Tagged Questions
7
votes
3answers
68 views
Equation of Cone vs Elliptic Paraboloid
I can't understand why $$\frac{z}{c} = \frac{x^2}{a^2} + \frac{y^2}{b^2} \tag{*}$$ corresponds to an elliptic paraboloid and $$\frac{z^2}{c^2} = \frac{x^2}{a^2} + \frac{y^2}{b^2} \tag{**}$$ to a cone, ...
-1
votes
1answer
59 views
Moment of inertia of a circle
A wire has the shape of the circle $x^2+y^2=a^2$. Determine the moment of inertia about a diameter if the density at $(x,y)$ is $|x|+|y|$
Thank you
2
votes
1answer
77 views
Using Cavalieri's Principle to find the volume of an ellipsoid
I understand how to use the triple integral + change of variable method to find the volume of an ellipsoid, but given an ellipse $$\frac{x^2}{a^2} + \frac{y^2}{b^2} \le 1$$ whose area is$$A= \pi ab$$
...
4
votes
2answers
67 views
Showing the function $f(x,y)$ is one by one
Yesterday, while teaching geometry, I was faced to a problem saying that the function below is an distance function: $$d(P,Q)=\Big|\ln\frac{\frac{x_1-c+r}{y_1}}{\frac{x_2-c+r}{y_2}}\Big|$$ where in ...
0
votes
0answers
22 views
Volume under intersection of 2D Gaussians
Given 2 2D Gaussian functions (probability cdfs, if that matters), is there a direct method to calculate the volume under their intersection?
If not, are there any quick ways to approximate it? ...
0
votes
1answer
32 views
Determining Volume by rotation of a plane
a) What is the area of the region R between the graphs of $y= \sin x$ and $y=\sin ^2 x$ for $x \in [0; \frac{\pi}{2}]$
I found $a(R)= 1-\frac{\pi}{4}$
b) Let R be the region from question a). ...
2
votes
3answers
92 views
Diagonals of parallelogram intersect at $90^\circ$ if and only if figure is rhombus
How can we use vectors and dot products to show that the diagonals of a parallelogram intersect at $90^\circ$ if and only if the figure is a rhombus?
I did the proof, but I realized my final answer ...
4
votes
2answers
78 views
Minimizing the length of a pipeline between cities
I have been trying to minimize piping going to two different cities. City A is located at $(0,4)$ and city B is located at $(6,3)$. The cities must connect to the $x$-axis (the main pipe line.) It ...
0
votes
2answers
36 views
Prove that 3 points are not on the same line
Given $P_1=(1, 1, 1)$, $P_2=(2, -1, 2)$ and $P_3=(3,0,1)$, I need to prove that these three points are not on the same line.
What I tried - I showed that $\vec{P_1P_2}$, $\vec{P_1P_3}$ and ...
0
votes
1answer
32 views
What happens to the Frenet-Serret frame when $\kappa=0$?
I was considering the following question for 3D curves: Does zero curvature imply zero torsion?
I think it's reasonable, because zero curvature implies the curve is a straight line, which lies in a ...
1
vote
1answer
78 views
Proof that cone not diffeomorphic to plane
What is the simplest way to show that the cone $\{(x,y,z)\in\mathbb R^3\,|\,z=\sqrt{x^2 + y^2}, z\geq 0\}$ is not diffeomorphic to $\mathbb R^2$?
After some comments, I realize that this question The ...
3
votes
1answer
48 views
Every closed $C^1$ curve in $\mathbb R^3 \setminus \{ 0 \}$ is the boundary of some $C^1$ 2-surface $\Sigma \subset \mathbb R^3 \setminus \{ 0 \}$
How can I prove it?
This problem looks similar to Plateau's problem - but it is much more specific. I believe there exists some elementary proof.
(Proving this will help me apply Stokes' theorem to ...
-2
votes
1answer
75 views
Volume of a pyramid.
Find the volume of the pyramid with base in the plane $z=−9$ and sides formed by the three planes $y=0$ and $y−x=3$ and $2x+y+z=3$.
1
vote
2answers
69 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 ...
5
votes
1answer
74 views
Volume of an n-simplex
It's rather tedious to show using Fubini's Theorem and induction on $n$ that the volume of the region $x_1+x_2+...+x_n \leq 1$ with $x_1,...,x_n$ nonnegative is $\frac{1}{n!}$. Is there an easier way ...
0
votes
1answer
57 views
Equation with 2 variables tricky problem
So let's say I have some random equation $6yx^3 - 3yx + 5 = 0$, but could be anything.
How would I go about finding a value for $y$, that makes it so that this equation only holds true if $x$ is ...
4
votes
2answers
170 views
Has anyone published a formula for the volume of the intersection of two balls
Math people:
I have Googled this question and searched Math Stack Exchange, and not found an answer. Given $r_1, r_0, t >0$, $r_0 \leq r_1$ I have found a formula for the volume of the ...
2
votes
1answer
94 views
What does the definition of curvature mean?
First question, am I right in saying that curvature measure how quickly the direction of a cruve changes?
Also, we have been given the "definition":
$$T'(t) = \kappa(t) | \gamma'(t) | U(t)$$
where ...
4
votes
2answers
318 views
Geometric intuition behind gradient, divergence and curl
I learned vector analysis and multivariate calculus about two years ago and right now I need to brush it up once again. So while trying to wrap my head around different terms and concepts in vector ...
2
votes
0answers
179 views
Parametrization of square to calculate Dot-product in line-integrals and area-integrals, electric field from $\frac{dB}{dt}$?
I am calculating 3A of Tfy-0.1064 in Aalto University. I realized here that I am misunderstanding something in vector calculus: the thing market in green particularly.
I know
$$\nabla\times E= ...
0
votes
2answers
51 views
Set of points reachable by the tip of a swinging sticks kinetic energy structure
This is an interesting problem that I thought of myself but I'm racking my brain on it. I recently saw this kinetic energy knick knack in a scene in Iron Man 2:
...
3
votes
1answer
693 views
Intersection between a rectangle and a circle?
I have a poor working knowledge of math. I would like to calculate collision detection between a 2D circle and a 2D rectangle for a simple game of Pong.
I thought of splitting the 2D rectangle into 4 ...
1
vote
1answer
125 views
What are the transformations of the plane called whose derivatives at each point are isometries?
Let $f:\Bbb R^2\to\Bbb R^2$ be a differentiable function. Are there names for the following two conditions?
$Df(p)$ is an isometry at each point $p\in\Bbb R^2$;
$Df(p)$ is a similarity at each point ...
0
votes
1answer
88 views
How to get the arc length of a real-valued function in 3D.
The definition of arc length of a parametric function is given by
$$\int|r'(t)|dt=\int\sqrt{[x'(t)]^2+[y'(t)]^2+[z'(t)]^2} dt$$
So I guess what I'm asking is how do I use a function like ...
1
vote
1answer
59 views
The length of a space curve =? The diagonal across the rectangle surface of a cylinder
I was looking through my multivariable calculus homework and I saw an example where we needed to find the arclength of a simple space curve. It's very simple:
$$r(t)= cos(t)\hat i + sin(t)\hat j + ...
0
votes
1answer
101 views
Paraboloidal Coordinates
How would one find the transforms for paraboloidal coordinate systems. ie) I want to find $x,y$, and $z$ in terms of other variables so that I can use the Jacobian to find the differential volume.
...
0
votes
0answers
73 views
Identities related normal unitary vector, normal tangent vector and curvature.
i need help with this homework, about vector calculus.
Given a soft parametrized curve $r(t)$, we got that $T(t)=\frac {r´(t)}{\|r´(t)\|}$, the curvature is $\kappa(t)=\|\frac {dT}{ds}\|$=$\frac ...
6
votes
0answers
154 views
Computing the volume of a region on the unit $n$-sphere
I would like to compute the surface volume of a region on the unit $n-1$-sphere:
$$x_1^2 + \dots + x_i^2 + \dots + x_n^2 = 1,$$
bounded by an ellipsoid
$$a_1x_1^2 + \dots + a_ix_i^2 + \dots + ...
1
vote
2answers
119 views
Dot product of vectors and projections
Can someone explain to me what this means (figure at the end of the post; original link to image: http://imgur.com/7vvDs). I understand the part that says the scalar projection of vector $u$ onto ...
1
vote
2answers
164 views
Why are two vectors that are parallel equivalent?
Why are two parallel vectors with the same magnitude equivalent?
Why is their start point irrelevant?
How can a vector starting at $\,(0, -10)\,$ going to $\,(10, 0)\,$ be the same as
a vector ...
0
votes
1answer
80 views
Directional derivative (muiltivariable calculus)
I have an encountered an example in my text book which I don't fully see the intuition of. I will write out the part of the example I'm struggling with:
A hiker is standing beside a stream on the ...
2
votes
0answers
152 views
Relationship between $\nabla\cdot n$ and the normal $n$
If $k=\nabla\cdot n$, what is the geometric relationship between $k$ and $n$?
In terms of size and direction?
Is it true that $n$ is an outward-pointing normal iff $k>0$ and $n$ is ...
5
votes
2answers
274 views
Existence of Saddle Point
Consider a function $g$ with the following properties.
It is smooth.
$g > 0$.
$g \to 0$ at infinity.
It has at least two critical points.
There are finitely many critical points.
Each critical ...
4
votes
0answers
157 views
Laplace-Beltrami Operator for Euclidean Space
Consider the space $\mathbb{R}^n$ and let $x_1,\ldots, x_n$ be the coordinates. Fix the orientation $dx_1\wedge dx_2\ldots\wedge dx_n$. Let $E^p$ denote the space of smooth $p$ forms and let $d$ ...
2
votes
1answer
1k views
Calculating Solid angle for a sphere in space
How can I calculate the solid angle that a sphere of radius R subtends at a point P? I would expect the result to be a function of the radius and the distance (which I'll call d) between the center of ...
8
votes
2answers
170 views
What is it that makes this proof about rational rectangles work fundamentally?
I saw this problem several years ago, and I discovered a solution to it. I've since learned a somewhat more efficient solution based on the same idea.
Call a rectangle in the $(x,y)$ plane ...
6
votes
0answers
131 views
Volume of a sphere by “adding” half-spheres of lower dimension
I'm wondering about different ways to compute the volume of an $n$-sphere. Please see the wikipedia page for one method to compute the volume via hyperspherical coordinates:
...
2
votes
1answer
571 views
When is the determinant of a Hessian matrix positive?
Let $f:\mathbb{R}^n\to \mathbb{R}^n$ be a $C^2$-function and let $H=\left(\frac{\partial^2f}{\partial x_i \partial x_j}\right)_{1\le i,j\le n}$ be its Hessian matrix. Suppose I know that $ \det ...
23
votes
4answers
769 views
Volume of Region in 5D Space
I need to find the volume of the region defined by
$$\begin{align*}
a^2+b^2+c^2+d^2&\leq1,\\
a^2+b^2+c^2+e^2&\leq1,\\
a^2+b^2+d^2+e^2&\leq1,\\
a^2+c^2+d^2+e^2&\leq1 &\text{ ...
1
vote
1answer
994 views
Parameterizing the upper hemisphere of a sphere with an upward pointing normal
Can someone explain how to do this?
area we're dealing with:
$x^2 + y^2 + z^2 = a^2, z \geq 0$
I'm aware that the answer is:
$x = a \sin(\phi) \cos(\theta)$
$y = a \sin(\phi) \sin(\theta)$
$z = ...
2
votes
1answer
227 views
3d axis rotation
I have a vector V= and several line segments Seg1, Seg2, Seg3, Seg4.
I want to know how to rotate each of the line segments so that the X axis is parallel to my given vector.
How can I do this?
...
1
vote
1answer
214 views
Get vector components from from magnitude and angle
I am given the length and the direction of a vector, and I need to get the the X,Y components. I can go one way, but going the other has me a little lost.
Example:
A man walks 3.50 m south, then ...
2
votes
1answer
189 views
How to compute Hyper-area?
The function $A=(\sin(y)\sin(z)+\cos(y)\cos(z))\sin(w)\sin(x)+\cos(w)\cos(x)$, given $w\in[0,\pi], x\in[0,\pi], y\in[0,2\pi], z\in[0,2\pi]$, defines a three-dimensional "surface" in 4D. ($A = ...
1
vote
3answers
5k views
Finding parametric equations for the tangent line at a point on a curve
Find parametric equations for the tangent line at the point $(\cos(-\frac{4 \pi}{6}), \sin(-\frac{4 \pi}{6}), -\frac{4 \pi}{6}))$ on the curve $x = \cos(t), y = \sin(t), z=t$
I understand that in ...
2
votes
1answer
2k views
Simple but Stuck: How do I find the point of intersection of two lines in Vector Calculus?
Simple but Stuck: How do I find the point of intersection of two lines in Vector Calculus? Given symmetric equations and deriving parametric equations.
Find the point of intersection of the lines L1: ...
4
votes
2answers
689 views
Create a trapping region for Lorenz Attractor
I would like to show that the quantity:
$-2\sigma\left(rx^{2}+y^{2}+b\left(z-r\right)^{2}-br^{2}\right)$
is negative on the surface:
$rx^{2}+\sigma y^{2}+\sigma\left(z-2r\right)^{2}=C$
for some ...
