Work along an oriented curve

Prove that the work

$$\mathcal{W}=- \oint \limits_{\gamma} \frac{(x, y, z)}{\left ( x^2+y^2+z^2 \right )^{3/2}} \cdot \, {\rm d}(x, y, z)$$

produced along a $\mathcal{C}^1$ oriented curve $\gamma$ of $\mathbb{R}^3 \setminus \{(0, 0, 0) \}$ depends only on the distances of starting and ending point of $\gamma$ about the origin.

Solution

On known formulae

1. Evaluate the area of the disk of center $(0, 0)$ and radius $R>0$
2. Let $f$ be a continuous function such that $f(z) \geq 0 , \; z \in [0, R]$. Prove , using the previous question as well as Cavalieri's principal that the volume $\mathcal{M}=\{(x,y, z) \in \mathbb{R}^3 : z \in [0, R] \mid x^2+y^2 \leq f^2(z) \}$ produced by an entire rotation by the graph of $f$ (which is a curve on the $xy$ plane) around the $z$ axis is equal to:
   
   $$\mathcal{V}\left ( \mathcal{M} \right )= \pi \int_{0}^{R}f^2(z) \, {\rm d}z$$


3.  If the function of the previous question is continuously differentiable , then prove that the area of the surface:
   
   $$\mathbb{S}=\left \{ \left ( f(z)\cos \varphi, f(z) \sin \varphi, z \right ) \in \mathbb{R}^3 : z \in [0, R], \varphi \in [0, 2\pi] \right \}$$
   
   produced by an entire rotation by the graph of $f$ around the $z$ axis is equal to:
   
   $$\mathcal{A}\left ( \mathbb{S} \right ) = 2\pi \int_{0}^{R} f(z) \sqrt{1+\big(f'(z)\big)^2} \, {\rm d}z$$

Solution

Smooth functions

Let $f:\mathbb{R}^n \rightarrow \mathbb{R}$ be a smooth function (i.e a function that has continuous partial derivatives of any order). Prove that there exist functions $g_i, \; i=1, 2, \dots, n$  such that:

$$f(x_1, x_2, \dots, x_n) -f (0, 0, \dots, 0) = \sum_{i=1}^{n} x_i g_i (x_1, x_2, \dots, x_n)$$

Solution

Closure of a set and extrema values of a function

Given the subset of $\mathbb{R}^2$

$$\mathcal{C}=\left \{ e^{-t}\left ( \cos t, \sin t \right ) \; \text{such that} \; t \geq 0 \right \}$$

we define the function $f:\overline{\mathcal{C}} \rightarrow \mathbb{R}$ as

$$f(x, y)= \left\{\begin{matrix} - \left ( x^2+y^2 \right )^{-1} &,  & (x, y) \in \mathcal{C} \\   0& , &\text{elsewhere}  \end{matrix}\right.$$

1. Find the closure $\overline{\mathcal{C}} \subset \mathbb{R}^2$ of $\mathcal{C}$ and prove that is compact.
2. Find all (local) extrema of $f$ and characterize them.

Solution

An $n$ dimensional integral

Let $n \geq 3$ be a natural number. Evaluate the integral:

$$\int \limits_{[0, 1]^n} \left \lfloor x_1+x_2+\cdots+x_n \right \rfloor\, {\rm d}\left ( x_1, x_2, \dots, x_n \right )$$

(Recurrent) volume of sphere

Let $V_n(R)$ be the volume of the sphere of center $0$ and radius $R>0$ in $\mathbb{R}$. Prove that for $n \geq 3$ holds:

$$V_n(1)= \frac{2\pi}{n} V_{n-2}(1)$$

Solution

Length of curve and line integral

Given the curve $\gamma(t)=\left( \frac{1}{2}t^2, \; \frac{1}{3}t^3 \right), \;\; t \in [0, \sqrt{3}]$ evaluate its length as well as the line integral
$$\oint_{\gamma} f(x, y)\,{\rm d}s$$

where $f(x, y)=1+2x$.

Solution

Volume of tetrahedron using integrals

Find the volume of the tetrahedron bounded by the planes $x=0, y=0, z=0, \; z = 2-2x-y$.

Solution

Conservative field

a) Let $D \subset \mathbb{R}^2$ be the unit disk and $\partial D$ be its positive oriented boundary. Evaluate the counter clockwise line integral:

$$\oint \limits_{\partial D} (x-y^3, x^3-y^2)\, {\rm d}(x, y)$$

b) Can you deduce if the function $f(x, y) =(x-y^3, x^3-y^2)$ is a conservative field using the above question?

Solution

On double integral

Let $a, b$ be positive real numbers. Let $E$ be the region defined as:

$$E=\left \{ (x, y): 0\leq x \leq \frac{\pi}{2}, \;\;\; 0\leq y\leq \min \left \{\frac{a}{\cos x}, \; \frac{b}{\sin x}  \right \} \right \}$$

Prove that $\displaystyle \iint \limits_{E} y \, {\rm d}x \, {\rm d}y =ab$.

Solution

Curve and line integrals

Let $\gamma$ be defined as $\gamma(t) = e^{-t} (\cos t, \sin t ), \;\; t \geq 0$.

a) Sketch the graph of the curve.
b) Evaluate the line integrals:

$$ \begin{matrix} &  \displaystyle ({\rm i})\; \oint_{\gamma}\left ( x^2 +y^2 \right )\, {\rm d}s &  & ({\rm ii}) \displaystyle \oint_{\gamma} (-y, x)\cdot {\rm d}(x, y) \end{matrix}$$

Solution