0
votes
1answer
36 views

The motion of the particle satisfies $\textbf{v} = \textbf{c}\times \textbf{r}$

Why is the path is contained in a circle that lies in a plane perpendicular to $\textbf{c}$ with centre on a line through the origin in the direction of $\textbf{c}$
3
votes
2answers
262 views

What is a French curve, as mentioned by Feynman?

I'm reading "Surely You're Joking, Mr. Feynman!", he says: I often liked to play tricks on people when I was at MIT. One time, in mechanical drawing class, some joker picked up a French curve (a ...
3
votes
0answers
145 views

Question on Moment of inertia about center of mass of a smooth plane curve.

This question is in continuation to this and motivated to answer this question. If I have an answer to it, then I can prove a special case of this question. Consider two smooth plane curves $C \equiv ...
0
votes
0answers
21 views

What is the meaning of “slope of ca”?

I'm reading a paper, when this article refers to the function: $$\beta(v)=\frac{(\frac{v}{I})^k}{1+(\frac{v}{I})^k}$$ It say that "around the $I$, $\beta$ is approximately linear in $I$, and has a ...
0
votes
2answers
25 views

Lengths of Plane Curves - Calculus 2: $\sqrt{1-x^2} ; x=-\frac{1}{2} \to x=\frac{1}{2}$

$$ \sqrt{1-x^2} ; x=-\frac{1}{2} \to x=\frac{1}{2} $$ I am having problems setting this up. Taking the derivative of $\sqrt{1-x^2}$. Leaves me with: $$ \frac{1}{2}\left(1-x^2 ...
0
votes
1answer
52 views

Do line integrals of non smooth curves exist?

Wolfram says that the theorem of conservative fields is : The following conditions are equivalent for a conservative vector field on a particular domain $D$: For any oriented simple closed ...
1
vote
1answer
25 views

Determing if a parametric curve is smooth

I have to determine whether the following curves are smooth or not and I'm having trouble with the following two functions: Consider $f(t) = (t^{2}-1,t^{2}+1)^{T}$ The solution states: $f'(t) = ...
0
votes
0answers
45 views

Creating a $C^2$ smooth convex curve which joins circular arcs

Suppose that I have many points $p_1$,..., $p_n$, all very close to origin 0. For R very, very large, I have circular arcs of radius R, $\alpha_i$, about $p_i$. I do not speak about their ...
0
votes
0answers
44 views

AP Calculus Integral Problem

A solid is formed by revolving the curve $y=x^{2/3}+1$, for $0 \leq x \leq 2.5$, about the $X$-axis Estimate the volume of the solid by partitioning $[0,2.5]$ into five sub-intervals of equal length, ...
0
votes
1answer
33 views

Domain and Range problem(plane)

Consider the function $z = \ln{(y + 1)}+\sqrt{x-3}$. Find the domain and range, and sketch the domain in the plane.
0
votes
1answer
27 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 ' ...
3
votes
2answers
130 views

find maximum area

Consider a problem here : There is a wall in your backyard. It is so long that you can’t see its endpoints. You want to build a fence of length L such that the area enclosed between the wall and the ...
0
votes
4answers
44 views

Finding the arc length of $r(t)=ti+j+(\frac{1}{6}t^3+\frac{1}{2}t^{-1})$

How would I find the arc length of the following curve from $t=0$ to $t=2$ $r(t)=ti+j+(\frac{1}{6}t^3+\frac{1}{2}t^{-1})$ I took the first derivative and got ...
11
votes
4answers
418 views

When is the moment of inertia of a smooth plane curve is maximum?

Given a smooth plane curve $(x(s),y(s))$, parameterized in arc length $s$, of fixed finite length $L$, its moment of inertia about its center of mass (axis perpendicular to the plane) is given as $$MI ...
2
votes
1answer
38 views

Equation of a plane (not sure if I got this)

When I was doing my calculus midterm, I came across a question that I didn't really know how to solve, I think I skipped over these problems in my studies. The question is: Find an equation of the ...
0
votes
0answers
154 views

Convex curve in R^2 pass through two point with fixed normal direction

Given two distinct point in $\mathbb{R}^2$ and two distinct normal directions what are some explicit convex $C^2$ curve (e.g. a map $[0,1] \to \mathbb{R}^2$ which satisfy many equivalent property) ...
1
vote
1answer
124 views

Show the curve $\alpha$ is differentiable and regular

Consider the map: $$ \alpha(t) = \left\{ \begin{array}{ll} (t,0,e^{-1/t^2}) & t>0 \\ (t,e^{-1/t^2},0) & t<0 \\ (0,0,0) & t=0 \end{array} \right. $$ a. Prove that ...
2
votes
2answers
130 views

Curves with “constant speed”?

I am new to the concept of curves. Let us a assume we have a simple function such as $f:\mathbb{R}^+\rightarrow \mathbb R^+\quad f(x) := \sqrt{x}$. (Or $f(x)=\exp(x)$ or a polynomial etc.). We can ...
3
votes
2answers
155 views

why $x^2 = y^3$ is not smooth?

I read a definition of a smooth curve on the plane: A smooth curve is a map from $[a,b] \to \mathbb R^2: t\mapsto ( f(t),g(t) )$, where $f$ and $g$ are infinitely differentiable functions. According ...
1
vote
1answer
113 views

Straightness measure for smooth 2-d plane curves of a given fixed length

Consider a smooth, 2-d plane curve of given fixed length $d$. Any straight line of length $d$, is also a curve of this type. What i am interested in is, How straight a curve of a fixed length, is? In ...
1
vote
2answers
136 views

Finding arc length and binormal vector for a given curve

Can somebody show me the arc length of a curve formula, and the binormal vector formula. The curve C with equation $r(t)=(\sqrt{3}\cos t,t,\sqrt{3}\sin t)$ How do you find the arc length from $t=0$ ...
2
votes
1answer
66 views

Separation of variables (ODEs)

Here is the question I am currently stuck on: Here is what I have done so far: My apologies as I understand this post seems fairly lengthy. However I cannot seem to get the final answer ...
1
vote
2answers
313 views

General way to find out whether a curve is positively oriented

I have a general question, that deals with the question how I am able to find out whether a particular curve(in $\mathbb{C}$) is positively oriented? Take e.g. $ y(t)=a+re^{it}$. Obviously this one ...
1
vote
3answers
203 views

Arc Length Formulas

Use the arc length formula to find the arc length of the upper half of the circle with center at $(0,0)$ and radius $3$. Also, find the arc length of the curve in the first question by using ...
1
vote
2answers
137 views

Proof of Astroid?

How can I prove that an astroid is an envelope of all line segments of length 1 from the x-axis to the y-axis? I read one proof of this online at the link ...
3
votes
1answer
1k views

How to calculate the area between 2 polar curves: $r=\frac{4}{2}-\sin\theta$ and $r=3\sin\theta$?

How to calculate the area between 2 polar curves: $r=2-\sin\theta$ and $r=3\sin\theta$? I know that one curve is a limaçon and the other is a circle. I have them drawn out as well, my only question ...
3
votes
3answers
727 views

How can I find the points of intersection between the curves $r=1+\sin\theta$ and $r=1-\sin\theta$?

Find the points of intersection for the curve $r=a(1+\sin\theta)$ and $r=a(1-\sin\theta)$ My book says the answer is $(0,0),(a,0),(a,\pi)$. However I calculated $ (a,0),(a,\pi),(a,2\pi)$.
0
votes
0answers
41 views

Curve is approximatable by function

I want to show that for $\gamma: [a,b] \subseteq \mathbb{R} \rightarrow V$ continously differentiable where V is a bounded subset of $\mathbb{R}^2$. There is always a sequence of functions ...
1
vote
0answers
37 views

Approximation theorem

I am looking for some theorem that gives me that each curve $x(t)=(x_1(t),x_2(t))$ that is continously differentiable and has $\dot{x_1}(t)\ge 0$ can be approximated by continously differentiable ...
0
votes
1answer
84 views

How To Write The equation for a line given a set of co-ordinates

I'm trying to learn how can I write equation for a line given all the points that belongs in the line. I'm looking to find equation for a curve. An example Set of points is: { (24,11) (25,11) ...
4
votes
1answer
1k views

Arc length formula for the lemniscate

This question can be homework for elementary calculus. The lemniscate of Bernoulli $C$ is a plane curve defined as follows. Let $a > 0$ be a real number. Let $F_1 = (a, 0)$ and $F_2 = (-a, 0)$ be ...
1
vote
1answer
268 views

Finding the equation of a curve that has a perpendicular distance of $d$ from another curve

Let's say we have an equation of a curve as $y = f(x)$. I want to find the curve $y = g(x)$ where $(x_1, f(x_1))$ has a perpendicular distance of $d$ from that curve. Doing this with straight lines ...
1
vote
1answer
81 views

Computing the gradient of the function $\psi(u) = f[ \phi(u)]$

This question is based on section 6 of the paper Kriging and splines with derivative information. A parametric curve $\phi(u)$ in three dimensions is deformed by the function $f$ to a new curve ...
2
votes
2answers
718 views

Parallel functions.

In 2 dimensions, we can draw 2 parallel lines that have the same distance from a line. I wanted to find parallel functions of a function and their distance is $d$ to the function for all inputs and ...
2
votes
1answer
50 views

Equivalence of two definitions of path (in $\mathbb{R}^3$) length

In a previews question I asked here I used the following definition of path length:$\gamma=(x(t),y(t),z(t))$ : $L(\gamma)=\intop_{a}^{b}\sqrt{(x'(t))^{2}+(y'(t))^{2}+(z'(t))^{2}}$. In the answer ...
0
votes
1answer
118 views

Prove if a polar function involves only the rational numbers and sin, cos, tan functions, it can be written in rectangular form.

Prove if a function only have including the rational numbers and sin, cos, tan function and $r$, you always could write it in rectangular form. Ex. For $r=2/(2+2\cos\theta)$ it could be represent in ...
3
votes
1answer
635 views

The area of the superellipse

I'm watching this video, where D. Knuth explains the connection of $\pi$ and factorials, and other matters (it is very interesting). Almost at the end of the talk he says the area of the superellipse ...
0
votes
1answer
555 views

Tractrix Tangent Length Problem

I am having trouble with a problem I am working on The trace of $\vec{r}(t):=\sin(t)\vec{i}+[\cos(t)+\ln[\tan(t/2)]]\vec{j}$ where $t\in(0,\pi)$ is called a tractrix. Show the length of the line ...
0
votes
1answer
356 views

Logarithmic Spiral Calculus Question

I am working on this problem. Even some of the notation has me confused (the vectors $\vec i$ and $\vec j$). Let $\vec r(t):=ae^{-bt}\cos(t)\vec i +ae^{-bt}\sin(t)\vec j$ where $a$ and $b$ are ...
2
votes
3answers
4k views

Polar to Parametric Equation?

I'm struggling with this problem, I'm still only on part (a). I tried X=rcos(theta) Y=rsin(theta) but I don't think I'm doing it right. Curve C has polar equation ...
1
vote
2answers
958 views

What does “the circle is tangent to the curve” mean?

I've got a math exercise where it's said I have to prove that a circle is tangent to a curve (described by a parametric plot). Here's the graph : So we can see that when $y=0$, the circle is really ...
11
votes
1answer
735 views

Folium of Descartes

A colleague came to me with an interesting observation: Consider the folium of Descartes, $$x^3+y^3=3axy$$ which upon implicit differentiation of the latter yields $$\frac{\mathrm dy}{\mathrm ...
1
vote
1answer
244 views

Why is this curve convex?

I am considering the curve traced by the equation $r=a\sin 3\theta$. Specifically as $\theta$ varies from $0$ to $\frac{\pi}{6}$, $r$ varies from $0$ to $a$. How do I conclude that the curve is convex ...
3
votes
2answers
1k views

How Can I Calculate Area of Astroid Represented by Parameter?

Let $x=2\cos^3\theta$ and $y=2\sin^3\theta$ known as the astroid. In this case, radius $r=2$. and gray part's $x$ range is $1/\sqrt{2}\leq x\leq 2$. this deal with $0\leq\theta\leq \pi/4$. ...
4
votes
1answer
2k views

Why does the focus of a rolling parabola trace a catenary?

I keep hearing that when one rolls a parabola on a straight line, the focus traces a catenary. I kept trying to find a proof on the Internet, but no dice. How does one prove this to be true?
1
vote
1answer
294 views

rolling wheel problem

To achieve this: http://en.wikipedia.org/wiki/Square_wheel, what should $L$ be?
5
votes
1answer
1k views

Formula for curve parallel to a parabola

I have a simple parabola in the form $y = a + bx^2$. I would like to find the formula for a curve which is parallel to this curve by distance $c$. By parallel I mean that there is an equal distance ...
2
votes
2answers
235 views

Looking for the name of a Rising/Falling Curve

I'm looking for a particular curve algorithm that is similar to to a bell curve/distribution, but instead of approaching zero at its ends, it stops at its length/limit. You specify the length of the ...
1
vote
1answer
242 views

Where is my (algebra) mistake? Converting parametric to Cartesian equation

I'm having a problem with my solution to a textbook exercise: Find the Cartesian equation of the curve given by this parametric equation: $$x = \frac{t}{2t-1}, y = \frac{t}{t+1}$$ The textbook's ...
36
votes
4answers
2k views

A circle rolls along a parabola

I'm thinking about a circle rolling along a parabola. Would this be a parametric representation? $(t + A\sin (Bt) , Ct^2 + A\cos (Bt) )$ A gives us the radius of the circle, B changes the frequency ...