2
votes
2answers
40 views

Geometric intuition: Seeing the regions in double integrals

Context: solving double integrals. I had the formula $$x^2+y^2=1-x-y$$ yet I could not see what shape it had. This is even more true with 3D pictures like $$2x^2+2y^2 \le 1+z^2.$$ Is there a summary ...
1
vote
2answers
37 views

the geometric explain of $t = x-\frac{a}{3}$ in the simplify of cubic equation $x^3+ax^2+bx+c=0$

Assume $$f(x) = x^3+ax^2+bx+c$$ we have $$f''(x)=2a+6x$$. we get $x = -\frac{a}{3}$ Magically, If we take the transformation: $$t = x -\left(-\frac{a}{3}\right)$$. we can transform the above ...
1
vote
0answers
31 views

Shifting a plot in polar coordinates

Say we have the plot of a function $r=f(\theta)$ and want to "relocate" it to $(h,k)$. Is there a general procedure for this? I have tried the following tactic to no avail on the following example: ...
1
vote
1answer
29 views

Area of intersection between line and circle?

I have a circle $\mathcal{C}$ and a line $\mathcal{L}$ in the euclidean plane. Let say that the equation of the circle and the line are given respectively by: $$E_{\mathcal{C}}: ...
3
votes
2answers
59 views

What is the most elementary but still correct according to the most rigorous standard proof of the isoperimetric inequality?

Can you write the most elementary proof of the isoperimetric inequality (but still correct according to the most rigorous standard )? $$l^2> 4πA$$
0
votes
0answers
34 views

Polygones inscribed with in a circle

Let's say that there is a circle in two dimension and the diameter of the circle is 1.First start with an equilateral triangle inscribed with in the circle and the measure of the angles are equal to ...
55
votes
6answers
4k views

Area covered by a constant length segment rotating around the center of a square.

This is an idea I have had in my head for years and years and I would like to know the answer, and also I would like to know if it's somehow relevant to anything or useless. I describe my thoughts ...
0
votes
0answers
22 views

Name of this Formula [Spherical Earth projected to a plane]

I am using a formula to calculate the distance between two coordinates. Basically this is the Pythagorean theorem. I saw this formula on Wikipedia and it works perfectly for my use case. However I ...
0
votes
0answers
24 views

Isoperimetric inequality proof [duplicate]

Can someone give me a neat clear proof (the most simple but rigorous avaiable) of the isoperimetric inequality $L^2> 4πA$?
11
votes
1answer
280 views

A variation of the isoperimetric problem in the plane

The isoperimetric problem in the plane: « The classical isoperimetric problem dates back to antiquity. The problem can be stated as follows: Among all closed curves in the plane of fixed ...
1
vote
1answer
41 views

I need to do math from ground up, so what is a good workbook?

Can you guys recommend me a workbook that begins with arithmetic and ends with calculus. Or from pre-algebra to calculus. Like all "Master Math Series" books but in one complete book. It would really ...
2
votes
2answers
136 views

Closest distance between two quadratic curves

I'm having trouble with the following problem : "find the closest distance between $x^2+4y^2=4$ and $xy=4$" I tried to solve using the properties of ellipse and hyperbola, but the relatively tilted ...
1
vote
1answer
62 views

Fitting a Circle Arc to a Parabola

Reading this paper for a project. In section 2.1 it says an approximate formula for the smooth curve described by the edge of the ski is y ≈ $x^2/2R_{SC} − d$. Why is the $x^2$ value divided by ...
3
votes
1answer
45 views

Two curvature formulas when equal arc-length

all. So with a parametric curve $\vec{r}=\langle x(t),y(t)\rangle$, curvature is given by $$\kappa=\frac{|x'y''-x''y'|}{(x'^2+y'^2)^{3/2}}.$$ When we have constant arc-length, an alternate ...
0
votes
0answers
66 views

find the distance between two non-parallel lines using calculus

Let $(x_1,x_2,x_3) + s(u_1,u_2,u_3),(y_1,y_2,y_3) + t(v_1,v_2,v_3)$ be two non-parallel lines in $\mathbb{R}^3,s,t \in\mathbb{R}$.the problem is to compute the distance between the two lines.By ...
2
votes
2answers
56 views

Find the maximum angle possible

$P$ is a point on the $Y-axis$ . Find the maximum possible value of $\angle APB$ where $A=(1,0)$ and $B=(3,0)$. Here is how I solved the problem. Suppose $P=(0,k)$ . Then using the cosine formula we ...
0
votes
1answer
23 views

Finding a paremetric curve traced out by certain lines around a circle

Sorry for the non-descriptive title: it was a rough choice between 3 lines or less description. Let $O$ be the origin and the centre of a circle with radius $a$. Let $T$ be a point on the circle so ...
0
votes
1answer
55 views

Proof: The coordinates of the witch of Agnesi curve

I need to prove that the coordinates ofthe witch of Agnesi curve is: $$x=2a\cot \theta$$ and $$y=2a\sin ^2 \theta$$ Any idea how to prove it? And I don't understand how we got $a$... (because the ...
3
votes
1answer
60 views

About limits and successions

This is a problem which I am not sure I solved correctly, mainly because there are some passages which are not very rigourous. Let $\{x_n\} $ be an increasing succession such that $x_n > 0$ and ...
0
votes
1answer
21 views

Prove the angle of $v \in \mathbb R^2 $ after rotation by $R_{\theta}$ (rotation matrix) change $\theta$-degrees.

Prove the angle of $v \in \mathbb R^2 $ after rotation by $R_{\theta}$ (rotation matrix) change $\theta$-degrees. We have $R_{\theta} =\begin{bmatrix}\cos(\theta) & ...
2
votes
1answer
106 views

Suppose $x_{\theta} = (\cos(\theta),\sin(\theta )) \in \mathbb R^2$. Prove $||x_{\theta + \theta^{'}} - x_{\theta}|| = ||x_{\theta^{'}} - x_{0}||$.

Suppose $x_{\theta} = (\cos(\theta),\sin(\theta )) \in \mathbb R^2$. Prove $||x_{\theta + \theta^{'}} - x_{\theta}|| = ||x_{\theta^{'}} - x_{0}||$. How do I prove that these values are ...
4
votes
1answer
27 views

Calculating Volume using Shell Method

Hey everyone, I'm trying to do this problem and I'm just a little confused. when it says to the right of x=0.5 does that mean I should subtract the integral by 0.5 and I'm confused on whether I ...
0
votes
2answers
26 views

Muddled Notions Regarding the Measurement of Quantities in Different Dimensions

In my calculus class, I've learned a lot about finding areas/volumes of various shapes by summing up infinitely small slices of 'something' and adding them all up. This is very interesting to me, but ...
1
vote
0answers
45 views

The distance between two distinct points in the upper half plane

I'm trying to derive the distance between two distinct points in hyperbolic space and I'm working on the upper half plane. So, with the parametrization $\sigma(t): x=r\cos(t), y=r\sin(t),\; ...
0
votes
1answer
30 views

Size of square formed by soap in a cube frame

So through the work of Plateau (as I understand it), we know that soap tries to find the shortest connection between points. At least, that's what I was taught. With this in mind, I had to solve the ...
0
votes
1answer
30 views

How to find the value of tangent vectors?

In the figure $z_1=a_1+ib_1$ and $z_2=a_2+ib_2$ are the two curves. $T_1$ and $T_2$ be the tangents on the curves $z_1$ and $z_2$. What I am interested to know what will be the tangent vectors?
0
votes
1answer
51 views

Surface area of a Hypersphere

Hypersphere in 4 dimensions, I am having problem with finding the surface area of it. please help. I know that surface area will have 3 dimensions in 4 dimensional space, I am having trouble to ...
0
votes
1answer
40 views

$\angle BAC=20^{\circ}, \angle BDA=70^{\circ}, \angle BCA=35^{\circ}, \angle BDC=40^{\circ}$. Then : $\angle AOD=?$

The quadrilateral $ABCD$, $AC\cap BD=\left \{ O \right \}$; $\angle BAC=20^{\circ}, \angle BDA=70^{\circ}, \angle BCA=35^{\circ}, \angle BDC=40^{\circ}$. Then : $\angle AOD=?$ Thanks ! :) P/s ...
1
vote
3answers
53 views

All points where line tangent to circle

I'm given that the circle has the equation $x^2+y^2=1$. There is a line through $(5,0)$ and $(x,y)$. How do you find all $x,y$ such that the line is tangent to the circle? $(x,y)$ is on the circle.
2
votes
2answers
98 views

parametrization of plane in $\mathbb R^3$

Parametrize the plane in $\mathbb R^3$ with direction vectors $\hat u$ and $\hat v$ and through the point $p$ as in representation as the range of a $C^1$ function $f:\mathbb R^2\to\mathbb R^3$. ...
0
votes
0answers
14 views

Identify the surface and find the traces of ellipsoid?

How can you find the traces of the following ellipsoid. $9x^2+4y^2+36z^2=36$ would not the xy trace be (0,0,0) as the graph passes through the origin.
1
vote
1answer
78 views

Finding the radius, distance of the center of circle inscribed in the square

I am trying to solve this question but can't figure out the last part. I was able to get answers for part A and B but i don't know how to approach/solve part C. Any help will be appreciated. The ...
0
votes
0answers
33 views

How can I maximize the area of a rectangle given a continuous range of values for the length and width?

All the examples I have for rectangle area maximization problems start by having one of the sides fixed. But suppose I have a continuous range such that length is between A and B and height is between ...
1
vote
3answers
115 views

Why can't I obtain values of Sine/Cosine with $f(x)=\sqrt{1-x^2}?$

Taking into account the circle equation $ x^2 + y^2 = 1$, I've made the following function on Mathematica : $ f (x) = \sqrt {1 - x^2}$ which yields this plot with the domain $ 0\leq x\leq 1$ : I' ...
0
votes
0answers
30 views

interior nodes of

I have a circle by center $(0,0)$ and radius $0.4$ that some points are spread on it. Now I want to choose those points that are inside the following domain. $$x(t)=(0.2+0.2 .sin(5t))cos(5t)$$ ...
1
vote
2answers
45 views

is there a problem in the answer? finiding an angle

Can you tell me if there's an error in the answer? Given isosceles triangle $ABC$ ($AB=AC$) and $AB=b$. $BD$ is perpendicular to $AC$ and $DE$ is perpendicular to $BC$. Angle $BAC=2x$. The ...
1
vote
2answers
43 views

Expressing a multivarible function as ∇ϕ [closed]

Show that $g(x,y,z)=(yz^2 +3, xz^2 +2z+1, 2xyz+2y)$ can be expressed as $∇ϕ$ for a function $ϕ$ and find $ϕ$. Thanks.
2
votes
1answer
104 views

tower height problem

Let's say we are measuring the height of a tower at two points in the same line from the tower. Point $A$ is at $50\pm1^\circ$ angle towards the tower and point $B$ at $35\pm1^\circ$. The distance ...
1
vote
1answer
33 views

Problem in conics question

A vertical line passing through the point ($h$,0) intersects the ellipse $$\frac{x^2}{4}+\frac{y^2}{3}=1$$ at the points P & Q.Let the tangents to ellipse at P & Q meet at the point R.If ...
0
votes
1answer
24 views

Count the surface limited by two curves.

I am not very familiar polar coordinate system and I am obliged to count surface limited by two curves $r=\frac{1}{\phi}$ and $r=\frac{1}{\sin\phi}$ and $\phi \in(0,\frac{\pi}{2})$ I know the formula ...
2
votes
1answer
79 views

Existence of a real which allows a observer to see a square in a circle

In $\mathbb{R}^2$ was disposed on each point coordinates in $\mathbb{Z}$, except $(0,0)$, a small open square of side $2r$ where $r\in]0,\frac{1}{2}[$. Prove that there is a real $R>0$ such that ...
2
votes
1answer
30 views

Counting the volume of block.

I am given function $f(x)=\frac{1}{\sqrt{(x^2+1)}}, x\in\mathbb{R}$ I am obliged to find the volume of block which is designated by rotation of $f(x)$. The rotation is made realtive to the axis $OX$. ...
0
votes
1answer
58 views

Volume of cube with faces laying flush with planes

The cube has faces laying flush with planes: $$P_1 = \{(x,y,z) \in \mathbb{R}^3 | x + 2(y-3) + 2z = 1\}$$ $$P2 = \text{the plane parallel to $P_1$ but containing point }(2, 2, 2)$$ I'm not really ...
1
vote
1answer
92 views

Revolve a 3D shape around an axis to create a 4D shape (and so on and so fourth)

You can revolve a 2 dimensional shape around an axis to make it a 3 dimensional shape, and finding the volume of this shape is pretty simple using the disk method. What I want to know is if it is ...
1
vote
1answer
68 views

Let L be the line of intersection of the planes $cx + y + z = c$ and $x - cy + cz = -1$, where c is a real number.

Find symmetric equations for $L$ As the number $c$ varies, the line $L$ sweeps out a surface $S$. Find an equation for the curve of intersection of $S$ with the horizontal plane $z = t$ (the trace of ...
0
votes
1answer
50 views

Solid of revolution volume.

I have the following problem. A quadratic curve have the following description: $$ \left\{ \begin{array}{l l} f(2) = 0 & \\ f'(2) = 0 & \\ f'(1) = -2 \end{array} \right. $$ ...
2
votes
1answer
87 views

Volume of an n-simplex (Without Probabilities) [duplicate]

Compute the volume of $$ S_n=\{(x_1,x_2,...,x_n)\in\mathbb{R^n},x_i\geq 0,\displaystyle\sum_{k=0}^{n} x_i<1\} $$ I don't really have an idea how to solve it. My 'work': Perhaps I could use $$ ...
3
votes
0answers
62 views

Why does a figure look the same in every coordinate system?

After reading Maximilian M. Answer here: Gauss' Theorem - Can't understand a parameterization I'm trying to figure out why does a figure look the same in every coordinate system I choose. For ...
1
vote
1answer
54 views

Volume of $n$-sphere with recurrence relations

On [Wikipedia][1], there is a discussion of recurrence relations between volumes $S_n$ of $n$-spheres and volumes $V_n$ of $n$-balls. Putting two recurrence relations together can give us the volume ...
1
vote
1answer
64 views

Extreme values of a function with a condition

Could you tell me how to find extreme values of this function? $f_p(x_1, x_2, x_3) = (\frac{x_1^p + x_2^p + x_3^p}{3})^{\frac{1}{p}}$, $p>0$ Here $x_1. x_2, x_3$ are lengths of the sides of a ...