Math Problems Vault

Problem 3211

Prove that

f'_{x}(x,b)=\dfrac {d}{dx}\left[f(x,b)\right] .

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Problem 3212

Let

f(x,y)=x+(y-1)\arcsin \sqrt{\dfrac {x}{y}} ,

find $f'_{x}(x,1)$ .

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Problem 3212.1

[NO TASK]

Problem 3212.2

[NO TASK]

Problem 3212.3

[NO TASK]

Problem 3213

Find the first- and second-order partial derivatives of the following function: $u=x^{4}+y^{4}-4x^{2}y^{2}$ .

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Problem 3214

Find the first- and second-order partial derivatives of the following function: $u=xy+\dfrac {x}{y}$ .

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Problem 3215

Find the first- and second-order partial derivatives of the following function: $u=\dfrac {x}{y^{2}}$ .

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Problem 3216

Find the first- and second-order partial derivatives of the following function: $u=\dfrac {x}{\sqrt{x^{2}+y^{2}}}$ .

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Problem 3217

Find the first- and second-order partial derivatives of the following function: $u=x\sin (x+y)$ .

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Problem 3218

Find the first- and second-order partial derivatives of the following function: $u=\dfrac {\cos x^{2}}{y}$ .

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Problem 3219

Find the first- and second-order partial derivatives of the following function: $u=\tan \dfrac {x^{2}}{y}$ .

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Problem 3220

Find the first- and second-order partial derivatives of the following function: $u=x^{y}$ .

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Problem 3221

Find the first- and second-order partial derivatives of the following function: $u=\ln (x+y^{2})$ .

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Problem 3222

Find the first- and second-order partial derivatives of the following function: $u=\arctan \dfrac {y}{x}$ .

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Problem 3223

Find the first- and second-order partial derivatives of the following function: $u=\arctan \dfrac {x+y}{1-xy}$ .

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Problem 3224

Find the first- and second-order partial derivatives of the following function: $u=\arcsin \dfrac {x}{\sqrt{x^{2}+y^{2}}}$ .

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Problem 3225

Find the first- and second-order partial derivatives of the following function: $u=\dfrac {1}{\sqrt{x^{2}+y^{2}+z^{2}}}$ .

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Problem 3226

Find the first- and second-order partial derivatives of the following function: $u=\left(\dfrac {x}{y}\right)^{z}$ .

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Problem 3227

Find the first- and second-order partial derivatives of the following function: $u=x^{\frac{y}{z}}$ .

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Problem 3228

Find the first- and second-order partial derivatives of the following function: $u=x^{y^{z}}$ .

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Problem 3229

(a)(b)(c)

Let (1) $u=x^{2}-2xy-3y^{2}$ ; (2) $u=x^{y^{2}}$ ; (3) $u=\arccos \sqrt{\dfrac {x}{y}}$ , verify the identity

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

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(a)

(b)

(c)

Problem 3230

Let $f(x,y)=xy\dfrac {x^{2}-y^{2}}{x^{2}+y^{2}}$ , if $x^{2}+y^{2} \neq 0$ and $f(0,0)=0$ . Prove that

f''_{xy}(0,0) \neq f''_{yx}(0,0) .

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Problem 3230.1

[NO TASK]

Problem 3231

(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

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

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 3234

Let $u=f(x,y,z)$ be a twice differentiable homogeneous function of degree $n$ . Prove that

\left(x\dfrac {\partial }{\partial x}+y\dfrac {\partial }{\partial y}+z\dfrac {\partial }{\partial z}\right)^{2} u=n(n-1)u .

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Problem 3235

Find the first- and second-order differentials of the following functions ( $x, y, z$ are independent variables): $u=x^{m}y^{n}$ .

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Problem 3236

Find the first- and second-order differentials of the following functions ( $x, y, z$ are independent variables): $u=\dfrac {x}{y}$ .

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Problem 3237

Find the first- and second-order differentials of the following functions ( $x, y, z$ are independent variables): $u=\sqrt{x^{2}+y^{2}}$ .

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Problem 3238

Find the first- and second-order differentials of the following functions ( $x, y, z$ are independent variables): $u=\ln \sqrt{x^{2}+y^{2}}$ .

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Problem 3239

Find the first- and second-order differentials of the following functions ( $x, y, z$ are independent variables): $u=e^{xy}$ .

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Problem 3240

Find the first- and second-order differentials of the following functions ( $x, y, z$ are independent variables): $u=xy+yz+zx$ .

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Problem 3241

Find the first- and second-order differentials of the following functions ( $x, y, z$ are independent variables): $u=\dfrac {z}{x^{2}+y^{2}}$ .

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Problem 3242

Let $f(x,y,z)=z\sqrt{\dfrac {x}{y}}$ , find $df(1,1,1)$ and $d^{2}f(1,1,1)$ .

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Problem 3243

Prove that if

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

then

d^{2}u \geqslant 0 .

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Problem 3244

(a)(b)(c)

Assuming the absolute values of $x, y$ are very small, derive approximate formulas for each of the following expressions:

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(a)

$(1+x)^{m}(1+y)^{n}$ ;

(b)

$\ln (1+x) \cdot \ln (1+y)$ ;

(c)

$\arctan \dfrac {x+y}{1+xy}$ .

Problem 3245

Using the differential to replace the increment of the function, compute approximately:

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(a)

$1.002 \cdot 2.003^{2} \cdot 3.004^{3}$ ;

(b)

$\dfrac {1.03^{2}}{\sqrt[3]{0.98}\sqrt[4]{1.05^{3}}}$ ;

(c)

$\sqrt{1.02^{3}+1.97^{3}}$ ;

(d)

$\sin 29^{\circ }\tan 46^{\circ }$ ;

(e)

$0.97^{1.05}$ .

Problem 3246

Let the sides of a rectangle be $x=6$ meters and $y=8$ meters. If the first side is increased by 2 millimeters and the second side is decreased by 5 millimeters, by how much do the diagonal and area of the rectangle change?

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Problem 3247

The central angle $\alpha =60^{\circ }$ of a sector is increased by $\Delta \alpha =1^{\circ }$ . In order to keep the area of the sector unchanged, by how much should the radius $R=20$ centimeters of the sector be decreased?

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Problem 3248

Prove that the relative error of a product is approximately equal to the sum of the relative errors of the factors.

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Problem 3249

When measuring the base radius $R$ and height $H$ of a cylinder, the following results are obtained:

R=2.5 \text{meters} \pm 0.1 \text{meters} ; \quad H=4.0 \text{meters} \pm 0.2 \text{meters} ,

what absolute error $\Delta$ and relative error $\delta$ can the computed volume of the cylinder have?

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Problem 3250

The sides of a triangle are $a=200$ meters $\pm 2$ meters, $b=300$ meters $\pm 5$ meters, and the angle between them is $C=60^{\circ } \pm 1^{\circ }$ ; what absolute error can the computed third side $c$ of the triangle have?

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Problem 3251

Prove that the function

f(x,y)=\sqrt{\left|xy\right|}

which is continuous at the point $(0,0)$ , has the two partial derivatives $f'_{x}(0,0)$ and $f'_{y}(0,0)$ at the point $(0,0)$ , but is not differentiable at the point $(0,0)$ .

Describe the behavior of the derivative functions $f'_{x}(x,y)$ and $f'_{y}(x,y)$ in a neighborhood of the point $(0,0)$ .

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Problem 3252

Prove that the function

f(x,y)=\dfrac {xy}{\sqrt{x^{2}+y^{2}}}, \text{if} \ x^{2}+y^{2} \neq 0 \text{, and} f(0,0)=0 ,

is continuous in a neighborhood of the point $(0,0)$ and has bounded partial derivatives $f'_{x}(x,y)$ and $f'_{y}(x,y)$ , but this function is not differentiable at the point $(0,0)$ .

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Problem 3253

Prove that the function

f(x,y)=(x^{2}+y^{2})\sin \dfrac {1}{x^{2}+y^{2}}, \text{if} \ x^{2}+y^{2} \neq 0

and $f(0,0)=0$

has partial derivatives $f'_{x}(x,y)$ and $f'_{y}(x,y)$ in a neighborhood of the point $(0,0)$ ; these partial derivatives are discontinuous at the point $(0,0)$ and are unbounded in any neighborhood of this point; however, this function is differentiable at the point $(0,0)$ .

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Problem 3254

Prove: a function $f(x,y)$ having bounded partial derivatives $f'_{x}(x,y)$ and $f'_{y}(x,y)$ in some convex domain $E$ is uniformly continuous in the domain $E$ .

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Problem 3255

Prove: if the function $f(x,y)$ is continuous in the variable $x$ (for each fixed value of $y$ ) and has a bounded derivative $f'_{y}(x,y)$ with respect to the variable $y$ , then this function is continuous in the variables $x$ and $y$ jointly.

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Problem 3256

In each of the following problems find the indicated partial derivative: find $\dfrac {\partial^{4}u}{\partial x^{4}}$ , $\dfrac {\partial^{4}u}{\partial x^{3}\partial y}$ , $\dfrac {\partial^{4}u}{\partial x^{2}\partial y^{2}}$ , if

u=x-y+x^{2}+2xy+y^{2}+x^{3}-3x^{2}y-y^{3}+x^{4}-4x^{2}y^{2}+y^{4} .

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Problem 3257

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

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Problem 3258

Find $\dfrac {\partial^{6} u}{\partial x^{3}\partial y^{3}}$ , if $u=x^{3}\sin y+y^{3}\sin x$ .

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Problem 3259

Find $\dfrac {\partial^{3} u}{\partial x \partial y \partial z}$ , if $u=\arctan \dfrac {x+y+z-xyz}{1-xy-xz-yz}$ .

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Problem 3260

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

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Problem 3261

Find $\dfrac {\partial^{4} u}{\partial x \partial y \partial \xi \partial \eta }$ , if $u=\ln \dfrac {1}{\sqrt{(x-\xi )^{2}+(y-\eta )^{2}}}$ .

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Problem 3262

Find $\dfrac {\partial^{p+q} u}{\partial x^{p}\partial y^{q}}$ , if $u=(x-x_{0})^{p}(y-y_{0})^{q}$ .

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Problem 3263

Find $\dfrac {\partial^{m+n} u}{\partial x^{m}\partial y^{n}}$ , if $u=\dfrac {x+y}{x-y}$ .

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Problem 3264

Find $\dfrac {\partial^{m+n} u}{\partial x^{m}\partial y^{n}}$ , if $u=(x^{2}+y^{2})e^{x+y}$ .

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Problem 3265

Find $\dfrac {\partial^{p+q+r} u}{\partial x^{p}\partial y^{q}\partial z^{r}}$ , if $u=xyze^{x+y+z}$ .

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Problem 3266

If $f(x,y)=e^{x}\sin y$ , find $f^{(m+n)}_{x^{m}y^{n}}(0,0)$ .

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Problem 3267

Prove: if

u=f(xyz) ,

then

\dfrac {\partial ^{3} u}{\partial x \partial y \partial z}=F(t) ,

where $t=xyz$ , and find the function $F$ .

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Problem 3268

Let $u=x^{4}-2x^{3}y-2xy^{3}+y^{4}+x^{3}-3x^{2}y-3xy^{2}+y^{3}+2x^{2}-xy+2y^{2}+x+y+1$ ; find $d^{4}u$ , and the derivatives $\dfrac {\partial^{4}u}{\partial x^{4}}$ , $\dfrac {\partial^{4}u}{\partial x^{3}\partial y}$ , $\dfrac {\partial^{4}u}{\partial x^{2}\partial y^{2}}$ , $\dfrac {\partial^{4}u}{\partial x \partial y^{3}}$ and $\dfrac {\partial^{4}u}{\partial y^{4}}$ — what are they equal to?

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Problem 3269

Find the differential of the indicated order: $d^{3}u$ , if $u=x^{3}+y^{3}-3xy(x-y)$ .

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Problem 3270

Find the differential of the indicated order: $d^{3}u$ , if $u=\sin (x^{2}+y^{2})$ .

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Problem 3271

Find the differential of the indicated order: $d^{10}u$ , if $u=\ln (x+y)$ .

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Problem 3272

Find the differential of the indicated order: $d^{6}u$ , if $u=\cos x \cosh y$ .

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Problem 3273

Find the differential of the indicated order: $d^{3}u$ , if $u=xyz$ .

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Problem 3274

Find the differential of the indicated order: $d^{4}u$ , if $u=\ln (x^{x}y^{y}z^{z})$ .

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Problem 3275

Find the differential of the indicated order: $d^{n}u$ , if $u=e^{ax+by}$ .

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Problem 3276

Find the differential of the indicated order: $d^{n}u$ , if $u=X(x)Y(y)$ .

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Problem 3277

Find the differential of the indicated order: $d^{n}u$ , if $u=f(x+y+z)$ .

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Problem 3278

Find the differential of the indicated order: $d^{n}u$ , if $u=e^{ax+by+cz}$ .

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Problem 3279

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 3280

(a)(b)

Let:

Au=x\dfrac {\partial u}{\partial x}+y\dfrac {\partial u}{\partial y} .

Find $Au$ and $A^{2}u=A(Au)$ , if

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(a)

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

(b)

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

Problem 3281

(a)(b)

Let:

\Delta u=\dfrac {\partial ^{2} u}{\partial x^{2}}+\dfrac {\partial ^{2} u}{\partial y^{2}} .

Find $\Delta u$ , if

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(a)

$u=\sin x \cosh y$ ;

(b)

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

Problem 3282

(a)(b)

Let:

\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}} .

Find $\Delta_{1}u$ and $\Delta_{2}u$ , if

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(a)

$u=x^{3}+y^{3}+z^{3}-3xyz$ ;

(b)

$u=\dfrac {1}{\sqrt{x^{2}+y^{2}+z^{2}}}$ .

Problem 3283

Find the first- and second-order derivatives of the following composite function: $u=f(x^{2}+y^{2}+z^{2})$ .

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Problem 3284

Find the first- and second-order derivatives of the following composite function: $u=f\left(x, \dfrac {x}{y}\right)$ .

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Problem 3285

Find the first- and second-order derivatives of the following composite function: $u=f(x, xy, xyz)$ .

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Problem 3286

Let $u=f(x+y,xy)$ , find $\dfrac {\partial^{2} u}{\partial x \partial y}$ .

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Problem 3287

Let $u=f(x+y+z, x^{2}+y^{2}+z^{2})$ , find $\Delta u=\dfrac {\partial^{2} u}{\partial x^{2}}+\dfrac {\partial^{2} u}{\partial y^{2}}+\dfrac {\partial^{2} u}{\partial z^{2}}$ .

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Problem 3288

Find the first- and second-order total differentials of the following composite function ( $x, y$ and $z$ are the independent variables): $u=f(t)$ , where $t=x+y$ .

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Problem 3289

Find the first- and second-order total differentials of the following composite function ( $x, y$ and $z$ are the independent variables): $u=f(t)$ , where $t=\dfrac {y}{x}$ .

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Problem 3290

Find the first- and second-order total differentials of the following composite function ( $x, y$ and $z$ are the independent variables): $u=f(\sqrt{x^{2}+y^{2}})$ .

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Problem 3291

Find the first- and second-order total differentials of the following composite function ( $x, y$ and $z$ are the independent variables): $u=f(t)$ , where $t=xyz$ .

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Problem 3292

Find the first- and second-order total differentials of the following composite function ( $x, y$ and $z$ are the independent variables): $u=f(x^{2}+y^{2}+z^{2})$ .

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Problem 3293

Find the first- and second-order total differentials of the following composite function ( $x, y$ and $z$ are the independent variables): $u=f(\xi ,\eta )$ , where $\xi =ax$ , $\eta =by$ .

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Problem 3294

Find the first- and second-order total differentials of the following composite function ( $x, y$ and $z$ are the independent variables): $u=f(\xi ,\eta )$ , where $\xi =x+y$ , $\eta =x-y$ .

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Problem 3295

Find the first- and second-order total differentials of the following composite function ( $x, y$ and $z$ are the independent variables): $u=f(\xi ,\eta )$ , where $\xi =xy$ , $\eta =\dfrac {x}{y}$ .

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Problem 3296

Find the first- and second-order total differentials of the following composite function ( $x, y$ and $z$ are the independent variables): $u=f(x+y,z)$ .

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Problem 3297

Find the first- and second-order total differentials of the following composite function ( $x, y$ and $z$ are the independent variables): $u=f(x+y+z, x^{2}+y^{2}+z^{2})$ .

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Problem 3298

Find the first- and second-order total differentials of the following composite function ( $x, y$ and $z$ are the independent variables): $u=f\left(\dfrac {x}{y}, \dfrac {y}{z}\right)$ .

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Problem 3299

Find the first- and second-order total differentials of the following composite function ( $x, y$ and $z$ are the independent variables): $u=f(x,y,z)$ , where $x=t$ , $y=t^{2}$ , $z=t^{3}$ .

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Problem 3300

Find the first- and second-order total differentials of the following composite function ( $x, y$ and $z$ are the independent variables): $u=f(\xi ,\eta ,\zeta )$ , where $\xi =ax$ , $\eta =by$ , $\zeta =cz$ .

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Problem 3301

Find the first- and second-order total differentials of the following composite function ( $x, y$ and $z$ are the independent variables): $u=f(\xi ,\eta ,\zeta )$ , where $\xi =x^{2}+y^{2}$ , $\eta =x^{2}-y^{2}$ , $\zeta =2xy$ .

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Problem 3302

Find $d^{n}u$ , given: $u=f(ax+by+cz)$ .

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Problem 3303

Find $d^{n}u$ , given: $u=f(ax,by,cz)$ .

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Problem 3304

Find $d^{n}u$ , given: $u=f(\xi ,\eta ,\zeta )$ , where $\xi =a_{1}x+b_{1}y+c_{1}z$ , $\eta =a_{2}x+b_{2}y+c_{2}z$ , $\zeta =a_{3}x+b_{3}y+c_{3}z$ .

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Problem 3305

Let $u=f(r)$ , where $r=\sqrt{x^{2}+y^{2}+z^{2}}$ and $f$ is a twice-differentiable function. Prove that:

\Delta u=F(r) ,

where $\Delta u=\dfrac {\partial^{2} u}{\partial x^{2}}+\dfrac {\partial^{2} u}{\partial y^{2}}+\dfrac {\partial^{2} u}{\partial z^{2}}$ , $\Delta$ is the Laplace operator, and find the function $F$ .

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Problem 3306

Let $u$ and $v$ be twice-differentiable functions and let $\Delta$ be the Laplace operator (see problem 3305: $\Delta u=\dfrac {\partial^{2} u}{\partial x^{2}}+\dfrac {\partial^{2} u}{\partial y^{2}}+\dfrac {\partial^{2} u}{\partial z^{2}}$ ). Prove that:

\Delta (uv)=u\Delta v+v\Delta u+2\Delta (u,v) ,

where $\Delta (u,v)=\dfrac {\partial u}{\partial x}\dfrac {\partial v}{\partial x}+\dfrac {\partial u}{\partial y}\dfrac {\partial v}{\partial y}+\dfrac {\partial u}{\partial z}\dfrac {\partial v}{\partial z}$ .

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Problem 3307

Prove that the function

u=\ln \sqrt{(x-a)^{2}+(y-b)^{2}}

( $a$ and $b$ are constants) satisfies the Laplace equation

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

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Problem 3308

Prove that if the function $u=u(x,y)$ satisfies the Laplace equation (see problem 3307: $\dfrac {\partial^{2} u}{\partial x^{2}}+\dfrac {\partial^{2} u}{\partial y^{2}}=0$ ), then the function

v=u\left(\dfrac {x}{x^{2}+y^{2}}, \dfrac {y}{x^{2}+y^{2}}\right)

also satisfies this equation.

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Problem 3309

Prove that the function

u=\dfrac {1}{2a\sqrt{\pi t}}e^{-\frac{(x-b)^{2}}{4a^{2}t}}

( $a$ and $b$ are constants) satisfies the heat conduction equation

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

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Problem 3310

Prove that if the function $u=u(x,t)$ satisfies the heat conduction equation (see problem 3309: $\dfrac {\partial u}{\partial t}=a^{2}\dfrac {\partial^{2} u}{\partial x^{2}}$ ), then the function

v=\dfrac {1}{a\sqrt{t}}e^{-\frac{x^{2}}{4a^{2}t}}u\left(\dfrac {x}{a^{2}t}, -\dfrac {1}{a^{4}t}\right) \quad (t>0)

also satisfies this equation.

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Problem 3311

Prove that the function

u=\dfrac {1}{r}

(where $r=\sqrt{(x-a)^{2}+(y-b)^{2}+(z-c)^{2}}$ ), when $r \neq 0$ , satisfies the Laplace equation

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

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Problem 3312

Prove that if the function $u=u(x,y,z)$ satisfies the Laplace equation (see problem 3311: $\Delta u=\dfrac {\partial^{2} u}{\partial x^{2}}+\dfrac {\partial^{2} u}{\partial y^{2}}+\dfrac {\partial^{2} u}{\partial z^{2}}=0$ ), then the function

v=\dfrac {1}{r}u\left(\dfrac {k^{2}x}{r^{2}}, \dfrac {k^{2}y}{r^{2}}, \dfrac {k^{2}z}{r^{2}}\right)

(where $k$ is a constant and $r=\sqrt{x^{2}+y^{2}+z^{2}}$ ) also satisfies this equation.

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Problem 3313

Prove that the function

u=\dfrac {C_{1}e^{-ar}+C_{2}e^{ar}}{r}

(where $r=\sqrt{x^{2}+y^{2}+z^{2}}$ and $C_{1}, C_{2}$ are constants) satisfies the Helmholtz equation

\dfrac {\partial ^{2} u}{\partial x^{2}}+\dfrac {\partial ^{2} u}{\partial y^{2}}+\dfrac {\partial ^{2} u}{\partial z^{2}}=a^{2}u .

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Problem 3314

Let the functions $u_{1}=u_{1}(x,y,z)$ and $u_{2}=u_{2}(x,y,z)$ satisfy the Laplace equation (see problem 3311), and suppose the function

v=u_{1}+(x^{2}+y^{2}+z^{2})u_{2}

also satisfies this equation. Prove that

\Delta (\Delta v)=0 .

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Problem 3315

Let $f(x,y,z)$ be an $n$ -th order homogeneous function that is $m$ times differentiable. Prove that

\begin{align} \left(x\dfrac {\partial }{\partial x}+y\dfrac {\partial }{\partial y}+z\dfrac {\partial }{\partial z}\right)^{m} f(x,y,z) \\ & =n(n-1)\cdots (n-m+1)f(x,y,z) . \end{align}

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Problem 3316

If $z=\sin y+f(\sin x-\sin y)$ , where $f$ is a differentiable function. Simplify the expression

\sec x\dfrac {\partial z}{\partial x}+\sec y\dfrac {\partial z}{\partial y} .

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Problem 3317

Prove that the function

z=x^{n}f\left(\dfrac {y}{x^{2}}\right)

(where $f$ is an arbitrary differentiable function) satisfies the equation

x\dfrac {\partial z}{\partial x}+2y\dfrac {\partial z}{\partial y}=nz .

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Problem 3318

Prove that

z=yf(x^{2}-y^{2})

(where $f$ is an arbitrary differentiable function) satisfies the equation

y^{2}\dfrac {\partial z}{\partial x}+xy\dfrac {\partial z}{\partial y}=xz .

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Problem 3319

If

\begin{align} u & =\dfrac {1}{12}x^{4}-\dfrac {1}{6}x^{3}(y+z)+\dfrac {1}{2}x^{2}yz +f(y-x,z-x) , \end{align}

where $f$ is a differentiable function. Simplify the expression

\dfrac {\partial u}{\partial x}+\dfrac {\partial u}{\partial y}+\dfrac {\partial u}{\partial z} .

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Problem 3320

Let:

x^{2}=vw , \quad y^{2}=uw , \quad z^{2}=uv

and

f(x,y,z)=F(u,v,w) ,

Prove that:

xf_{x}'+yf_{y}'+zf_{z}'=uF_{u}'+vF_{v}'+wF_{w}' .

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Problem 3321

Verify the following identity: $y\dfrac {\partial z}{\partial x}-x\dfrac {\partial z}{\partial y}=0$ , if $z=\varphi (x^{2}+y^{2})$ .

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Problem 3322

Verify the following identity: $x^{2}\dfrac {\partial z}{\partial x}-xy\dfrac {\partial z}{\partial y}+y^{2}=0$ , if $z=\dfrac {y^{2}}{3x}+\varphi (xy)$ .

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Problem 3323

Verify the following identity: $(x^{2}-y^{2})\dfrac {\partial z}{\partial x}+xy\dfrac {\partial z}{\partial y}=xyz$ , if $z=e^{x}\varphi \left(ye^{\frac{x^{2}}{2y^{2}}}\right)$ .

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Problem 3324

Verify the following identity: $x\dfrac {\partial u}{\partial x}+\alpha y\dfrac {\partial u}{\partial y}+\beta z\dfrac {\partial u}{\partial z}=nu$ , if $u=x^{n}\varphi \left(\dfrac {y}{x^{\alpha }},\dfrac {z}{x^{\beta }}\right)$ .

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Problem 3325

Verify the following identity: $x\dfrac {\partial u}{\partial x}+y\dfrac {\partial u}{\partial y}+z\dfrac {\partial u}{\partial z}=u+\dfrac {xy}{z}$ , if

u=\dfrac {xy}{z}\ln x+x\varphi \left(\dfrac {y}{x},\dfrac {z}{x}\right) .

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Problem 3326

Verify the following identity: $\dfrac {\partial^{2} u}{\partial t^{2}}=a^{2}\dfrac {\partial^{2} u}{\partial x^{2}}$ , if $u=\varphi (x-at)+\psi (x+at)$ .

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Problem 3327

Verify the following identity: $\dfrac {\partial^{2} u}{\partial x^{2}}-2\dfrac {\partial^{2} u}{\partial x \partial y}+\dfrac {\partial^{2} u}{\partial y^{2}}=0$ , if

u=x\varphi (x+y)+y\psi (x+y) .

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Problem 3328

Verify the following identity: $x^{2}\dfrac {\partial^{2} u}{\partial x^{2}}+2xy\dfrac {\partial^{2} u}{\partial x \partial y}+y^{2}\dfrac {\partial^{2} u}{\partial y^{2}}=0$ , if

u=\varphi \left(\dfrac {y}{x}\right)+x\psi \left(\dfrac {y}{x}\right) .

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Problem 3329

Verify the following identity: $x^{2}\dfrac {\partial^{2} u}{\partial x^{2}}+2xy\dfrac {\partial^{2} u}{\partial x \partial y}+y^{2}\dfrac {\partial^{2} u}{\partial y^{2}}=n(n-1)u$ , if

u=x^{n}\varphi \left(\dfrac {y}{x}\right)+x^{1-n}\psi \left(\dfrac {y}{x}\right) .

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Problem 3330

Verify the following identity: $\dfrac {\partial u}{\partial x}\dfrac {\partial^{2} u}{\partial x \partial y}=\dfrac {\partial u}{\partial y}\dfrac {\partial^{2} u}{\partial x^{2}}$ , if $u=\varphi \left[x+\psi (y)\right]$ .

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Problem 3331

Using successive differentiation, eliminate the arbitrary function $\varphi$ and find the partial differential equation satisfied by $z=x+\varphi (xy)$ .

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Problem 3332

Using successive differentiation, eliminate the arbitrary function $\varphi$ and find the partial differential equation satisfied by $z=x\varphi \left(\dfrac {x}{y^{2}}\right)$ .

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Problem 3333

Using successive differentiation, eliminate the arbitrary function $\varphi$ and find the partial differential equation satisfied by $z=\varphi \left(\sqrt{x^{2}+y^{2}}\right)$ .

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Problem 3334

Using successive differentiation, eliminate the arbitrary function $\varphi$ and find the partial differential equation satisfied by $u=\varphi (x-y,y-z)$ .

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Problem 3335

Using successive differentiation, eliminate the arbitrary function $\varphi$ and find the partial differential equation satisfied by $u=\varphi \left(\dfrac {x}{y},\dfrac {y}{z}\right)$ .

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Problem 3336

Using successive differentiation, eliminate the arbitrary functions $\varphi$ and $\psi$ , and find the partial differential equation satisfied by $z=\varphi (x)+\psi (y)$ .

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Problem 3337

Using successive differentiation, eliminate the arbitrary functions $\varphi$ and $\psi$ , and find the partial differential equation satisfied by $z=\varphi (x)\psi (y)$ .

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Problem 3338

Using successive differentiation, eliminate the arbitrary functions $\varphi$ and $\psi$ , and find the partial differential equation satisfied by $z=\varphi (x+y)+\psi (x-y)$ .

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Problem 3339

Using successive differentiation, eliminate the arbitrary functions $\varphi$ and $\psi$ , and find the partial differential equation satisfied by $z=x\varphi \left(\dfrac {x}{y}\right)+y\psi \left(\dfrac {x}{y}\right)$ .

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Problem 3340

Using successive differentiation, eliminate the arbitrary functions $\varphi$ and $\psi$ , and find the partial differential equation satisfied by $z=\varphi (xy)+\psi \left(\dfrac {x}{y}\right)$ .

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Problem 3341

Find the derivative of the function

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

at the point $M(1,1)$ along the direction $l$ making an angle $\alpha =60°$ with the positive direction of the $Ox$ axis.

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Problem 3342

(a)(b)(c)

Find the derivative of the function

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

at the point $M(1,1)$ along the direction $l$ making an angle $\alpha$ with the positive direction of the $Ox$ axis. In what direction does this derivative have:

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(a)

its maximum value;

(b)

its minimum value;

(c)

the value $0$ .

Problem 3343

Find the derivative of the function

z=\ln (x^{2}+y^{2})

at the point $M_{0}(x_{0},y_{0})$ along the direction perpendicular to the level line passing through this point.

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Problem 3344

Find the derivative of the function

z=1-\left(\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}\right)

at the point $M\left(\dfrac {a}{\sqrt{2}},\dfrac {b}{\sqrt{2}}\right)$ along the inward normal direction of the curve $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}=1$ at this point.

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Problem 3345

Find the derivative of the function

u=xyz

at the point $M(1,1,1)$ along the direction $l\left\{ \cos \alpha ,\cos \beta ,\cos \gamma \right\}$ . What is the magnitude of the gradient of the function at this point?

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Problem 3346

Find the magnitude and direction of the gradient of the function

u=\dfrac {1}{r}

(where $r=\sqrt{x^{2}+y^{2}+z^{2}}$ ) at the point $M_{0}(x_{0},y_{0},z_{0})$ .

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Problem 3347

Find the angle between the gradients of the function

u=x^{2}+y^{2}-z^{2}

at the two points $A(\varepsilon ,0,0)$ and $B(0,\varepsilon ,0)$ .

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Problem 3348

At the point $M(1,2,2)$ , by how much does the magnitude of the gradient of the function

u=x+y+z

differ from the magnitude of the gradient of the function

v=x+y+z+0.001\sin (10^{6}\pi \sqrt{x^{2}+y^{2}+z^{2}})

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Problem 3349

Prove that at the point $M_{0}(x_{0},y_{0},z_{0})$ , the angle between the gradients of the functions

u=ax^{2}+by^{2}+cz^{2}

and

v=ax^{2}+by^{2}+cz^{2}+2mx+2ny+2pz

( $a,b,c,m,n,p$ constants with $a^{2}+b^{2}+c^{2} \neq 0$ ) tends to zero as the point $M_{0}$ recedes to infinity.

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Problem 3350

Let $u=f(x,y,z)$ be a twice differentiable function. If $\cos \alpha ,\cos \beta ,\cos \gamma$ are the direction cosines of the direction $l$ , find $\dfrac {\partial^{2} u}{\partial l^{2}}=\dfrac {\partial }{\partial l}\left(\dfrac {\partial u}{\partial l}\right)$ .

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Problem 3351

(a)(b)

Let $u=f(x,y,z)$ be a function twice differentiable, and let

l_{1}\left\{ \cos \alpha _{1},\cos \beta _{1},\cos \gamma _{1}\right\} , \quad l_{2}\left\{ \cos \alpha _{2},\cos \beta _{2},\cos \gamma _{2}\right\} ,

l_{3}\left\{ \cos \alpha _{3},\cos \beta _{3},\cos \gamma _{3}\right\}

be three mutually perpendicular directions. Prove that:

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(a)

$\left(\dfrac {\partial u}{\partial l_{1}}\right)^{2}+\left(\dfrac {\partial u}{\partial l_{2}}\right)^{2}+\left(\dfrac {\partial u}{\partial l_{3}}\right)^{2}=\left(\dfrac {\partial u}{\partial x}\right)^{2}+\left(\dfrac {\partial u}{\partial y}\right)^{2}+\left(\dfrac {\partial u}{\partial z}\right)^{2}$ ;

(b)

$\dfrac {\partial^{2} u}{\partial l_{1}^{2}}+\dfrac {\partial^{2} u}{\partial l_{2}^{2}}+\dfrac {\partial^{2} u}{\partial l_{3}^{2}}=\dfrac {\partial^{2} u}{\partial x^{2}}+\dfrac {\partial^{2} u}{\partial y^{2}}+\dfrac {\partial^{2} u}{\partial z^{2}}$ .

Problem 3352

Let $u=u(x,y)$ be a differentiable function such that when $y=x^{2}$ we have

u(x,y)=1

and

\dfrac {\partial u}{\partial x}=x .

Find $\dfrac {\partial u}{\partial y}$ when $y=x^{2}$ .

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Problem 3353

Let the function $u=u(x,y)$ satisfy the equation

\dfrac {\partial ^{2} u}{\partial x^{2}}-\dfrac {\partial ^{2} u}{\partial y^{2}}=0

together with the following conditions:

u(x,2x)=x , \quad u_{x}'(x,2x)=x^{2} .

Find: $u_{xx}''(x,2x)$ , $u_{xy}''(x,2x)$ , $u_{yy}''(x,2x)$ .

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Problem 3354

Assume $z=z(x,y)$ ; solve the following equation: $\dfrac {\partial^{2} z}{\partial x^{2}}=0$ .

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Problem 3355

Assume $z=z(x,y)$ ; solve the following equation: $\dfrac {\partial^{2} z}{\partial x \partial y}=0$ .

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Problem 3356

Assume $z=z(x,y)$ ; solve the following equation: $\dfrac {\partial^{n} z}{\partial y^{n}}=0$ .

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Problem 3357

Assume $u=u(x,y,z)$ ; solve the equation

\dfrac {\partial ^{3} u}{\partial x \partial y \partial z}=0 .

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Problem 3358

Find the solution $z=z(x,y)$ of the equation

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

satisfying the condition $z(x,x^{2})=1$ .

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Problem 3359

Find the solution $z=z(x,y)$ of the equation

\dfrac {\partial ^{2} z}{\partial y^{2}}=2

satisfying the conditions $z(x,0)=1$ , $z_{y}'(x,0)=x$ .

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Problem 3360

Find the solution $z=z(x,y)$ of the equation

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

satisfying the conditions $z(x,0)=x$ , $z(0,y)=y^{2}$ .

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