Math Problems Vault

Problem 4401

(a)(b)(c)

Find the magnitude and direction of the gradient of the field

u = x^{2}+2y^{2}+3z^{2}+xy+3x-2y-6z

at the points: (1) $O(0,0,0)$ ; (2) $A(1,1,1)$ ; (3) $B(2,1,1)$ . At which points of the field does the gradient equal zero?

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(a)

(b)

(c)

Problem 4401.1

[NO TASK]

Problem 4402

(a)(b)(c)

At which points of the space $Oxyz$ is the gradient of the field

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

(1) perpendicular to the $Oz$ axis; (2) parallel to the $Oz$ axis; (3) equal to zero?

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(a)

(b)

(c)

Problem 4403

Given the scalar field

u = \ln \dfrac {1}{r} ,

where $r=\sqrt{(x-a)^{2}+(y-b)^{2}+(z-c)^{2}}$ . At which points of the space $Oxyz$ does the following equality hold:

\left|\operatorname {grad} u\right| = 1 \ ?

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Problem 4404

Construct the level surfaces of the scalar field

u = \sqrt{x^{2}+y^{2}+(z+8)^{2}}+\sqrt{x^{2}+y^{2}+(z-8)^{2}}

and find the level surface passing through the point $M(9,12,28)$ . What is $\max u$ in the domain $x^{2}+y^{2}+z^{2}\leqslant 36$ ?

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Problem 4405

Find the angle $\theta$ between the gradients of the field

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

at the points $A(1,2,2)$ and $B(-3,1,0)$ .

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Problem 4406

Let a scalar field be given

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

Construct the level surfaces of the field and the level surfaces of equal gradient magnitude.

In the region $1<z<2$ find $\inf u$ , $\sup u$ , $\inf \left|\operatorname {grad} u\right|$ , $\sup \left|\operatorname {grad} u\right|$ .

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Problem 4407

Find, accurate to a higher-order infinitesimal, the distance at the point $M_{0}(x_{0},y_{0},z_{0})$ between two infinitely close level surfaces

u(x,y,z) = c \qquad \text{and} \qquad u(x,y,z) = c+\Delta c

where $u(x_{0},y_{0},z_{0})=c$ .

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Problem 4408

Prove the formulas

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(a)

$\operatorname {grad}(u+c) = \operatorname {grad} u$ ( $c$ a constant);

(b)

$\operatorname {grad} cu = c\operatorname {grad} u$ ( $c$ a constant);

(c)

$\operatorname {grad}(u+v) = \operatorname {grad} u+\operatorname {grad} v$ ;

(d)

$\operatorname {grad} uv = v\operatorname {grad} u+u\operatorname {grad} v$ ;

(e)

$\operatorname {grad}(u^{2}) = 2u\operatorname {grad} u$ ;

(f)

$\operatorname {grad} f(u) = f'(u)\operatorname {grad} u$ .

Problem 4409

(a)(b)(c)

Compute: (1) $\operatorname {grad} r$ , (2) $\operatorname {grad} r^{2}$ , (3) $\operatorname {grad}\dfrac {1}{r}$ , where $r=\sqrt{x^{2}+y^{2}+z^{2}}$ .

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(a)

(b)

(c)

Problem 4410

Find $\operatorname {grad} f(r)$ , where $r=\sqrt{x^{2}+y^{2}+z^{2}}$ .

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Problem 4411

Find $\operatorname {grad}(\overrightarrow {c}\cdot \overrightarrow {r})$ , where $\overrightarrow {c}$ is a constant vector and $\overrightarrow {r}$ is the position vector from the origin.

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Problem 4412

Find $\operatorname {grad}\left\{ \left|\overrightarrow {c}\times \overrightarrow {r}\right|^{2}\right\}$ ( $\overrightarrow {c}$ a constant vector).

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Problem 4413

Prove the formula

\operatorname {grad} f(u,v) = \dfrac {\partial f}{\partial u}\operatorname {grad} u+\dfrac {\partial f}{\partial v}\operatorname {grad} v .

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Problem 4414

Prove the formula

\nabla ^{2}(uv) = u\nabla ^{2}v+v\nabla ^{2}u+2\nabla u\nabla v ,

where

\nabla = \overrightarrow {i}\dfrac {\partial }{\partial x}+\overrightarrow {j}\dfrac {\partial }{\partial y}+\overrightarrow {k}\dfrac {\partial }{\partial z} ,

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

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Problem 4415

Prove: if the function $u=u(x,y,z)$ is differentiable in a convex region $\Omega$ and $\left|\operatorname {grad} u\right|\leqslant M$ , where $M$ is a constant, then for any two points $A$ , $B$ in $\Omega$ we have:

\left|u(A)-u(B)\right| \leqslant M\rho (A,B) ,

where $\rho (A,B)$ is the distance between the points $A$ and $B$ .

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Problem 4415.1

(a)(b)

(a)

(b)

Problem 4416

Find the derivative of the field $u=\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}+\dfrac {z^{2}}{c^{2}}$ at the given point $M(x,y,z)$ along the direction of the position vector $\overrightarrow {r}$ of this point.

Under what conditions will this derivative equal the magnitude of the gradient?

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Problem 4417

Find the derivative of the field $u=\dfrac {1}{r}$ (where $r=\sqrt{x^{2}+y^{2}+z^{2}}$ ) in the direction $\overrightarrow {l}\left\{ \cos \alpha ,\cos \beta ,\cos \gamma \right\}$ .

Under what conditions does this derivative equal zero?

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Problem 4418

Find the derivative of the field $u=u(x,y,z)$ in the direction of the gradient of the field $v=v(x,y,z)$ .

Under what conditions does this derivative equal zero?

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Problem 4419

Let:

u = \arctan \dfrac {z}{\sqrt{x^{2}+y^{2}}} \quad \text{and} \quad \overrightarrow {c} = \overrightarrow {i}+\overrightarrow {j}+\overrightarrow {k} ,

compute

\overrightarrow {a} = \overrightarrow {c}\times \operatorname {grad} u .

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Problem 4420

Determine the field lines of the vector field

\overrightarrow {a} = x\overrightarrow {i}+y\overrightarrow {j}+2z\overrightarrow {k}

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Problem 4421

Prove by direct computation that the divergence of the vector $\overrightarrow {a}$ is independent of the choice of Cartesian coordinate system.

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Problem 4422

Prove

\operatorname {div}\overrightarrow {a}(M) = \lim _{d(S)\to 0} \dfrac {1}{V}\iint _{S} \overrightarrow {a}\cdot \overrightarrow {n}\, dS ,

where $S$ is a closed surface enclosing the point $M$ and bounding a volume $V$ , $\overrightarrow {n}$ is the outward normal of the surface $S$ , and $d(S)$ is the diameter of the surface $S$ .

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Problem 4422.1

[NO TASK]

Problem 4423

Find:

\operatorname {div} \begin{vmatrix} \overrightarrow {i} & \overrightarrow {j} & \overrightarrow {k} \\ \dfrac {\partial }{\partial x} & \dfrac {\partial }{\partial y} & \dfrac {\partial }{\partial z} \\ \omega _{x} & \omega _{y} & \omega _{z} \end{vmatrix} .

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Problem 4424

(a)(b)(c)

Prove:

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(a)

$\operatorname {div}(\overrightarrow {a}+\overrightarrow {b}) = \operatorname {div}\overrightarrow {a}+\operatorname {div}\overrightarrow {b}$ ;

(b)

$\operatorname {div}(u\overrightarrow {c}) = \overrightarrow {c}\cdot \operatorname {grad} u$ (where $\overrightarrow {c}$ is a constant vector and $u$ is a scalar);

(c)

$\operatorname {div}(u\overrightarrow {a}) = u\operatorname {div}\overrightarrow {a}+\overrightarrow {a}\cdot \operatorname {grad} u$ .

Problem 4425

Find $\operatorname {div}(\operatorname {grad} u)$ .

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Problem 4426

Find $\operatorname {div}\left[\operatorname {grad} f(r)\right]$ , where $r=\sqrt{x^{2}+y^{2}+z^{2}}$ . Under what condition is $\operatorname {div}\left[\operatorname {grad} f(r)\right]=0$ ?

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Problem 4427

(a)(b)

Compute: (1) $\operatorname {div}\overrightarrow {r}$ ; (2) $\operatorname {div}\dfrac {\overrightarrow {r}}{r}$ .

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(a)

(b)

Problem 4428

Compute $\operatorname {div}\left[f(r)\overrightarrow {c}\right]$ , where $\overrightarrow {c}$ is a constant vector.

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Problem 4429

Find $\operatorname {div}\left[f(r)\overrightarrow {r}\right]$ . Under what condition is the divergence of this vector equal to zero?

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Problem 4430

(a)(b)

Find: (1) $\operatorname {div}(u\operatorname {grad} u)$ ; (2) $\operatorname {div}(u\operatorname {grad} v)$ .

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(a)

(b)

Problem 4431

A body rotates counterclockwise about the $Oz$ axis with a certain angular velocity $\overrightarrow {\omega }$ . Find the divergence of the velocity vector $\overrightarrow {v}$ and the acceleration vector $\overrightarrow {w}$ at the point $M(x,y,z)$ in space, and at a given moment in time.

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Problem 4432

Find the divergence of the force field produced by a finite system of gravitational centers.

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Problem 4433

Find the expression for the divergence of a plane vector $\overrightarrow {a}=\overrightarrow {a}(r,\varphi )$ represented in polar coordinates $r$ and $\varphi$ .

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Problem 4434

Let

x=f(u,v,w) , \qquad y=g(u,v,w) , \qquad z=h(u,v,w) ,

express $\operatorname {div}\overrightarrow {a}(x,y,z)$ in orthogonal curvilinear coordinates $u$ , $v$ , $w$ .

As special cases, find the expressions for $\operatorname {div}\overrightarrow {a}$ in cylindrical coordinates and in spherical coordinates.

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Problem 4435

(a)(b)

Prove:

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(a)

$\operatorname {rot}(\overrightarrow {a}+\overrightarrow {b}) = \operatorname {rot}\overrightarrow {a}+\operatorname {rot}\overrightarrow {b}$ ;

(b)

$\operatorname {rot}(u\overrightarrow {a}) = u\operatorname {rot}\overrightarrow {a}+\operatorname {grad} u\times \overrightarrow {a}$ .

Problem 4436

(a)(b)

Find: (1) $\operatorname {rot}\overrightarrow {r}$ ; (2) $\operatorname {rot}\left[f(r)\overrightarrow {r}\right]$ .

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(a)

(b)

Problem 4436.1

[NO TASK]

Problem 4437

(a)(b)

Find: (1) $\operatorname {rot}\overrightarrow {c}f(r)$ , (2) $\operatorname {rot}\left[\overrightarrow {c}\times f(r)\overrightarrow {r}\right]$ (where $\overrightarrow {c}$ is a fixed vector).

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(a)

(b)

Problem 4438

Prove that $\operatorname {div}(\overrightarrow {a}\times \overrightarrow {b}) = \overrightarrow {b}\cdot \operatorname {rot}\overrightarrow {a}-\overrightarrow {a}\cdot \operatorname {rot}\overrightarrow {b}$ .

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Problem 4439

(a)(b)

Find: (1) $\operatorname {rot}(\operatorname {grad} u)$ ; (2) $\operatorname {div}(\operatorname {rot}\overrightarrow {a})$ .

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(a)

(b)

Problem 4440

A body rotates with constant angular velocity $\omega$ about the axis $\overrightarrow {l}\left\{ \cos \alpha ,\cos \beta ,\cos \gamma \right\}$ . Find the curl of the velocity vector $\overrightarrow {v}$ at the point $M(x,y,z)$ of space at a given instant of time.

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Problem 4440.1

[NO TASK]

Problem 4440.2

(a)(b)

(a)

(b)

Problem 4441

(a)(b)

Find the flux of the vector $\overrightarrow {r}$ : (1) through the lateral surface of the cone $x^{2}+y^{2}\leqslant z^{2}$ $(0\leqslant z\leqslant h)$ ; (2) through the base of this cone.

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(a)

(b)

Problem 4442

(a)(b)

Find the flux of the vector $\overrightarrow {a}=yz\overrightarrow {i}+zx\overrightarrow {j}+xy\overrightarrow {k}$ : (a) through the lateral surface of the cylinder $x^{2}+y^{2}\leqslant a^{2}$ $(0\leqslant z\leqslant h)$ ; (б) through the entire surface of this cylinder.

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(a)

(b)

Problem 4443

Find the flux of the radius vector $\overrightarrow {r}$ through the surface

z = 1-\sqrt{x^{2}+y^{2}} \qquad (0\leqslant z\leqslant 1)

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Problem 4444

Find the flux of the vector $\overrightarrow {a}=x^{2}\overrightarrow {i}+y^{2}\overrightarrow {j}+z^{2}\overrightarrow {k}$ through the part of the sphere $x^{2}+y^{2}+z^{2}=1$ , $x\geqslant 0$ , $y\geqslant 0$ , $z\geqslant 0$ , lying in the first octant.

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Problem 4445

Find the flux of the vector $\overrightarrow {a}=y\overrightarrow {i}+z\overrightarrow {j}+x\overrightarrow {k}$ through the entire surface of the pyramid bounded by the planes $x=0$ , $y=0$ , $z=0$ , $x+y+z=a$ $(a>0)$ .

Verify the result using the Ostrogradsky formula.

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Problem 4445.1

[NO TASK]

Problem 4446

Prove that the flux of the vector $\overrightarrow {a}$ through the surface $S$ given by the equation $\overrightarrow {r}=\overrightarrow {r}(u,v)$ $((u,v)\in \Omega )$ equals

\iint _{S} \overrightarrow {a}\cdot \overrightarrow {n}\, dS = \iint _{\Omega } \left(\overrightarrow {a}\ \dfrac {\partial \overrightarrow {r}}{\partial u}\ \dfrac {\partial \overrightarrow {r}}{\partial v}\right)\, du\, dv ,

where $\overrightarrow {n}$ is the unit normal vector of the surface $S$ .

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Problem 4447

Find the flux of the vector $\overrightarrow {a}=m\dfrac {\overrightarrow {r}}{r^{3}}$ ( $m$ a constant) through a closed surface $S$ surrounding the origin.

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Problem 4448

Given the vector

\overrightarrow {a}(\overrightarrow {r}) = \overrightarrow {a}\sum _{i=1}^{n} \operatorname {grad}\left(-\dfrac {e_{i}}{4\pi r_{i}}\right) ,

where $e_{i}$ are constants and $r_{i}$ is the distance from the point $M_{i}$ (the starting point) to the moving point $M(\overrightarrow {r})$ . Find the flux of this vector through a closed surface $S$ surrounding the points $M_{i}$ $(i=1,2,\ldots ,n)$ .

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Problem 4449

Prove that

\iint _{S} \dfrac {\partial u}{\partial n}\, dS = \iiint _{V} \nabla ^{2}u\, dx\, dy\, dz ,

where the surface $S$ encloses the volume $V$ .

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Problem 4450

The quantity of heat entering a temperature field $u$ per unit time through a surface element $dS$ equals

dQ = -k\overrightarrow {n}\cdot \operatorname {grad} u\, dS ,

where $k$ is the coefficient of internal thermal conductivity and $\overrightarrow {n}$ is the unit normal vector of the surface $S$ . Find the quantity of heat accumulated per unit time by a body $V$ . Study the rate of temperature rise so as to derive the equation satisfied by the temperature of the body (the heat conduction equation).

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Problem 4451

An incompressible fluid in motion occupies a volume $V$ . Assuming that there are no sources or sinks in the region $V$ , derive the continuity equation:

\dfrac {\partial \rho }{\partial t}+\operatorname {div}(\rho \overrightarrow {v}) = 0 ,

where $\rho =\rho (x,y,z,t)$ is the density of the fluid, $\overrightarrow {v}$ is the velocity vector, and $t$ is time.

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Problem 4452

Find the work of the vector $\overrightarrow {a}=\overrightarrow {r}$ along the segment of the helix

\overrightarrow {r} = a\cos t\, \overrightarrow {i}+a\sin t\, \overrightarrow {j}+bt\, \overrightarrow {k} \qquad (0\leqslant t\leqslant 2\pi )

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Problem 4452.1

[NO TASK]

Problem 4452.2

[NO TASK]

Problem 4452.3

[NO TASK]

Problem 4453

Find the work of the vector $\overrightarrow {a}=f(r)\overrightarrow {r}$ (where $f$ is a continuous function) along the arc $AB$ .

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Problem 4454

(a)(b)

Find the circulation of the vector

\overrightarrow {a} = -y\overrightarrow {i}+x\overrightarrow {j}+c\overrightarrow {k}

( $c$ a constant): (1) along the circle $x^{2}+y^{2}=1$ , $z=0$ ; (2) along the circle $(x-2)^{2}+y^{2}=1$ , $z=0$ .

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(a)

(b)

Problem 4455

(a)(b)

Find the circulation $\Gamma$ of the vector $\overrightarrow {a}=\operatorname {grad}\left(\arctan \dfrac {y}{x}\right)$ along a closed contour $C$ : (1) $C$ does not encircle the $Oz$ axis; (2) $C$ encircles the $Oz$ axis.

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(a)

(b)

Problem 4455.1

[NO TASK]

Problem 4456

The steady incompressible flow of a plane is described by the velocity vector

\overrightarrow {w} = u(x,y)\overrightarrow {i}+v(x,y)\overrightarrow {j}

Find: (1) the amount $Q$ of fluid flowing out through a closed contour $C$ enclosing a region $S$ (the fluid discharge); (2) the circulation $\Gamma$ of the velocity vector along the contour $C$ . If the flow field has no sources, no sinks, and no vorticity, what equations must the functions $u$ and $v$ satisfy?

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Problem 4457

Prove that the field

\begin{align} \overrightarrow {a} & = yz(2x+y+z)\overrightarrow {i}+xz(x+2y+z)\overrightarrow {j} +xy(x+y+2z)\overrightarrow {k} \end{align}

is a potential field, and find the potential of this field.

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Problem 4457.1

[NO TASK]

Problem 4458

Find the potential of the gravitational field

\overrightarrow {a} = -\dfrac {m}{r^{3}}\overrightarrow {r}

produced by a mass $m$ located at the origin of coordinates.

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Problem 4459

Find the potential of the gravitational field produced by a system of masses $m_{i}$ $(i=1,2,\ldots ,n)$ located at points $M_{i}$ $(i=1,2,\ldots ,n)$ .

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Problem 4460

Prove that the field $\overrightarrow {a}=f(r)\overrightarrow {r}$ (where $f(r)$ is a single-valued continuous function) is a potential field. Find the potential of this field.

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Problem 4461

Prove the formula

\begin{align} \operatorname {grad}_{P}\left\{ \iiint _{V} \rho (Q)\dfrac {dV}{r}\right\} & = -\iint _{S} \rho (Q)\overrightarrow {n}\dfrac {dS}{r} +\iiint _{V} \operatorname {grad}_{Q}\rho (Q)\dfrac {dV}{r} , \end{align}

where $S$ is the surface enclosing the volume $V$ , $\overrightarrow {n}$ is the outward normal of the surface $S$ , and $r$ is the distance between the points $P(x,y,z)$ and $Q(\xi ,\eta ,\zeta )$ .

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Problem 4462

Prove: if $\overrightarrow {a}=\operatorname {grad} u$ , where

u(x,y,z) = -\dfrac {1}{4\pi }\iiint _{-\infty }^{+\infty } \dfrac {\rho (\xi ,\eta ,\zeta )}{r}\, d\xi \, d\eta \, d\zeta

and

r = \sqrt{(\xi -x)^{2}+(\eta -y)^{2}+(\zeta -z)^{2}} ,

then

\operatorname {div}\overrightarrow {a} = \rho (x,y,z)

(assuming the corresponding integrals are meaningful).

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