-1
votes
1answer
22 views

Euclidean space and vector field

Can someone explain me what a Euclidean space is? and more detailed what a vector field is? Or a continuous vector field
-2
votes
1answer
41 views

Why is $\sum^{n}_{i=1} \int_{t_{i-1}}^{t_i} = \int_{a}^{b} $?

This is a part of my proof: $$\begin{align} \left| \sum^{n}_{i=1} V(r(\tau_i)) \cdot \int_{t_{i-1}}^{t_i} r'(t) dt - \int_{a}^{b} V(r(t)) \cdot r'(t) dt \right| &\leq \sum^{n}_{i=1} ...
0
votes
0answers
22 views

Limit of an Expression on a Plane Curve

Say we have a curve $\mathcal{C}$ like this: $$ x^5 + y^5 = 5xy.$$ Say then that we want to find (and prove) the limit of the quantity $x/y$ (if it exists) as $x \to + \infty$ on the curve ...
1
vote
2answers
41 views

Find the distance of the point $( 1,2,3 )$ to ,,,,,,,,,,,,,,

Find the distance of the point $( 1,2,3 )$ of the plane $3x-2y+5z+17=0$ .
0
votes
1answer
24 views

show that $\frac{d}{dt}\langle \gamma(t),\eta(t)\rangle =\langle\gamma '(t),\eta (t)\rangle +\langle \gamma (t),\eta ' (t)\rangle$

Let $\gamma,\eta:[a,b]\to \mathbb R^n$ be continuous, differentiable, curves. show that $$\frac{d}{dt}\langle \gamma(t),\eta(t)\rangle =\langle\gamma '(t),\eta (t)\rangle +\langle \gamma (t),\eta ' ...
1
vote
2answers
50 views

Find a parametrization of the intersection curve between surfaces

Find a parametrization of the intersection curve between the surfaces $−3x^2+2z=10$ and $4x^2+10y^2=5$. You should parametrize such that $y=k\sin(t)$ for some constant k. The answer should be in ...
2
votes
2answers
65 views

When does this parametric curve cross itself?

Find the points where the curve given parametrically by$$\mathbf{r}(t)=\left(2+\cos\frac{3}{2}t\right)\left(\begin{matrix}\cos t\\\sin t\end{matrix}\right)$$crosses itself. So, I understand that ...
1
vote
1answer
38 views

Finding the orthogonal family of curves to a given family of curves. I am missing some.

Given the family of curves: $F(x,y,x_0)=0$ where $F(x,y,x_0) = (x-x_0)^2 + y^2 - R^2\ ,\ x_0 \in \mathbb{R}$ find the orthogonal family. This is my attempt: I first get the differential equation ...
1
vote
2answers
61 views

Prove using an example that there is no plane on R3 that contains every group of 4 points

Well, this is a homewrok question (which I know I should not be asking, but I cannot find an answer to this anywhere): The exercise is as follows: i) Find the equation of the plane of R3 that ...
2
votes
2answers
46 views

Parameterized curve describing trajectory of thrown object

We describe the trajectory of a thrown object (neglecting friction and similiar effects) with the curve $$k(t) = \left(v_0\cos(\beta)t,\,v_0\sin(\beta)t-\frac{g}{2}t^2\right)$$ with ...
1
vote
1answer
349 views

Find the area of the region determined by two curves

Find the area of the region $R$ given by two curves. So the region $R$ describes the area that is common between the two curves: $$\begin{align*} \text{Function 1: } r&= 2\sin(\theta)\\ ...
5
votes
2answers
1k views

Is $r=2\cos(\theta)$ a one-petal polar function?

I'm currently learning about polar functions and their graphs in precalculus, and one of the questions on my homework is to identify the shape of the function $r=2\cos(\theta)$. We were taught that ...
1
vote
0answers
119 views

Vector equation and curvature

Vector equation $r(t)=2\cos(t)\mathbf i + 3\sin(t)\mathbf j\ \ (0 \le t \le 2\pi)$ represents ellipse. I need to find curvature of this ellipse on endpoints of x and y axis that are given with ...
6
votes
2answers
271 views

Need help with Curves and parameterizations

I'm having some trouble solving a couple of problems: I know this one must be pretty easy but can't find the way to solve it. I need to find the arc length of a curve described by $ r=1- ...