0
votes
0answers
42 views

Arc length for a function $f:\mathbb{R}^2 \to \mathbb{R}^2.$

Assume $f:\mathbb{R}^2 \to \mathbb{R}^2$ is $C^1.$ Is there a formula for the length of the subset of $\mathbb{R}^2$ given by $$ \{f(x,y) \in \mathbb{R}^2:a_1\leq x\leq b_1, a_2\leq y \leq b_2\} ? $$ ...
0
votes
3answers
23 views

Finding a vector normal to the plane with position point and parallel to two vectors

A cross product of the two parallel vectors will get the vector normal to the plane. But I'm looking for a specific normal vector. The plane with point $A(3,2,1)$ and parallel to $u = 5i+2k$ and $v= ...
3
votes
1answer
48 views

Relationship between Surface Area and Volume

Question: Is there a general relationship between surface area and volume analogous to the below examples? Example 1. Consider a ball $B$ centered at the origin of a spherical coordinate system. The ...
3
votes
3answers
61 views

Easiest way to find the (shortest) distance between a point and a line in $3$-space

I have tried doing some research on this and am looking for the easiest way to compute this distance. For example, Let $l$ be the line determined by $x=y=z$. Find the shortest distance from this line ...
1
vote
0answers
34 views

From 2 to 3 dimensions: integrating a force along a contour/surface.

I am studying the following problem: Consider a closed contour $\mathcal{C}$ in $\mathbb{R}^2$ defined by $r(\theta)$ where $\theta\in[0,2\pi)$ and $r(0)=r(2\pi)$ (let the center to be zero for ...
0
votes
0answers
35 views

What is the density of a homogeneous disk with mass $m$ and radius $a$?

Could someone help me understand why the density of a homogeneous disk is $\dfrac{m}{(\pi a)^2}$? I am trying to understand an example about finding the moment of inertia of an object. The question ...
0
votes
2answers
24 views

Having a bit of trouble with min/max distance from sphere to point

The sphere is $x^2 + y^2 + z^2 = 81$ and the point is $(5,6,9)$ I am using Langrane multipliers , but the answers I am getting are so far off. I will post my system of equations soon. I found ...
1
vote
1answer
43 views

Surface area of lateral section of paraboloid

Here is my 2D parabola curve. The x,y locations of each point of interest are displayed, and the equation of the parabola and the line are given as well. I want to create a hollow 3D Paraboloid by ...
1
vote
1answer
29 views

The normal of a surface that passes through the origin

this is my first question here. Suppose I have a surface as follow: $$x^2+y^2+z^2=9$$ The gradient of the surface at a particular point $P = (x_0,y_0,z_0)$ is just $(2x_0,2y_0,2z_0)$ and the ...
1
vote
1answer
33 views

surface area of the graph of a convex function

I started out with the following question: Say $\Omega$ is a nice bounded domain in $\mathbb{R}^{n-1}$. (One can imagine it being a unit ball in $\mathbb{R}^{n-1}$.) Let $f:\Omega\rightarrow ...
1
vote
1answer
79 views

The shortest path connecting three points

I have 3 points X,Y,Z, lets call them buildings. I need to find the shortest amount of path that connects the 3 buildings, these buildings can be in any sort of shape and any distance from each ...
1
vote
2answers
91 views

Constructing a Cone and its Normal Vectors in Spherical Coordinates

I am attempting to construct a right circular cone of maximum radius $R$ and angle $\theta$ in spherical coordinates, then find the normal vector of the surface of this cone at all points. Here's what ...
2
votes
2answers
74 views

Area of Intersection of Circle and Square

Given a point $(x,y)\in [0,1]^2$ and $r > 0$, I would like to derive a general formula for the area of the intersection of the circle of radius $r$ centered at $(x,y)$ and the unit square. What is ...
0
votes
1answer
16 views

Equation of a cone in $\mathbb{R}^n$ formed by taking all points within angle $\alpha$ of a line.

What is the equation of a cone in $\mathbb{R}^n$ formed by taking all points within angle $\alpha$ of $e_1$ (the first standard normal vector)? I would be interested in a parametric equation of its ...
0
votes
1answer
12 views

Computing volume of an ellipsoid with a cone about its major axis removed?

Let $E$ be an $n$-dimensional ellipsoid in $\mathbb{R}^n$ centered at $0$, and let $C$ be the cone formed by the set of all points within angle $\alpha$ of the major axis of $E$. How can one compute ...
0
votes
1answer
61 views

Find the equation of a plane which is perpendicular to another plane

Find the equation of a plane which is perpendicular to the plane $\pi\equiv x+2y-2z+3=0$ and it intersects it through the line that lies in the XOZ plane. Normal vector of the given plane is ...
0
votes
1answer
60 views

Regular Parametrization of a Sphere

Is there a function $f:U→ \mathbb{R^3}$, such that: (1) U is an open connected subset of $ \mathbb{R^2} $; (2) f is $ C^r , r≥1$; (3) the Jacobian of f is of maximal rank at all points of U; (4) ...
0
votes
1answer
77 views

How to find the equation of a line which intersects these lines at 90 degrees?

How to find the equation of a line which intersects these lines at 90 degrees? $p\equiv \dfrac{x}{2}=\dfrac{y+1}{0}=\dfrac{z-2}{1}$ $q\equiv \dfrac{x-1}{1}=\dfrac{y-2}{1}=\dfrac{z+5}{0}$ Since the ...
0
votes
1answer
177 views

Transformation of ellipsoid to sphere

So I need to find an volume-preservating mapping from an ellipsoid to a ball (solid sphere). (Specifically: x^2/9 + y^2 + z^2 <= 3, but I'd rather understand the general case than just get how to ...
0
votes
0answers
36 views

Can every smooth path be parametrized?

How should I go prove this, namely, if $C\in R^n$ is a smooth path, then $\exists r(t),t\in[a,b]$ which describes C. My textbook on line integrals, multi-variable chain rule, etc takes this as ...
0
votes
2answers
65 views

How to calculate volume of a solid under a given surface with double intergrals?

How can I calculate the volume of the solid under the surface $z = 6x + 4y + 7$ and above the plane $z = 0$ over a given rectangle $R = \{ (x, y): -4 \leq x \leq 1, 1 \leq y \leq 4 \}$? I know I have ...
2
votes
1answer
92 views

Volume of ellipsoid using Linear Algebra

Can someone tell me how to find the volume of an ellipsoid of dimension $\mathbb{R}^3$ by using linear algebra? I know the formula is $\frac{4}{3}\pi abc$. I am given the equation ...
0
votes
0answers
55 views

How much can it be?

How to estimate from above (as well as possible) the $n-1$-dimensional Lebesque measure of the intersection of the $n-$dimensional unit cube $\{(x_1,x_2,\dots,x_n): |x_1| \le \frac 1 2,\dots, |x_n| ...
3
votes
3answers
276 views

Calculate the mass of the earth's atmosphere give the density of air.

Assume the desity of air $\rho$ is given by $\rho(r)=\rho_0$$e^{-(r-R_0)/h_0}$ for $r\ge R_0$ where $r$ is the distance from the centre of the earth, $R_0$ is the radius of the earth in meters, ...
0
votes
1answer
43 views

Finding the distance between a plane and $(0,0,0)$

Given the lines: $ \frac{x+1}{4} = \frac{y-3}{1} = \frac{z}{k} $ and $\frac{x-1}{3} = \frac{y+2}{-2} = \frac{z}{1} $ that lie on the same plane. How can I find the parameter $k$ ? (I guess ...
0
votes
2answers
35 views

Double integral of a region.

Could someone help with the following question please: For shape one I think it is just $ \int_{-1}^{1} \mathrm \int_{-1}^{1} \mathrm{f(x,y)}\,\mathrm{d}xdy $
1
vote
2answers
229 views

Find the volume bounded by a cylinder.

Find the volume bounded by the cylinder $x^2 + y^2=1$ and the planes $y=z , x=0 ,z=0$ in the first octant. How do I go about doing this?
0
votes
1answer
65 views

Find all points at which the direction of steepest ascent of a function is in a given direction.

How do I find all the points at which the direction of steepest ascent of the function $f(x,y)$=$x^2+y^2-2x-4y$ is in the direction $(\sqrt{\frac{1}{2}},\sqrt{\frac{1}{2}}))$
3
votes
0answers
67 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 ...
0
votes
1answer
75 views

Gauss' Theorem - Can't understand a parameterization

Given the following file (example 6): http://math.bard.edu/~mbelk/math601/GaussExamples.pdf Can someone please explain to me how did he move from the parameterization $ t=\theta, r=cos(u), z=sin(2u) ...
2
votes
2answers
494 views

Surface area of intersection of two cylinders

Let $$R=\{(x,y,z):y^2+z^2\leq 1\,\, \text{and}\,\, x^2+z^2\leq 1\}.$$ Compute the volume of $R$. Compute the area of its boundary $\partial R$. I'm fine with #1. For #2, I have a ...
0
votes
0answers
87 views

Question about spherical curvature ( binormal, tangent vectors)

Let $a:I\mapsto R^3$ be a unit speed curve. if $\rho^2+(\rho'\sigma)^2$ is constant and equal to $r^2$, show that a curve is on a sphere with r radius. answer is: define $\gamma:I\mapsto R^3$ ...
1
vote
0answers
134 views

The curvature and torsion of the tangent indicatrix

Let $\alpha$ be a unit speed curve. Its tangent indicatrix $\sigma$ is defined by $\sigma(t)=T(t)$. Find torsion and curvature of $\sigma$ with respect to the torsion and curvature of $\alpha$. ...
1
vote
1answer
38 views

Change of Variables from Sphere to Plane

Say we are in the space $\mathbb{R}^{n}$. Consider the $n-1$ dimensional surface measure $dS$ on the boundary of the upper half sphere. We can define the coordinates on this sphere by ...
1
vote
1answer
121 views

Calculate normal vector to $2$-face of polytope in $\Bbb R^n$

I am trying to work through a divergence theorem application for a function integrated over an $n$-dimensional convex polytope, but I can't seem to figure out how to properly calculate the normal ...
1
vote
1answer
85 views

Stokes' Theorem: line integrals around 2-faces of n-dimensional surface?

Suppose we have a convex polytope in $n$ dimensions and are trying to calculate the surface integral (over this polytope) of some scalar function $f:R^n \rightarrow R$. Suppose all edges and vertices ...
1
vote
0answers
40 views

A question on the minimums

For any given $a_{i},\ a_{i}',\ c\in\mathbb{R},a_{i}\le\ a_{i}',\ i=1,2,\ 3$, let $$ S:=\left\{ \left(x_{1},x_{2},x_{3}\right)\in\mathbb{R}^{3}:\ x_{i}\in[a_{i},a_{i}'],\ i=1,2,3,\ ...
3
votes
4answers
979 views

Finding distance between two parallel 3D lines

I can handle non-parallel lines and the minimum distance between them (by using the projection of the line and the normal vector to both direction vectors in the line), however, in parallel lines, I'm ...
1
vote
1answer
171 views

Navier-Stokes equations in tensorial form on a general coordinate system

How to write the classical Navier-Stokes equations in tensorial form on a general coordinate system? Any references?
0
votes
2answers
137 views

Volume of a triangular prism with non parallel bases

Consider an $\mathbf{(v_1,v_2,v_3)}$ triangle and its $\mathbf{\hat{n}}$ unit normal. Let $\mathbf{p_i}=\lambda_i\mathbf{\hat{n}} + \mathbf{v_i}$, $i=\overline{1,3}$. Is it possible to compute the ...
0
votes
1answer
89 views

Area in the spherical coordinate system

I need to configure a constant infinitesimal area (constant magnitude) on the surface of a sphere. But the elemental area $dA=r^2\sin \theta d\theta \,d\phi$ which depends on $\theta$. Does it mean, ...
3
votes
1answer
55 views

Show that $\dot{n_s}=-\kappa_s t$

I found the question in a differential geometry textbook while studying. This question seems so intesting to me. So please help me solving it. Show that, if $\gamma$ is a unit-speed plane curve, ...
3
votes
2answers
53 views

Volume of a Special Pyramid

Let $P$ be a plane in $\mathbb{R}^3$ parallel to the $xy$-plane. Let $\Omega$ be a closed, bounded set in the $xy$-plane with $2$-volume $B$. Pick a point $Q$ in $P$ and make a pyramid by joining ...
2
votes
2answers
87 views

Find the volume of the intersection of the cylinders $\{(x,y,z)\in \mathbb{R}^3: x^2+z^2\leq 1\} \cap \{(x,y,z)\in \mathbb{R}^3:y^2 + z^2 \leq 1\}$

Find the volume of the intersection of the cylinders $$\{(x,y,z)\in \mathbb{R}^3: x^2+z^2\leq 1\} \cap \{(x,y,z)\in \mathbb{R}^3:x^2 + y^2 \leq 1\}.$$ My first approach led me into contradiction, and ...
1
vote
2answers
85 views

Volume of Generalized Tetrahedron in $R^n$

I'm having difficulty finding the volume of a tetrahedron in $\mathbb{R}^n$. Find the volume of a generalized tetrahedron in $\mathbb{R}^n$ bounded by the coordinate hyperplanes and the hyperplane ...
0
votes
1answer
70 views

Unit speed curves and Frenet frames

Let $\alpha(s)$ and $\beta(s)$ be two unit speed curves and assume that $\kappa_{\alpha}(s)=\kappa_{\beta}(s)$ and $\tau_{\alpha}(s)=\tau_{\beta}(s)$, where $\kappa$ and $\tau$ are ...
2
votes
2answers
1k views

Why does the derivative not exist at a cusp?

I'm trying to grasp what's going on at a cusp geometrically. For instance, $y^2=x^3$ is not differentiable at the origin. In $y$ things appear fine: differentiate $y = \pm x^{3/2}$ and we get $y'=0$ ...
1
vote
0answers
20 views

Angle between two centered noisy vectors

Let $\mathcal{H} = \lbrace u\in \mathbb{R}^n \mid \langle x, (1, 1, ...., 1) \rangle = 0 \rbrace $, the hyperplain where the avarage is zero i.e. $\frac{1}{n}\sum\limits_{i=1}^n x_i = 0$. Given two ...
0
votes
1answer
64 views

Point of tangency for a circle between two vector

I'm having two vectors p and q starting at point O (origin). These vectors are known, as well as the origin point is. I know the angle α (alpha). Given a circle with arbitrary radius r, I want to be ...
6
votes
2answers
327 views

Why does the volume of a hypersphere decrease in higher dimensions? [duplicate]

First let us define an $n$-ball as the euclidean sphere in $\mathbb{R}^n$ including its interior and its surface where $n$ refers to the number of coordinates needed to describe the object (the ...