Questions on the use of algebraic techniques to prove geometric theorems.
0
votes
1answer
38 views
Proving the Poincare Lemma for $1$ forms on $\mathbb{R}^2$
I am trying to prove the Poincare Lemma for $1$ forms on $\mathbb{R^2}$. So I said that if I doing this, I start of with
$$\omega = f_1(x_1,x_2) dx_1 + f_2(x_1,x_2)dx_2.$$
First thing I want to ...
0
votes
0answers
24 views
Are my definitions of cotangent space, differential and differential forms and coboundary operator correct?
Define the cotangent space $T_a^*\mathbb{R}^n$. Define the differential of a dunction $f$ at the point $a, df \in T_a^*\mathbb{R}^n$. Write down the explicit formula for the deffertial $df$ in ...
1
vote
1answer
43 views
Prove that a multilinear function $f$ is skew-symmetric if and only if $f = \mathrm{Alt}(f)$
Prove that a multilinear function $f$ is skew-symmetric if and only if $f = \mathrm{Alt}(f)$.
I said the first thing is to prove that $\mathrm{Alt}(f)$ is skew-symmetric. In other words, we want ...
1
vote
2answers
53 views
Computing wedge products
Compute $\omega = (e_1^* + e_2^* + \cdots+ e_n^*) \wedge (e_1^* + e_2^*) \wedge (e_1^* + e_3^*) \wedge \cdots \wedge(e_1^* + e_n^*)$ in the standard form.
I first thuoght I'd pick a value from ...
1
vote
1answer
23 views
Putting the wedge product in standard/normal form
I have to compute the wedge product of
$$(e_1^* + ze_2^*) \wedge (e_2^* + ze_3^*) \wedge \cdots \wedge (e_{n-1}^* + ze_n^*) \wedge (e_n^* + ze_1^*),$$
and then put it in normal/standard form.
So I ...
0
votes
1answer
35 views
How to determine if two points lie in a particular section of circle.
I'll take assistance from the figure below.
O is the center of the circle, and A,B,C are the points on the circle, and are known. i.e. the x,y coordinates of these three points are known. I want to ...
2
votes
1answer
50 views
Very basic geometry question about vectors
I have a very basic question about geometry. The problem is: let $\vec{r}$, $\vec{u}$ and $\vec{v}$ be vectors in the plane such that $\vec{r} = \vec{u} + \vec{v}$ and such that $|\vec{r}|=10$. If the ...
0
votes
1answer
25 views
Intersection Point of a Line and four Planes
Let's assume a helicopter crashes into a wall after flying in a straight line:
$$g : \overrightarrow {OX} = \begin{pmatrix}2\\5\\28 \end{pmatrix}+ \lambda*\begin{pmatrix}1\\\frac{1}{3}\\\frac{-1}{11} ...
1
vote
2answers
45 views
Find area of triangle (given its equations)
Find the area of the triangle in the plane $R^2$ bounded by the lines $y = x$, $y = -3x+8$, and $3y + 5x = 0$
I know that I can find the area of the triangle by taking the half of the area of the ...
0
votes
1answer
49 views
Finding a,b of elipse
Given $x^{2}+y^{2}=R^{2}$, so that we multiply every $x$ by $a$ and every $y$ by $b$, $(a>b)$
And the distance between the focuses of this locus is $48R$, and the area of the rhombus which ...
2
votes
3answers
55 views
Distance between point and line in the complex plane
Let $a,b$ be fixed complex numbers and let $L$ be the line
$$L=\{a+bt:t\in\Bbb R\}.$$
Let $w\in\Bbb C\setminus L$. Let's calculate $$d(w,L)=\inf\{|w-z|:z\in L\}=\inf_{t\in\Bbb R}|w-(a+bt)|.$$
The ...
1
vote
0answers
28 views
Equation of a general conic from 3 points and the major axis
I have read that given 3 points on a conic and the equation ($ax+by+c=0$) of its major axis, we can write the equation of the conic ($Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$). I've seen it done by ...
0
votes
0answers
36 views
Parabolic segment problem
I have a problem. I have tried to solve it but I get $125/6$ instead of $9/2$ (textbook result).
Find the area of the parts of plane given by the solutions of the following system:
$$ ...
4
votes
2answers
71 views
Cross Product Intuition
I know the cross product between a vector $a$ and a vector $b$ is just a vector whose magnitude is the product of magnitude of $b$ times the magnitude of the perpendicular component of $a$ in relation ...
3
votes
1answer
63 views
Intersection between a plane and a sphere
We have a sphere $(x-1)^2 + (y-1)^2 + (z-1)^2 = 1$ and a point $A = (1;1;-1)$. Find all equations of planes which contain the line $OA$ and intersections with the sphere are circles of radius ...
2
votes
2answers
79 views
Circles of radius $2$ passing through origin with centers on $x=1$
There are two circles of radius $2$ that have centers on the line $x=1$
and pass through the origin. Find their equations.
Please explain to me what the problem is really saying.
0
votes
2answers
73 views
Check my answers to the problems related to analytic-geometry
1) Find the equation of the circle of radius $2$ with center at $(3, 0)$.
My answer: $\sqrt{(x-3)^2 + y^2} = 2$
2) Find the equation of the circle of radius $\sqrt3$ with center at (-1, -2). ...
0
votes
1answer
35 views
Problem of sketching a circle
I've to solve a problem in which I've been given this equation: $x^2$ + $y^2$ $=$ $4$ and I've to sketch a circle which is the locus of the equation. Here $'2'$ is the radius $r$ of the circle. 2 ...
2
votes
2answers
84 views
Detecting an Intersection between Simple Shapes
I have a circle, ellipse, square or a rectangle, and I want to determine if it intersects a given triangle.
I am looking for the easiest way to determine if there exists a geometrical intersection ...
1
vote
3answers
54 views
Locus of the equation
One way to describe a set of points in the plane is by an equation or inequality in two variables, say $x$ and $y$. A solution of an equation in $x$ and $y$ is point $(x_0, y_0)$ in the plane for ...
0
votes
3answers
90 views
Distance between two points
The distance between the two points $P(x_1, y_1)$ and $Q(x_2, y_2)$ is the quantity
$$\mathrm{distance}(P, Q) = \sqrt{(\Delta x)^2 + (\Delta y)^2}.$$
Is $(P, Q)$ above indicating an open ...
3
votes
2answers
58 views
Difference of two points on a plane
If $P(x_1, y_1)$ and $Q(x_2, y_2)$ are the two points on a plane, then
the change in $x$ and $y$ coordinates is denoted by $∆x$ and $∆y$ respectively.
Therefore, $x = ∆x = x_2 - x_1$ and $y = ...
3
votes
1answer
46 views
Confusion between an ordered pair, and open interval
The $(x, y)$ plane is the set of all ordered pairs $(x, y)$ of real numbers. The origin is the point $(0, 0)$. The $x$-axis is the set of all points of the form $(x, 0)$, and the $y$-axis is ...
3
votes
1answer
131 views
How is the formula for the focal point of a ball lens derived?
In Optical Design Fundamentals for Infrared Systems 2nd ed., Mr. Riedl writes:
A sphere or ball performs surprisingly well as a lens. At closer
scrutiny, one fmds that such an element can be ...
0
votes
0answers
37 views
Geometric calculation: two kneading discs
I have two kneading discs of a screw overlapping with each other at 60 deg. I know the cross section area of one disc and I want to know what will be the overlap area if the other disc is rotated at ...
1
vote
2answers
95 views
Algebraic solution to find circle radius given distance of three external points from perimeter
I have an engineering problem, which involves math. The reason it's "engineering" is that I don't need a pure mathematical solution, but a good-enough approximation could work - the only constraint is ...
0
votes
1answer
29 views
Equation of a plane passing through a line and a separate point.
Is it possible to find the equation of a plane that passes through a line and a point not on the line? For example, the line $y=7x-7$ and the point $x=3,y=0,z=8$. I've tagged this as homework, but ...
2
votes
0answers
25 views
How does this method to find the centre work?
Say we have a conic with equation $f(x,y)=c$.
My teacher says that it's centre satisfies the equations :
$f_x(x,y)=f_y(x,y)=0$ (If it has a centre).
She didn't give any explanation. I thought this ...
0
votes
3answers
52 views
Prove that $|PC|^2 + |PD|^2 = |AB|^2$ if
We have an angle of 90° so that there are 2 points A, B on each side of the angle, O is the vertex and |OA| = |OB|. On the arc AB with it's center being in O, we pick an arbitrary point P and draw a ...
0
votes
0answers
26 views
Distance functions
Lets consider three points $p_{0},p_{1},p_{2}$ and the coordinates are given by $(x_{0},y_{0},z_{0})$ , $(x_{1},y_{1},z_{1})$ and $(x_{2},y_{2},z_{2})$. Those three coordinates are given. Lets pick ...
0
votes
1answer
49 views
Find a locus of points to satisfy these conditions?
So, we have to straight lines:
$$3x-4y+5=0$$
$$2x+3y-4=0$$
You have to find a locus of points from which all perpendiculars to the two lines given are in a 2:3 ratio.
3
votes
2answers
48 views
proving or disproving that two tangent lines are parallel to a curve
Im trying to prove or disprove that given the function, $f(x)=0.5\sqrt{1-x^{2}}$,
There are two different tangent lines to $f(x)$ so they are parallel.
I tried to derivative but with no success.
5
votes
2answers
79 views
If $\left |z-3\right |=\left |z+i\right |$, where $z=x+iy$, prove that $3x+y=4$
If $\left |z-3 \right |=\left |z+i\right |$, where $z=x+iy$, prove that $3x+y=4$.
I have got to the point where I have $\left |z \right |= \sqrt{x^2+(y+1)^2} = \sqrt{(x-3)^2+y^2}$
But really ...
2
votes
2answers
49 views
Compute the value of the exterior $2$-form
Compute the value of the exterior $2$-form
$$\omega = (x_1 + x_2)e_1^* \wedge e_2^* + (x_2 + x_3)e_2^* \wedge e_3^* + \cdots + (x_i, x_{i+1})e_i^* \wedge ...
0
votes
0answers
32 views
Parametrizing a section of a torus
Consider the torus obtained by rotating the circle $(x-R)^2+z^2=r^2$ about the $z$-axis, where $R>r>0$. Parametrize the part of this torus where $z>x+y$.
My approach to this so far is to ...
1
vote
2answers
80 views
Max and min value of $7x+8y$ in a given half-plane limited by straight lines?
So, there are four inequalities:
$$\begin{eqnarray*}
y &\geq &-3x+15; \\
y &\leq &-11/3x+56/3; \\
x &\geq &0; \\
y &\geq &0.
\end{eqnarray*}$$
If we draw all those ...
1
vote
0answers
26 views
Is there a way to check if my Taylor Expansion is correct?
I have an exam later and I need to do Taylor expansions of functions. I have questions like:
Consider the map $F:\mathbb{R}^2_x \rightarrow \mathbb{R}^2_y$, given by the equations
$$y_1 = ...
1
vote
4answers
52 views
A triangle has corners with coordinates (1,2), (-2,3) and (0,-1). Please help me determine the equations of the lines.
A triangle has corners with coordinates (1,2), (-2,3) and (0,-1). Please help me determine the equations of the lines that form the sides of the triangle.
0
votes
1answer
48 views
finding the locus of quotient lengths of tangents to circles
I'm trying to find the locus of all points so that their quotient of the Tangents lengths
to circles: $x^2+y^2-12x=0, x^2+y^2+8x-3y=0$ is $2:3$, respectively.
i tried to use the formula: ...
10
votes
2answers
83 views
Prove that $|PF_{1}|+|PF_{2}|$ is Constant in an Elipse
Given an elipse with two focus $F_{1}$ an $F_{2}$, and $A$ is an arbitrary point at the elipse. Stright line $AF_{1}$ has another intersection point $B$ with the elipse, and $AF_{2}$ has another ...
3
votes
1answer
61 views
The intuition behind the definition of geodesics on a Riemannian manifold. (A non-technical question)
In the text I'm studying, the idea behind the definition of a geodesic on a Riemannian manifold was sketched via paths in $\mathbb{R}^n$. I have trouble understanding some aspects of it.
Let $\gamma: ...
3
votes
1answer
101 views
Draw an arc in 3d coordinate system
I have some legacy code which is supposed to draw an arc with constant radius in 3d space however it is drawing the arc in the wrong position. I would like to know and understand the mathematical ...
0
votes
0answers
58 views
two points on a unit sphere
Consider the two vectors to the points on the unit sphere,
$${\bf v}_i=(\sin\theta_i\cos\varphi_i,\sin\theta_i\sin\varphi_i,\cos\theta_i)$$
with $i=1,2$. Use the dot product to get the angle $\psi$ ...
1
vote
1answer
137 views
shortest distance between two points on $S^2$
Length of Curve in $2D$ is $l_{\gamma}(\mathbb{R}^2)=\int_{0}^{1}\sqrt{(dr/dt)^2+r^2(d\theta/dt)^2}$
Length of a curve in $3D$ is ...
4
votes
2answers
72 views
Length of curve in 3D spherical coordinate
let $r$ be the magnitude of a vector in 3D with Spherical co-ordinate $(r,\theta,\phi)$ and cartesian coordinates is $(x,y,z)$, whose angle with $z$ axis is $\phi$ and projection of the vector makes ...
1
vote
2answers
70 views
How can I calculate the Euclidian displacement of two places on a sphere (earth in this case ) and calculate the
I would like to get the formula on how to calculate the distance between two geographical co-ordinates on earth and heading angle relative to True North. Say from New York to New Dehli , I draw a ...
-1
votes
1answer
91 views
Length of a curve on $S^2$
$1.$ Could any one tell me what is the shortest distance between $2$ points on $S^2$?
$2.$ Could any one tell me how to measure explicitly a length of a curve on the $S^2$ using polar co-ordinates?
...
3
votes
1answer
89 views
Common area of two rectangles
Suppose we have a rectangle at the center of the coordinates. One top point of the rectangle has the coordinates (a, b), the second (-a, b), third (-a, -b) and (a, -b). We rotate this rectangle with ...
0
votes
2answers
36 views
Points in common
I have the following problem:
How many points do the graphs of $4x^2-9y^2=36$ and $x^2-2x+y^2=15$ have in common?
I know that the answer is in the system of two equations, but how should I solve it?
...
1
vote
3answers
78 views
Proof: Two non identical circles have at most 2 same points
I'm struggeling with an analytic proof for the fact, that two different circles have at most 2 same points. (I try to solve it analytical, because geometrical I already prooved it).
I tried to start ...
