-a)(x-b)(x-c)$ .

Problem 3613

Investigate the singular points of the transcendental curve: $y^{2}=1-e^{-x^{2}}$ .

Problem 3614

Investigate the singular points of the transcendental curve: $y^{2}=1-e^{-x^{3}}$ .

Problem 3615

Investigate the singular points of the transcendental curve: $y=x\ln x$ .

Problem 3616

Investigate the singular points of the transcendental curve: $y=\dfrac {x}{1+e^{\frac{1}{x}}}$ .

Problem 3617

Investigate the singular points of the transcendental curve: $y=\operatorname {arctg}\left(\dfrac {1}{\sin x}\right)$ .

Problem 3618

Investigate the singular points of the transcendental curve: $y^{2}=\sin \dfrac {\pi }{x}$ .

Problem 3619

Investigate the singular points of the transcendental curve: $y^{2}=\sin x^{2}$ .

Problem 3620

Investigate the singular points of the transcendental curve: $y^{2}=\sin^{3}x$ .

Section 6.7Extrema of Functions of Several Variables[93/97%]

Problem 3621

Investigate the extrema of the following function of several variables: $z=x^{2}+(y-1)^{2}$ .

Problem 3622

Investigate the extrema of the following function of several variables: $z=x^{2}-(y-1)^{2}$ .

Problem 3623

Investigate the extrema of the following function of several variables: $z=(x-y+1)^{2}$ .

Problem 3624

Investigate the extrema of the following function of several variables: $z=x^{2}-xy+y^{2}-2x+y$ .

Problem 3625

Investigate the extrema of the following function of several variables: $z=x^{2}y^{3}(6-x-y)$ .

Problem 3626

Investigate the extrema of the following function of several variables: $z=x^{3}+y^{3}-3xy$ .

Problem 3627

Investigate the extrema of the following function of several variables: $z=x^{4}+y^{4}-x^{2}-2xy-y^{2}$ .

Problem 3627.1

[NO TASK]

Problem 3627.1

[NO TASK]

Problem 3628

Investigate the extrema of the following function of several variables: $z=xy+\dfrac {50}{x}+\dfrac {20}{y}$ $(x>0, y>0)$ .

Problem 3629

Investigate the extrema of the following function of several variables: $z=xy\sqrt{1-\dfrac {x^{2}}{a^{2}}-\dfrac {y^{2}}{b^{2}}}$ $(a>0, b>0)$ .

Problem 3630

Investigate the extrema of the following function of several variables: $z=\dfrac {ax+by+c}{\sqrt{x^{2}+y^{2}+1}}$ $(a^{2}+b^{2}+c^{2} \neq 0)$ .

Problem 3631

Investigate the extrema of the following function of several variables: $z=1-\sqrt{x^{2}+y^{2}}$ .

Problem 3632

Investigate the extrema of the following function of several variables: $z=e^{2x+3y}(8x^{2}-6xy+3y^{2})$ .

Problem 3633

Investigate the extrema of the following function of several variables: $z=e^{x^{2}-y}(5-2x+y)$ .

Problem 3634

Investigate the extrema of the following function of several variables: $z=(5x+7y-25)e^{-(x^{2}+xy+y^{2})}$ .

Problem 3635

Investigate the extrema of the following function of several variables: $z=x^{2}+xy+y^{2}-4\ln x-10\ln y$ .

Problem 3636

Investigate the extrema of the following function of several variables: $z=\sin x+\cos y+\cos (x-y)$ $\left(0 \leqslant x \leqslant \dfrac {\pi }{2}; 0 \leqslant y \leqslant \dfrac {\pi }{2}\right)$ .

Problem 3637

Investigate the extrema of the following function of several variables: $z=\sin x \sin y \sin (x+y)$ $(0 \leqslant x \leqslant \pi ; 0 \leqslant y \leqslant \pi )$ .

Problem 3638

Investigate the extrema of the following function of several variables: $z=x-2y+\ln \sqrt{x^{2}+y^{2}}+3\operatorname {arctg}\dfrac {y}{x}$ .

Problem 3639

Investigate the extrema of the following function of several variables: $z=xy\ln (x^{2}+y^{2})$ .

Problem 3640

Investigate the extrema of the following function of several variables: $z=x+y+4\sin x\sin y$ .

Problem 3641

Investigate the extrema of the following function of several variables: $z=(x^{2}+y^{2})e^{-(x^{2}+y^{2})}$ .

Problem 3642

Investigate the extrema of the following function of several variables: $u=x^{2}+y^{2}+z^{2}+2x+4y-6z$ .

Problem 3643

Investigate the extrema of the following function of several variables: $u=x^{3}+y^{2}+z^{2}+12xy+2z$ .

Problem 3644

Investigate the extrema of the following function of several variables: $u=x+\dfrac {y^{2}}{4x}+\dfrac {z^{2}}{y}+\dfrac {2}{z}$ $(x>0, y>0, z>0)$ .

Problem 3645

Investigate the extrema of the following function of several variables: $u=xy^{2}z^{3}(a-x-2y-3z)$ $(a>0)$ .

Problem 3646

Investigate the extrema of the following function of several variables: $u=\dfrac {a^{2}}{x}+\dfrac {x^{2}}{y}+\dfrac {y^{2}}{z}+\dfrac {z^{2}}{b}$ $(x>0, y>0, z>0, a>0, b>0)$ .

Problem 3647

Investigate the extrema of the following function of several variables:

u=\sin x+\sin y+\sin z-\sin (x+y+z)

$(0 \leqslant x \leqslant \pi ; 0 \leqslant y \leqslant \pi ; 0 \leqslant z \leqslant \pi )$ .

Problem 3648

Investigate the extrema of the following function of several variables:

u=x_{1}x_{2}^{2}\cdots x_{n}^{n}(1-x_{1}-2x_{2}-\cdots -nx_{n})

$(x_{1}>0, x_{2}>0, \ldots , x_{n}>0)$ .

Problem 3649

Investigate the extrema of the following function of several variables:

u=x_{1}+\dfrac {x_{2}}{x_{1}}+\dfrac {x_{3}}{x_{2}}+\cdots +\dfrac {x_{n}}{x_{n-1}}+\dfrac {2}{x_{n}}

$(x_{i}>0, i=1,2,\ldots ,n)$ .

Problem 3650

Between two positive numbers $a$ and $b$ , insert $n$ numbers $x_{1}, x_{2}, \ldots , x_{n}$ so that the fraction

u=\dfrac {x_{1}x_{2}\cdots x_{n}}{(a+x_{1})(x_{1}+x_{2})\cdots (x_{n}+b)}

attains its maximum value.

Problem 3651

Find the extrema of the implicit function $z$ of the variables $x$ and $y$ : $x^{2}+y^{2}+z^{2}-2x+2y-4z-10=0$ .

Problem 3652

Find the extrema of the implicit function $z$ of the variables $x$ and $y$ : $x^{2}+y^{2}+z^{2}-xz-yz+2x+2y+2z-2=0$ .

Problem 3653

Find the extrema of the implicit function $z$ of the variables $x$ and $y$ : $(x^{2}+y^{2}+z^{2})^{2}=a^{2}(x^{2}+y^{2}-z^{2})$ .

Problem 3654

Find the conditional extremum points of the following function: $z=xy$ , if $x+y=1$ .

Problem 3655

Find the conditional extremum points of the following function: $z=\dfrac {x}{a}+\dfrac {y}{b}$ , if $x^{2}+y^{2}=1$ .

Problem 3656

Find the conditional extremum points of the following function: $z=x^{2}+y^{2}$ , if $\dfrac {x}{a}+\dfrac {y}{b}=1$ .

Problem 3657

Find the conditional extremum points of the following function: $z=Ax^{2}+2Bxy+Cy^{2}$ , if $x^{2}+y^{2}=1$ .

Problem 3657.1

[NO TASK]

Problem 3657.1

[NO TASK]

Problem 3658

Find the conditional extremum points of the following function: $z=\cos^{2}x+\cos^{2}y$ , if $x-y=\dfrac {\pi }{4}$ .

Problem 3659

Find the conditional extremum points of the following function: $u=x-2y+2z$ , if $x^{2}+y^{2}+z^{2}=1$ .

Problem 3660

Find the conditional extremum points of the following function: $u=x^{m}y^{n}z^{p}$ , if $x+y+z=a$ $(m>0, n>0, p>0, a>0)$ .

Problem 3661

Find the conditional extremum points of the following function: $u=x^{2}+y^{2}+z^{2}$ , if $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}+\dfrac {z^{2}}{c^{2}}=1$ $(a>b>c>0)$ .

Problem 3662

Find the conditional extremum points of the following function: $u=xy^{2}z^{3}$ , if $x+2y+3z=a$ $(x>0, y>0, z>0, a>0)$ .

Problem 3663

Find the conditional extremum points of the following function: $u=xyz$ , if $x^{2}+y^{2}+z^{2}=1$ , $x+y+z=0$ .

Problem 3663.1

[NO TASK]

Problem 3663.1

[NO TASK]

Problem 3664

Find the conditional extremum points of the following function: $u=\sin x \sin y \sin z$ , if $x+y+z=\dfrac {\pi }{2}$ $(x>0, y>0, z>0)$ .

Problem 3665

Find the conditional extrema of the following function: $u=\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}+\dfrac {z^{2}}{c^{2}}$ , subject to $x^{2}+y^{2}+z^{2}=1$ , $x\cos \alpha +y\cos \beta +z\cos \gamma =0$ $(a>b>c>0$ , $\cos^{2}\alpha +\cos^{2}\beta +\cos^{2}\gamma =1)$ .

Problem 3666

Find the conditional extrema of the following function: $u=(x-\xi )^{2}+(y-\eta )^{2}+(z-\zeta )^{2}$ , subject to $Ax+By+Cz=0$ , $x^{2}+y^{2}+z^{2}=R^{2}$ , $\dfrac {\xi }{\cos \alpha }=\dfrac {\eta }{\cos \beta }=\dfrac {\zeta }{\cos \gamma }$ , where $\cos^{2}\alpha +\cos^{2}\beta +\cos^{2}\gamma =1$ .

Problem 3667

Find the conditional extrema of the following function: $u=x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}$ , subject to $\dfrac {x_{1}}{a_{1}}+\dfrac {x_{2}}{a_{2}}+\cdots +\dfrac {x_{n}}{a_{n}}=1$ $(a_{i}>0 ; i=1,2,\ldots ,n)$ .

Problem 3668

Find the conditional extrema of the following function: $u=x_{1}^{p}+x_{2}^{p}+\cdots +x_{n}^{p}$ $(p>1)$ , subject to $x_{1}+x_{2}+\cdots +x_{n}=a$ $(a>0)$ .

Problem 3669

Find the conditional extrema of the following function: $u=\dfrac {\alpha_{1}}{x_{1}}+\dfrac {\alpha_{2}}{x_{2}}+\cdots +\dfrac {\alpha_{n}}{x_{n}}$ , subject to $\beta_{1}x_{1}+\beta_{2}x_{2}+\cdots +\beta_{n}x_{n}=1$ $(\alpha_{i}>0, \beta_{i}>0 ; i=1,2,\ldots ,n)$ .

Problem 3670

Find the conditional extrema of the following function: $u=x_{1}^{\alpha_{1}}x_{2}^{\alpha_{2}}\cdots x_{n}^{\alpha_{n}}$ , subject to $x_{1}+x_{2}+\cdots +x_{n}=a$ $(a>0, \alpha_{i}>1, i=1,2,\ldots ,n)$ .

Problem 3671

If $\displaystyle \sum_{i=1}^{n}x_{i}^{2}=1$ , find the extrema of the quadratic form $u=\displaystyle \sum_{i,j=1}^{n}a_{ij}x_{i}x_{j}$ $(a_{ij}=a_{ji})$ .

Problem 3672

If $n \geqslant 1$ and $x \geqslant 0$ , $y \geqslant 0$ , prove the inequality

\dfrac {x^{n}+y^{n}}{2} \geqslant \left(\dfrac {x+y}{2}\right)^{n} .

Problem 3673

Prove Hölder's inequality

\sum _{i=1}^{n}a_{i}x_{i} \leqslant \left(\sum _{i=1}^{n}a_{i}^{k}\right)^{\frac{1}{k}}\left(\sum _{i=1}^{n}x_{i}^{k'}\right)^{\frac{1}{k'}}

$(a_{i} \geqslant 0, x_{i} \geqslant 0, i=1,2,\ldots ,n$ ; $k>1$ , $\dfrac {1}{k}+\dfrac {1}{k’}=1)$ .

Problem 3674

For the $n$ -th order determinant $A=\left|a_{ij}\right|$ , prove Hadamard's inequality

A^{2} \leqslant \prod _{i=1}^{n}\left(\sum _{j=1}^{n}a_{ij}^{2}\right) .

Problem 3675

Find the supremum (sup) and infimum (inf) of the following function on the specified domain: $z=x-2y-3$ , if $0 \leqslant x \leqslant 1$ , $0 \leqslant y \leqslant 1$ , $0 \leqslant x+y \leqslant 1$ .

Problem 3676

Find the supremum (sup) and infimum (inf) of the following function on the specified domain: $z=x^{2}+y^{2}-12x+16y$ , if $x^{2}+y^{2} \leqslant 25$ .

Problem 3677

Find the supremum (sup) and infimum (inf) of the following function in the indicated domain: $z=x^{2}-xy+y^{2}$ , if $\left|x\right|+\left|y\right| \leqslant 1$ .

Problem 3678

Find the supremum (sup) and infimum (inf) of the following function in the indicated domain: $u=x^{2}+2y^{2}+3z^{2}$ , if $x^{2}+y^{2}+z^{2} \leqslant 100$ .

Problem 3679

Find the supremum (sup) and infimum (inf) of the following function in the indicated domain: $u=x+y+z$ , if $x^{2}+y^{2} \leqslant z \leqslant 1$ .

Problem 3680

Find the infimum (inf) and supremum (sup) of the function $u=(x+y+z)e^{-(x+2y+3z)}$ in the domain $x \geqslant 0$ , $y \geqslant 0$ , $z \geqslant 0$ .

Problem 3681

Prove that the function $z=(1+e^{y})\cos x-ye^{y}$ has infinitely many local maxima and not a single local minimum.

Problem 3682

Is it a sufficient condition for the function $f(x,y)$ to have a local minimum at the point $M_{0}(x_{0},y_{0})$ that this function have a local minimum along every straight line through $M_{0}$ ?

Problem 3683

Decompose a given positive number $a$ into $n$ positive factors so that the sum of their reciprocals is a minimum.

Problem 3684

Decompose a given positive number $a$ into $n$ addends so that the sum of their squares is a minimum.

Problem 3685

Decompose a given positive number $a$ into $n$ positive factors so that the sum of their given positive powers is a minimum.

Problem 3686

In the plane there are given $n$ mass points $P_{1}(x_{1},y_{1})$ , $P_{2}(x_{2},y_{2})$ , $\ldots$ , $P_{n}(x_{n},y_{n})$ , with masses $m_{1}$ , $m_{2}$ , $\ldots$ , $m_{n}$ respectively. At what position of the point $P(x,y)$ is the moment of inertia of this system with respect to that point smallest?

Problem 3687

An open-topped rectangular bathtub has volume $V$ . What dimensions give it the smallest surface area?

Problem 3688

An open cylindrical bathtub with a semicircular cross-section has surface area $S$ . What dimensions give it the largest volume?

Problem 3689

On the sphere $x^{2}+y^{2}+z^{2}=1$ find a point such that the sum of the squares of its distances to $n$ given points $M_{i}(x_{i},y_{i},z_{i})$ $(i=1,2,\ldots ,n)$ is smallest.

Problem 3690

A solid is formed by a right circular cylinder capped with a right circular cone. Given that the total surface area of the solid equals $Q$ , find its dimensions so that the volume of the solid is largest.

Problem 3691

A solid of volume $V$ has the shape of a right rectangular prism whose top and bottom faces are capped by regular quadrilateral pyramids. At what inclination of the pyramids' lateral faces to their bases is the total surface area of the solid smallest?

Problem 3692

A rectangle has perimeter $2p$ . It is rotated about one of its sides to generate a solid of revolution. Find the rectangle that gives the largest volume.

Problem 3693

A triangle has perimeter $2p$ . Find the triangle such that, when rotated about one of its own sides, the resulting volume is largest.

Problem 3694

Inscribe in a hemisphere of radius $R$ a right rectangular prism of maximum volume.

Problem 3695

Inscribe in a given right circular cone a rectangular parallelepiped of maximum volume.

Problem 3696

Inscribe in the ellipsoid

\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}+\dfrac {z^{2}}{b^{2}}=1

a rectangular parallelepiped of maximum volume.

Problem 3697

The slant generator $l$ of a right circular cone makes an angle $\alpha$ with the base plane. Inscribe in this cone a rectangular parallelepiped of maximum total surface area.

Problem 3698

Inscribe in the segment of the elliptic paraboloid $\dfrac {z}{c}=\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}$ , $z=c$ , a rectangular parallelepiped of maximum volume.

Problem 3699

Find the shortest distance from the point $M_{0}(x_{0},y_{0},z_{0})$ to the plane $Ax+By+Cz+D=0$ .

Problem 3700

Find the shortest distance between the two lines in space $\dfrac {x-x_{1}}{m_{1}}=\dfrac {y-y_{1}}{n_{1}}=\dfrac {z-z_{1}}{p_{1}}$ and $\dfrac {x-x_{2}}{m_{2}}=\dfrac {y-y_{2}}{n_{2}}=\dfrac {z-z_{2}}{p_{2}}$ .

Problem 3701

Find the shortest distance between the parabola $y=x^{2}$ and the line $x-y-2=0$ .

Problem 3702

Find the semi-axes of the central conic

Ax^{2}+2Bxy+Cy^{2}=1

Problem 3703

Find the semi-axes of the central quadric surface

Ax^{2}+By^{2}+Cz^{2}+2Dxy+2Eyz+2Fxz=1

Problem 3704

Find the area of the ellipse formed by the intersection of the plane $Ax+By+Cz=0$ with the cylinder $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}=1$ .

Problem 3705

Find the area of the cross-section cut from the ellipsoid $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}+\dfrac {z^{2}}{c^{2}}=1$ by the plane $x\cos \alpha +y\cos \beta +z\cos \gamma =0$ (where $\cos^{2}\alpha +\cos^{2}\beta +\cos^{2}\gamma =1$ ).

Problem 3706

By Fermat's principle, a ray of light leaving point $A$ and arriving at point $B$ travels along the curve requiring the least time. Suppose points $A$ and $B$ lie in different optical media separated by a plane, and that the speed of light propagation is $v_{1}$ in the first medium and $v_{2}$ in the second medium. Derive the law of refraction of light.

Problem 3707

For what angle of incidence is the deviation of a ray of light (that is, the angle between the incident ray and the emergent ray) a minimum, where the ray passes through a prism of refracting angle $\alpha$ and refractive index $n$ ? Find this minimum deviation.

Problem 3708

The variables $x$ and $y$ satisfy the linear equation $y=ax+b$ , whose coefficients need to be determined. From a series of equally precise measurements, the values $x_{i}$ , $y_{i}$ $(i=1,2,\ldots ,n)$ are obtained for the quantities $x$ and $y$ . Using the method of least squares, find the most reliable values of the coefficients $a$ and $b$ .

Problem 3709

In the plane, $n$ points $M_{i}(x_{i},y_{i})$ $(i=1,2,\ldots ,n)$ are given. In what position must the line $x\cos \alpha +y\sin \alpha -p=0$ be so that the sum of squared deviations of the given points from this line is a minimum?

Problem 3710

On the interval $(1,3)$ , approximately replace the function $x^{2}$ by a linear function $ax+b$ so that the absolute deviation

\Delta =\sup \left|x^{2}-(ax+b)\right| \quad (1 \leqslant x \leqslant 3)

is minimal.