Path independence, exact differentials and potential fields

In $\mathbb {R}^{2}$, consider the region $\mathcal{A}$ defined by $x^2+y^2 > 1$. Find differentiable real valued functions $f$ and $g$ on $\mathcal{A}$ such that $\partial f/\partial x = \partial g/\partial y$ but there is no real valued function $h$ on $\mathcal{A}$ such that $f = \partial h/\partial y$ and $g = \partial h/\partial x$.

Let the vector field $F$ in $\mathbb {R}^{3}$ have the form

where $g$ is a real valued smooth function on $(0,\infty )$ and $\left\| \cdot \right\|$ denotes the Euclidean norm. ($F$ is undefined at $(0,0,0)$.) Prove that

for any smooth closed path $C$ in $\mathbb {R}^{3}$ that does not pass through the origin.

For the paths indicated in problem 4248, compute $\displaystyle \int_{OA} x\, dy+y\, dx$, where $O$ is the origin of coordinates, the coordinates of point $A$ are $(1,2)$, and:

Verify that the integrand is a total differential, and compute the following curve integral: $\displaystyle \int_{(-1,2)}^{(2,3)} x\, dy+y\, dx$.

Verify that the integrand is a total differential, and compute the following curve integral: $\displaystyle \int_{(0,1)}^{(3,-4)} x\, dx+y\, dy$.

Verify that the integrand is a total differential, and compute the following curve integral: $\displaystyle \int_{(0,1)}^{(2,3)} (x+y)\, dx+(x-y)\, dy$.

Verify that the integrand is a total differential, and compute the following curve integral: $\displaystyle \int_{(1,-1)}^{(1,1)} (x-y)(dx-dy)$.

Having verified that the integrand is an exact differential, compute the curvilinear integral along the curve $\Gamma$ with initial point $A$ and terminal point $B$. $\int_{\Gamma } x d y+y d x, A(-1 ; 3), B(2 ; 2)$.

Having verified that the integrand is an exact differential, compute the curvilinear integral along the curve $\Gamma$ with initial point $A$ and terminal point $B$. $\int_{\Gamma } x d x+y d y, A(-1 ; 0), B(-3 ; 4)$.

Having verified that the integrand is an exact differential, compute the curvilinear integral along the curve $\Gamma$ with initial point $A$ and terminal point $B$. $\int_{\Gamma }(x+y) d x+(x-y) d y, A(2 ;-1), B(1 ; 0)$.

Having verified that the integrand is an exact differential, compute the curvilinear integral along the curve $\Gamma$ with initial point $A$ and terminal point $B$. $\int_{\Gamma } 2 x y d x+x^{2} d y, A(0 ; 0), B(-2 ;-1)$.

Having verified that the integrand is an exact differential, compute the curvilinear integral along the curve $\Gamma$ with initial point $A$ and terminal point $B$. $\int_{\Gamma }\left(x^{4}+4 x y^{3}\right) d x+\left(6 x^{2} y^{2}-5 y^{4}\right) d y, A(-2 ;-1), B(0 ; 3)$.

Having verified that the integrand is an exact differential, compute the curvilinear integral along the curve $\Gamma$ with initial point $A$ and terminal point $B$. $\int_{\Gamma }\left(x^{2}+2 x y-y^{2}\right) d x+\left(x^{2}-2 x y-y^{2}\right) d y, \quad A(3 ; 0), \quad A(0 ;-3)$.

Having verified that the integrand is an exact differential, compute the curvilinear integral along the curve $\Gamma$ with initial point $A$ and terminal point $B$. $\int_{\Gamma }\left(3 x^{2}-2 x y+y^{2}\right) d x+\left(2 x y-x^{2}-3 y^{2}\right) d y$,

Having verified that the integrand is an exact differential, compute the curvilinear integral along the curve $\Gamma$ with initial point $A$ and terminal point $B$. $\int_{\Gamma } f(x+y)(d x+d y), f(t)$ — a continuous function, $A(0 ; 0)$, $B\left(x_{0} ; y_{0}\right)$.

Having verified that the integrand is an exact differential, compute the curvilinear integral along the curve $\Gamma$ with initial point $A$ and terminal point $B$. $\int_{\Gamma } \varphi (x) d x+\psi (y) d y, \quad \varphi (t), \quad \psi (t)$ — continuous functions, $A\left(x_{1} ; y_{1}\right), B\left(x_{2} ; y_{2}\right)$.

Having verified that the integrand is an exact differential, compute the curvilinear integral along the curve $\Gamma$ with initial point $A$ and terminal point $B$. $\int_{\Gamma } e^{x} \cos y d x-e^{x} \sin y d y, A(0 ; 0), B\left(x_{0} ; y_{0}\right)$.

Having verified that the integrand is an exact differential, compute the curvilinear integral along the curve $\Gamma$ with initial point $A$ and terminal point $B$. $\int_{\Gamma } x d x+y^{2} d y-z^{3} d z, A(-1 ; 0 ; 2), B(0 ; 1 ;-2)$.

Having verified that the integrand is an exact differential, compute the curvilinear integral along the curve $\Gamma$ with initial point $A$ and terminal point $B$. $\int_{\Gamma } y z d x+x z d y+x y d z, A(2 ;-1 ; 0), B(1 ; 2 ; 3)$.

Having verified that the integrand is an exact differential, compute the curvilinear integral along the curve $\Gamma$ with initial point $A$ and terminal point $B$. $\int_{\Gamma } \frac{x d x+y d y+z d z}{\sqrt{x^{2}+y^{2}+z^{2}}}, A \in S_{1}, B \in S_{2}$, where $S_{1}$ — the sphere $x^{2}+y^{2} + z^{2} = R_{1}^{2}, S_{2}$ — the sphere $x^{2}+y^{2}+z^{2} = R_{2}^{2}\left(R_{1} > 0, R_{2} > 0\right)$.

Find the function $u$ from the given exact differential of this function. $d u = x^{2} d x+y^{2} d y$.

Find the function $u$ from the given exact differential of this function. $d u = \left(e^{2 y}-5 y^{3} e^{x}\right) d x+\left(2 x e^{2 y}-15 y^{2} e^{x}\right) d y$.

Find the function $u$ from the given exact differential of this function. $d u = e^{x-y}[(1+x+y) d x+(1-x-y) d y]$.

Find the function $u$ from the given exact differential of this function. $d u = \frac{2 x\left(1-e^{y}\right)}{\left(1+x^{2}\right)^{2}} d x+\left(\frac{e^{2}}{1+x^{2}}+1\right) d y$.

Find the function $u$ from the given exact differential of this function. $d u = \frac{d x+d y+d z}{x+y+z}$.

Find the function $u$ from the given exact differential of this function. $d u = \frac{y z d x+x z d y+x y d z}{1+x^{2} y^{2} z^{2}}$.

Find the function $u$ from the given exact differential of this function. $d u = \left(x^{2}-2 y z\right) d x+\left(y^{2}-2 x z\right) d y+\left(z^{2}-2 x y\right) d z$.

Find the function $u$ from the given exact differential of this function. $d u = \left(1-\frac{1}{y}+\frac{y}{z}\right) d x+\left(\frac{x}{z}+\frac{x}{y^{2}}\right) d y-\frac{x y}{z^{2}} d z$.

Find the function $u$ from the given exact differential of this function. $d u = \frac{(x+y-z) d x+(x+y-z) d y+(x+y+z) d z}{x^{2}+y^{2}+z^{2}+2 x y}$

What condition must the differentiable function $F(x ; y)$ satisfy in order for the curvilinear integral

to be independent of the path of integration $\Gamma_{A B}$?

Find the work done by the field $\mathbf{F} = -\mu \mathbf{r} / r^{3}, \mathbf{r} = (x ; y), r = \left|\mathbf{r}\right|, \mu =$ const:

Find the work done by the field of central forces $\mathbf{F} = f(r) \mathbf{r}$, where $\mathbf{r} = x \mathbf{i} + y \mathbf{j}+z \mathbf{k}, r = \left|\mathbf{r}\right|, f(r)$ is a function continuous for $r > 0$, along a smooth contour $\Gamma$ with initial point $A\left(x_{1} ; y_{1} ; z_{1}\right)$ and terminal point $B\left(x_{2} ; y_{2} ; z_{2}\right)$, not containing the origin.

Let $(p ; v)$ be the coordinates determining, on the plane $O p v$, the state of one mole of an ideal gas (pressure and volume). The equation of state of one mole of such a gas has the form $p v = R T$, where $R =$ const $> 0, T$ is the absolute temperature. When passing from the state $\left(p_{1} ; v_{1}\right)$ to the state $\left(p_{2} ; v_{2}\right)$ along the curve $\Gamma$, the amount of heat received (or given off) by the gas is determined by the formula

where $c_{v} = \mathrm{const}, c_{p} = c_{v}+R$. The curve given by the equation $p v^{\gamma } =$ const, where $\gamma = c_{p} / c_{v}$, is called an adiabat (and the process of change of state along this curve — adiabatic).

Which of the indicated fields are potential in $R^{3}$ :

Is the field $\mathbf{H} = 2 I \frac{-y \mathbf{i}+x \mathbf{j}}{x^{2}+y^{2}}, \quad (x ; y) \neq (0 ; 0)$, potential:

Verify that the field $\mathbf{H} = 2 I(-y \mathbf{i}+x \mathbf{j}) /\left(x^{2}+y^{2}\right)$ is potential in the half-space $y > 0$, and find its potential.

Verify the potentiality and find the potential of the field:

Let $f(r), r > 0$, be a differentiable function. Prove that the (central) field $\mathbf{a} = f(r) \mathbf{r}$ is potential for $r > 0(\mathbf{r} = x \mathbf{i} + y \mathbf{j}+z \mathbf{k}, r = \left|\mathbf{r}\right|)$. Find the potential of $\mathbf{a}$.

Prove that the potential $u$ of a continuously differentiable field a satisfies the equation $\Delta u = \operatorname {div} \mathbf{a}$.

Prove that if the field a is potential in a region $\Omega$ that is star-shaped (see problem 30) with respect to the point $M_{0}\left(r_{0}\right)$, then its potential at the point $M(\mathbf{r})$ is given by the formula

Prove that the positions of stable equilibrium of a particle in a potential force field $\mathbf{F} = -\operatorname {grad} u$ are located at the points of minimum of the potential $u$.

Prove that a potential field has no closed vector lines.

Is the field a potential, solenoidal, if $(\mathbf{r} = x \mathbf{i}+y \mathbf{j}+z \mathbf{k}, r = \left|\mathbf{r}\right|)$ :

Prove the harmonicity of the plane field

Prove the harmonicity of the gravitational field of a point mass and of the Coulomb field of a point charge.

Prove that the potential of a harmonic field is a harmonic function, i.e. $\Delta u = 0$.

Let the velocity field $\mathbf{v}$ of a moving continuous medium be potential. Prove that if the medium is incompressible, then the potential $u$ of the field $\mathbf{v}$ is harmonic (one may use the fact that the volumetric flow rate of the medium through any closed surface equals zero).